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23 a Fs(Density)18 b(and)g(current)h(of)f(a)g(dissip)l(a)l (tive)g(Schr)o(\366dinger)h(opera)l(tor)211 b Fz(1)89 464 y Fr(Abstract)89 608 y Fz(A)21 b(net)f(curren)o(t)g(\035o)o(w)h (through)h(an)g(op)q(en)f(1-dimensional)f(Sc)o(hr\366)q(dinger-P)o (oisson)i(system)d(is)89 668 y(mo)q(deled)d(b)o(y)g(replacing)g (self-adjoin)o(t)h(b)q(oundary)h(conditions)f(b)o(y)f(dissipativ)o(e)g (ones.)23 b(T)l(o)18 b(giv)o(e)89 728 y(a)f(rigorous)g(de\034nition)f (of)h(carrier)f(and)g(curren)o(t)g(densities)f(the)i(w)o(ell-kno)o(wn)e (dilation)h(theory)89 788 y(of)23 b(dissipativ)o(e)d(op)q(erators)k(is) e(used)g(where)f(the)h(self-adjoin)o(t)g(dilation)f(is)h(regarded)g(as) h(the)89 848 y(Hamiltonian)d(of)h(a)h(larger)f(closed)g(system)f(whic)o (h)h(con)o(tains)g(the)h(op)q(en)f(one.)37 b(The)22 b(carrier)89 909 y(densit)o(y)g(turns)h(out)g(to)g(b)q(e)g(p)q(erformed)e(b)o(y)i (the)f(generalized)f(eigenstates)i(of)g(the)f(dilation)89 969 y(while)17 b(the)g(curren)o(t)g(densit)o(y)f(is)h(related)g(to)h (the)f(c)o(haracteristic)f(function)h(of)h(the)f(dissipativ)o(e)89 1029 y(op)q(erator.)23 b(A)15 b(rigorous)j(setup)e(of)g(a)h(dissipativ) o(e)e(Sc)o(hr\366)q(dinger-P)o(oisson)i(system)e(is)h(outlined.)89 1173 y Fr(2000)22 b(Mathematics)d(Sub)t(ject)h(Classi\034cation)89 1317 y Fz(34B24,)d(47A20,)g(47A55,)g(47B44)89 1461 y Fr(Keyw)n(ords)89 1605 y Fz(op)q(en)25 b(Sc)o(hr\366dinger-P)o(oisson)f (system,)g(carrier)g(and)g(curren)o(t)f(densit)o(y)l(,)i(dissipativ)o (e)e(Sc)o(hr\366-)89 1665 y(dinger)13 b(op)q(erator,)h(self-adjoin)o(t) f(dilation,)f(generalized)g(eigenfunctions,)h(c)o(haracteristic)e (func-)89 1725 y(tion,)16 b(densit)o(y)f(matrix)89 1915 y Fx(Con)n(ten)n(ts)89 2048 y Fq(1)45 b(In)n(tro)r(duction)1363 b(2)89 2181 y(2)45 b(Sc)n(hr\366)r(dinger-t)n(yp)r(e)18 b(op)r(erators)1001 b(5)89 2314 y(3)45 b(Carrier)19 b(densit)n(y)1272 b(11)89 2446 y(4)45 b(Curren)n(t)19 b(densit)n(y)1255 b(20)89 2579 y(5)45 b(Remarks)1426 b(24)p eop %%Page: 2 4 2 3 bop 89 23 a Fz(2)716 b Fs(H.-Ch.)25 b(Kaiser,)18 b(H.)h(Neidhardt,)g(J.)g(Rehber)o(g)89 155 y Fx(1)77 b(In)n(tro)r(duction)89 287 y Fz(Sc)o(hr\366)q(dinger-P)o(oisson)19 b(systems)d(are)h(of)h(great)g(in)o(terest)e(in)h(semiconductor)f(ph)o (ysics.)24 b(In)17 b(the)89 347 y(follo)o(wing)c(w)o(e)f(are)h(in)o (terested)e(in)h(a)h(Sc)o(hr\366)q(dinger-P)o(oisson)h(system)d(on)j (the)e(in)o(terv)m(al)g Fp(\012)i(=)g([)p Fo(a;)8 b(b)p Fp(])89 408 y Fz(on)16 b(the)g(real)f(axis)g Fn(R)p Fz(.)j(Systems)c (of)i(this)g(t)o(yp)q(e)f(w)o(ere)f(considered)h(in)h([9].)k(By)15 b Fo(')h Fz(w)o(e)f(denote)g(the)89 468 y(electrostatic)g(p)q(oten)o (tial)h(on)h Fp(\012)f Fz(whic)o(h)g(is)g(determined)d(b)o(y)j(P)o (oisson's)h(equation)484 602 y Fm(\000)542 568 y Fo(d)p 528 591 54 2 v 528 636 a(dx)586 602 y(\017)p Fp(\()p Fo(x)p Fp(\))690 568 y Fo(d)p 677 591 V 677 636 a(dx)735 602 y(')c Fp(=)h Fo(q)864 562 y Fl(\000)887 602 y Fo(C)t Fp(\()p Fo(x)p Fp(\))c(+)h Fo(u)1079 581 y Fk(+)1119 602 y Fm(\000)g Fo(u)1197 581 y Fj(\000)1226 562 y Fl(\001)1258 602 y Fo(;)56 b(x)14 b Fm(2)g Fp(\012)p Fo(;)295 b Fz(\(1.1\))89 731 y(where)16 b Fo(u)258 712 y Fk(+)303 731 y Fz(and)h Fo(u)426 712 y Fj(\000)472 731 y Fz(are)f(the)g(densities)f(of)h(holes) g(and)h(electrons,)e Fo(q)j Fz(is)e(the)g(magnitude)f(of)h(the)89 791 y(elemen)o(tary)d(c)o(harge,)i Fo(C)t Fp(\()p Fm(\001)p Fp(\))h Fz(is)f(the)h(so-called)g(doping)g(pro\034le)g(of)g(the)g (semiconductor)e(devices)89 851 y(and)j Fo(\017)d Fp(=)f Fo(\017)p Fp(\()p Fo(x)p Fp(\))g Fo(>)h Fp(0)j Fz(denotes)f(the)g (dielectric)e(p)q(ermittivit)n(y)f(whic)o(h)j(satis\034es)h(the)f (condition)729 985 y Fo(\017)p Fp(\()p Fm(\001)p Fp(\))11 b(+)889 952 y(1)p 866 974 72 2 v 866 1020 a Fo(\017)p Fp(\()p Fm(\001)p Fp(\))956 985 y Fm(2)j Fo(L)1036 965 y Fj(1)1074 985 y Fp(\([)p Fo(a;)8 b(b)p Fp(]\))p Fo(:)538 b Fz(\(1.2\))89 1126 y(The)16 b(P)o(oisson)h(equation)f(\(1.1\))h(is)f (completed)e(b)o(y)i(the)g(b)q(oundary)h(conditions)660 1238 y Fo(')p Fp(\()p Fo(a)p Fp(\))c(=)h Fo(')853 1245 y Fi(a)874 1238 y Fo(;)56 b Fz(and)49 b Fo(')p Fp(\()p Fo(b)p Fp(\))14 b(=)g Fo(')1260 1245 y Fi(b)1277 1238 y Fo(:)470 b Fz(\(1.3\))89 1350 y(The)12 b(second)g(imp)q(ortan)o(t)f (ingredien)o(t)g(of)h(the)g(Sc)o(hr\366)q(dinger-P)o(oisson)h(system)d (are)i(Sc)o(hr\366dinger-)89 1410 y(t)o(yp)q(e)20 b(op)q(erators)h Fo(H)k Fz(whic)o(h)19 b(are)h(de\034ned)g(on)g(the)g(Hilb)q(ert)f (space)h Fh(H)g Fp(=)h Fo(L)1500 1392 y Fk(2)1519 1410 y Fp(\([)p Fo(a;)8 b(b)p Fp(]\))19 b Fz(and)h(lo)q(ok)89 1471 y(lik)o(e)702 1545 y Fo(H)e Fp(:=)c Fm(\000)870 1512 y Fp(1)p 870 1534 25 2 v 870 1580 a(2)918 1512 y Fo(d)p 904 1534 54 2 v 904 1580 a(dx)1009 1512 y Fp(1)p 967 1534 109 2 v 967 1580 a Fo(m)p Fp(\()p Fo(x)p Fp(\))1099 1512 y Fo(d)p 1085 1534 54 2 v 1085 1580 a(dx)1154 1545 y Fp(+)d Fo(V)s(;)513 b Fz(\(1.4\))89 1660 y(where)19 b(the)g(real)f(p)q(oten)o(tial)h Fo(V)30 b Fz(is)19 b(sligh)o(tly)f (di\033eren)o(t)g(for)h(holes)g(and)g(electrons,)g(see)f(Section)89 1720 y(5,)25 b(and)e Fo(m)g Fz(equals)f(either)g(to)i(the)e Fo(x)p Fz(-dep)q(enden)o(t)h(e\033ectiv)o(e)d(mass)j(of)g(the)g(holes)g (or)g(of)g(the)89 1781 y(electrons.)d(Since)13 b(the)h(formalism)e(of)i (quan)o(tum)f(mec)o(hanics)f(is)i(w)o(ell)f(dev)o(elop)q(ed)g(only)h (for)g(self-)89 1841 y(adjoin)o(t)j(op)q(erators)h(usually)f (self-adjoin)o(t)f(b)q(oundary)i(conditions,)f(for)g(instance)f(Neuman) f(or)89 1901 y(Diric)o(hlet)e(b)q(oundary)k(conditions)e(at)h Fo(a)f Fz(and)h Fo(b)p Fz(,)f(are)g(c)o(hosen.)21 b(Op)q(erators)16 b(of)g(this)f(t)o(yp)q(e)g(ha)o(v)o(e)f(a)89 1961 y(discrete)g(sp)q (ectrum.)19 b(Let)c(us)g(denote)f(b)o(y)g Fm(fE)917 1968 y Fi(l)930 1961 y Fm(g)955 1943 y Fj(1)955 1974 y Fi(l)p Fk(=1)1028 1961 y Fz(and)h Fm(f)p Fo( )1178 1968 y Fi(l)1191 1961 y Fm(g)1216 1943 y Fj(1)1216 1974 y Fi(l)p Fk(=1)1288 1961 y Fz(the)g(sequence)e(of)i(eigen)o(v)m(alues)89 2021 y(and)i(eigenfunctions)f(coun)o(ting)g(m)o(ultipli)o(cit)o(ies.) 148 2105 y(The)g(carrier)g(densities)f(used)h(in)g(the)f(P)o(oisson)i (equation)f(are)g(no)o(w)h(p)q(erformed)e(b)o(y)g(expres-)89 2165 y(sions)i(of)f(the)g(form)613 2253 y Fo(u)644 2260 y Fk(^)-21 b Fi(\045)661 2253 y Fp(\()p Fo(x)p Fp(\))14 b(=)811 2191 y Fj(1)792 2206 y Fl(X)800 2312 y Fi(l)p Fk(=1)873 2253 y Fo(\045)898 2260 y Fi(l)911 2253 y Fm(j)p Fo( )957 2260 y Fi(l)969 2253 y Fp(\()p Fo(x)p Fp(\))p Fm(j)1049 2233 y Fk(2)1069 2253 y Fo(;)56 b(x)14 b Fm(2)g Fp([)p Fo(a;)8 b(b)p Fp(])p Fo(;)422 b Fz(\(1.5\))89 2388 y(where)20 b Fp(^)-28 b Fo(\045)14 b Fp(:=)f Fm(f)p Fo(\045)384 2395 y Fi(l)397 2388 y Fm(g)422 2370 y Fj(1)422 2401 y Fi(l)p Fk(=1)497 2388 y Fz(is)j(a)g(sequence)f(of)i(o)q (ccupation)g(n)o(um)o(b)q(ers)e(suc)o(h)h(that)847 2539 y Fo(N)j Fp(:=)989 2477 y Fj(1)971 2492 y Fl(X)979 2598 y Fi(l)p Fk(=1)1051 2539 y Fo(\045)1076 2546 y Fi(l)1089 2539 y Fo(;)658 b Fz(\(1.6\))89 2694 y(is)16 b(the)g(total)h(n)o(um)o (b)q(er)d(of)j(carriers)e(of)i(the)f(system.)k(Usually)l(,)14 b(the)i(sequence)k Fp(^)-29 b Fo(\045)17 b Fz(is)f(giv)o(en)f(b)o(y)701 2806 y Fo(\045)726 2813 y Fi(l)753 2806 y Fp(:=)f Fo(f)5 b Fp(\()p Fm(E)893 2813 y Fi(l)906 2806 y Fp(\))p Fo(;)57 b(l)14 b Fp(=)g(1)p Fo(;)8 b Fp(2)p Fo(;)g(:)g(:)g(:)g(:)512 b Fz(\(1.7\))p eop %%Page: 3 5 3 4 bop 89 23 a Fs(Density)18 b(and)g(current)h(of)f(a)g(dissip)l(a)l (tive)g(Schr)o(\366dinger)h(opera)l(tor)211 b Fz(3)89 147 y(where)15 b Fo(f)k Fm(\025)14 b Fp(0)h Fz(is)g(a)g(equilibrium)d (distribution)i(functions)h(whic)o(h)g(ma)o(y)e(b)q(e)i(di\033eren)o(t) g(for)g(holes)89 208 y(and)i(electrons.)j(Let)d(us)f(in)o(tro)q(duce)g (the)g(op)q(erator)860 312 y Fo(\045)e Fp(:=)g Fo(f)5 b Fp(\()p Fo(H)t Fp(\))p Fo(:)671 b Fz(\(1.8\))89 417 y(The)24 b(op)q(erator)h Fo(\045)f Fz(is)g(self-adjoin)o(t)f(and)i (non-negativ)o(e.)44 b(If)23 b(the)h(equilibrium)c(distribution)89 477 y(function)15 b Fo(f)21 b Fz(tends)16 b(to)g(zero)f(su\036cien)o (tly)e(fast)j(as)g Fo(x)e Fm(!)g Fp(+)p Fm(1)p Fz(,)g(then)i Fo(\045)f Fz(is)h(n)o(uclear.)k(F)l(rom)14 b Fo(\045)i Fz(one)89 538 y(computes)h(the)h(total)h(n)o(um)o(b)q(er)d Fo(N)24 b Fz(of)19 b(carriers)e(b)o(y)h Fo(N)23 b Fp(=)17 b Fz(tr)p Fp(\()p Fo(\045)p Fp(\))g(=)h Fz(tr)p Fp(\()p Fo(f)5 b Fp(\()p Fo(H)t Fp(\)\))p Fz(.)28 b(Usually)l(,)17 b(non-)89 598 y(negativ)o(e)11 b(self-adjoin)o(t)h(n)o(uclear)f(op)q (erators)i(are)f(called)f(densit)o(y)g(matrices.)18 b(Since)11 b Fo(\045)h Fz(comm)o(utes)89 658 y(with)18 b Fo(H)j Fz(the)d(densit)o(y)e(matrix)g Fo(\045)i Fz(remains)e(unc)o(hanged)i (in)f(time.)23 b(In)18 b(this)f(case)h Fo(\045)g Fz(is)f(called)f(a)89 718 y(steady)j(state.)28 b(If)18 b(the)g(densit)o(y)f(matrix)g(is)h (kno)o(wn,)h(then)f(one)h(restores)f(the)g(carrier)g(densit)o(y)89 778 y(as)f(follo)o(ws:)k(the)16 b(n)o(um)o(b)q(er)f(of)h(carriers)g Fo(N)858 785 y Fi(\045)878 778 y Fp(\()p Fo(!)r Fp(\))g Fz(in)g(the)g(set)g Fo(!)g Fm(\022)e Fp([)p Fo(a;)8 b(b)p Fp(])14 b Fz(is)i(computed)f(b)o(y)792 883 y Fo(N)831 890 y Fi(\045)852 883 y Fp(\()p Fo(!)r Fp(\))f(=)f Fz(tr)p Fp(\()p Fo(\045\037)1100 890 y Fi(!)1125 883 y Fp(\))p Fo(;)603 b Fz(\(1.9\))89 988 y(where)11 b Fo(\037)256 995 y Fi(!)281 988 y Fp(\()p Fm(\001)p Fp(\))f Fz(is)h(the)g(indicator) g(function)f(of)i(the)e(subset)i Fo(!)r Fz(.)19 b(It)11 b(turns)g(out)h(that)f(\(1.9\))g(de\034nes)g(an)89 1048 y(absolutely)j(con)o(tin)o(uous)f(measure)g(\(with)g(resp)q(ect)h(to)g (the)f(Leb)q(esgue)i(measure\).)k(One)13 b(v)o(eri\034es)89 1108 y(that)k(its)f(Radon-Nik)o(o)q(dym)f(deriv)m(ativ)o(e)g Fo(u)880 1115 y Fi(\045)916 1108 y Fz(coincides)h(with)g(the)g(carrier) g(densit)o(y)f(de\034ned)h(b)o(y)89 1169 y(\(1.5\),)g(i.e.)k Fo(u)334 1176 y Fk(^)-21 b Fi(\045)365 1169 y Fp(=)14 b Fo(u)445 1176 y Fi(\045)465 1169 y Fz(.)148 1252 y(The)22 b(system)e(describ)q(ed)i(b)o(y)f Fo(H)26 b Fz(is)c(closed.)38 b(Hence)20 b(there)i(is)f(no)i(in)o(teraction)d(with)i(the)89 1313 y(en)o(vironmen)o(t,)15 b(in)i(particular,)g(no)h(carrier)f(exc)o (hange.)24 b(Consequen)o(tly)l(,)17 b(the)g(curren)o(t)g(densit)o(y)89 1373 y Fo(j)112 1380 y Fk(^)-21 b Fi(\045)129 1373 y Fp(\()p Fo(x)p Fp(\))16 b Fz(de\034ned)g(b)o(y)640 1472 y Fo(j)663 1479 y Fk(^)-21 b Fi(\045)680 1472 y Fp(\()p Fo(x)p Fp(\))14 b(:=)843 1410 y Fj(1)825 1425 y Fl(X)833 1531 y Fi(l)p Fk(=1)905 1472 y Fo(\045)930 1479 y Fi(l)943 1472 y Fo(j)963 1479 y Fi(l)977 1472 y Fp(\()p Fo(x)p Fp(\))p Fo(;)56 b(x)13 b Fm(2)h Fp([)p Fo(a;)8 b(b)p Fp(])p Fo(;)424 b Fz(\(1.10\))89 1603 y(is)24 b(iden)o(tical)e(zero,)j (where)e(in)g(accordance)h(with)g([15])f(the)h(curren)o(t)f(densit)o(y) f Fo(j)1627 1610 y Fi(l)1641 1603 y Fp(\()p Fo(x)p Fp(\))h Fz(of)h(the)89 1663 y(eigenstate)16 b Fo( )349 1670 y Fi(l)378 1663 y Fz(is)g(giv)o(en)f(b)o(y)379 1793 y Fo(j)399 1800 y Fi(l)412 1793 y Fp(\()p Fo(x)p Fp(\))f(:=)f Fm(=)p Fz(m)642 1723 y Fl(\022)725 1760 y Fp(1)p 684 1782 109 2 v 684 1828 a Fo(m)p Fp(\()p Fo(x)p Fp(\))797 1793 y Fo( )831 1773 y Fj(0)829 1806 y Fi(l)842 1793 y Fp(\()p Fo(x)p Fp(\))p 908 1750 111 2 v Fo( )940 1800 y Fi(l)952 1793 y Fp(\()p Fo(x)p Fp(\))1018 1723 y Fl(\023)1063 1793 y Fo(;)22 b(x)13 b Fm(2)h Fp([)p Fo(a;)8 b(b)p Fp(])p Fo(;)20 b(l)15 b Fp(=)e(1)p Fo(;)8 b Fp(2)p Fo(;)g(:)g(:)g(:)g(:)165 b Fz(\(1.11\))89 1924 y(Ho)o(w)o(ev)o(er,)12 b(from)f(the)i(p)q(oin)o (t)g(of)h(view)e(of)h(semiconductor)e(ph)o(ysics)i(this)g(consequence)f (is)g(totally)89 1984 y(unacceptable)19 b(since)f(a)i(net)f(curren)o(t) f(\035o)o(w)h(through)i(the)d(b)q(oundary)j(is)e(natural.)30 b(Th)o(us)20 b(one)89 2044 y(has)d(to)g(devise)e(b)q(oundary)i (conditions)g(whic)o(h)e(allo)o(w)h(those)h(\035o)o(ws.)148 2128 y(A)23 b(simple)f(prop)q(osal)j(w)o(as)f(made)e(in)h([9])g(whic)o (h)g(adds)i(up)e(to)h(replace)f(the)g(self-adjoin)o(t)89 2188 y(b)q(oundary)18 b(conditions)e(b)o(y)g(non-selfadjoin)o(t)g (ones,)g(i.e.)k(to)d(consider)f Fo(H)k Fz(on)d(the)f(domain)360 2365 y(dom)p Fp(\()p Fo(H)t Fp(\))e(=)600 2248 y Fl(8)600 2293 y(>)600 2308 y(<)600 2397 y(>)600 2412 y(:)645 2365 y Fo(g)i Fm(2)e Fo(W)784 2344 y Fk(1)p Fi(;)p Fk(2)831 2365 y Fp(\([)p Fo(a;)8 b(b)p Fp(]\))k(:)1061 2269 y Fk(1)p 1031 2277 79 2 v 1031 2306 a Fi(m)p Fk(\()p Fi(x)p Fk(\))1114 2289 y Fo(g)1139 2271 y Fj(0)1151 2289 y Fp(\()p Fo(x)p Fp(\))i Fm(2)g Fo(W)1331 2271 y Fk(1)p Fi(;)p Fk(2)1378 2289 y Fp(\([)p Fo(a;)8 b(b)p Fp(]\))p Fo(;)1070 2341 y Fk(1)p 1031 2349 96 2 v 1031 2378 a(2)p Fi(m)p Fk(\()p Fi(a)p Fk(\))1131 2361 y Fo(g)1156 2343 y Fj(0)1168 2361 y Fp(\()p Fo(a)p Fp(\))13 b(=)h Fm(\000)p Fo(\024)1364 2368 y Fi(a)1384 2361 y Fo(g)r Fp(\()p Fo(a)p Fp(\))p Fo(;)1068 2413 y Fk(1)p 1031 2421 92 2 v 1031 2450 a(2)p Fi(m)p Fk(\()p Fi(b)p Fk(\))1127 2433 y Fo(g)1152 2415 y Fj(0)1164 2433 y Fp(\()p Fo(b)p Fp(\))g(=)f Fo(\024)1316 2440 y Fi(b)1334 2433 y Fo(g)r Fp(\()p Fo(b)p Fp(\))1545 2248 y Fl(9)1545 2293 y(>)1545 2308 y(=)1545 2397 y(>)1545 2412 y(;)1736 2365 y Fz(\(1.12\))89 2541 y(where)k Fo(\024)259 2548 y Fi(a)279 2541 y Fo(;)8 b(\024)329 2548 y Fi(b)361 2541 y Fm(2)15 b Fn(C)9 b Fz(.)26 b(If)16 b(at)h(least)g(one)g(of)g (the)f(imaginary)g(parts)h(is)f(di\033eren)o(t)g(from)g(zero,)g(then)89 2602 y(the)g(op)q(erator)i Fo(H)j Fz(is)16 b(non-selfadjoin)o(t.)22 b(Ho)o(w)o(ev)o(er,)14 b(this)i(non-selfadjoin)o(tness)h(implies)d(sev) o(eral)89 2662 y(complications.)20 b(In)15 b(particular,)h(the)g (notion)h(of)f(carrier)g(densities)f(b)q(ecomes)g(unclear.)148 2746 y(This)e(situation)g(can)g(b)q(e)h(handled)e(if)h(w)o(e)f (restrict)g(ourselv)o(es)g(to)i(dissipativ)o(e)d(op)q(erators.)22 b(Let)89 2806 y(us)15 b(recall)f(that)h(an)h(op)q(erator)g(is)e(called) g(dissipativ)o(e)g(if)g(the)g(imaginary)g(part)h(of)g(its)g(asso)q (ciated)p eop %%Page: 4 6 4 5 bop 89 23 a Fz(4)716 b Fs(H.-Ch.)25 b(Kaiser,)18 b(H.)h(Neidhardt,)g(J.)g(Rehber)o(g)89 147 y Fz(quadratic)f(form)e(is)h (non-p)q(ositiv)o(e.)26 b(In)17 b(the)g(presen)o(t)g(case)h(the)f(op)q (erator)i Fo(H)j Fz(is)17 b(dissipativ)o(e)f(if)89 208 y Fo(\024)117 215 y Fi(a)138 208 y Fo(;)8 b(\024)188 215 y Fi(b)225 208 y Fm(2)19 b Fn(C)310 215 y Fk(+)363 208 y Fp(:=)g Fm(f)p Fo(z)i Fm(2)f Fn(C)31 b Fp(:)19 b Fm(=)p Fz(m)o Fp(\()p Fo(z)r Fp(\))g Fo(>)h Fp(0)p Fm(g)p Fz(,)g(see)f([11)q(].)30 b(Moreo)o(v)o(er,)19 b(under)h(this)f(assumption)89 268 y(the)h(op)q(erator)h Fo(H)j Fz(b)q(ecomes)18 b(maximal)f(dissipativ)o(e,)h(i.e.)31 b(it)19 b(admits)g(no)h(prop)q(er)g(dissipativ)o(e)89 328 y(extension,)12 b(see)f([11].)19 b(The)12 b(main)e(tec)o(hnical)f (to)q(ol)j(to)g(o)o(v)o(ercome)d(the)i(di\036culties)f(is)h(the)g (dilation)89 388 y(theory)k(for)h(maximal)c(dissipativ)o(e)i(op)q (erator.)22 b(In)15 b([12])g(the)g(minim)o(al)d(self-adjoin)o(t)j (dilation)g Fo(K)89 448 y Fz(of)j Fo(H)j Fz(w)o(as)d(explicitly)d (constructed)i(and)h(analysed)f(in)g(detail.)24 b(F)l(rom)16 b(the)h(ph)o(ysical)f(p)q(oin)o(t)i(of)89 509 y(view)f(the)g(minim)o (al)e(self-adjoin)o(t)h(dilation)h(pla)o(ys)g(the)g(role)g(of)h(the)f (Hamiltonian)f(of)h(a)h(larger)89 569 y(closed)h(system)e(whic)o(h)i (con)o(tains)g(the)f(original)h(system)f(describ)q(ed)g(b)o(y)h Fo(H)t Fz(.)29 b(Using)19 b(this)g(fact)89 629 y(one)f(de\034nes)g (steady)g(states,)h(carrier)e(densities)g(and)i(curren)o(t)e (densities.)26 b(It)17 b(turns)i(out)f(that)89 689 y(the)f(curren)o(t)f (densit)o(y)g(is)h(indep)q(enden)o(t)g(of)g Fo(x)e Fm(2)h Fp([)p Fo(a;)8 b(b)p Fp(])15 b Fz(and,)j(in)f(general,)f(di\033eren)o (t)h(from)f(zero.)89 749 y(So)22 b(w)o(e)e(ha)o(v)o(e)g(a)i(constan)o (t)f(curren)o(t)f(through)i Fp([)p Fo(a;)8 b(b)p Fp(])20 b Fz(whic)o(h)g(is)h(quite)f(satisfactory)h(from)f(the)89 809 y(ph)o(ysical)15 b(p)q(oin)o(t)i(of)f(view.)148 893 y(It)i(turns)h(out)g(that)f(this)h(formal)e(approac)o(h)i(to)g(the)f (net)g(curren)o(t)f(\035o)o(w)i(problem)e(\034ts)h(in)o(to)89 953 y(mo)q(dels)e(discussed)i(in)f(the)g(literature.)23 b(Indeed,)17 b(a)g(w)o(ell)f(adopted)i(mo)q(del)e(in)h(semiconductor)89 1014 y(ph)o(ysics)d(is)g(the)g(so-called)g(Kirkner-Len)o(t)f(mo)q(del)g ([14])h(whic)o(h)g(w)o(as)g(mathematically)d(analysed)89 1074 y(in)k([4],)f([5])g(and)h([6].)20 b(In)15 b(this)f(mo)q(del)g(one) h(replaces)f(the)g(maximal)e(dissipativ)o(e)h(op)q(erator)j Fo(H)j Fz(b)o(y)89 1134 y(a)d(family)d(of)i(maximal)d(dissipativ)o(e)i (op)q(erators)i Fm(f)p Fo(H)t Fp(\()p Fo(z)r Fp(\))p Fm(g)1138 1147 y Fi(z)q Fj(2)p 1179 1120 50 2 v Fg(C)1201 1151 y Ff(+)1246 1134 y Fz(de\034ned)f(on)h Fh(H)d Fp(=)h Fo(L)1614 1116 y Fk(2)1634 1134 y Fp(\([)p Fo(a;)8 b(b)p Fp(]\))13 b Fz(and)89 1194 y(giv)o(en)j(b)o(y)269 1335 y(dom)o Fp(\()p Fo(H)t Fp(\()p Fo(z)r Fp(\)\))41 b(:=)640 1264 y Fl(\032)678 1335 y Fo(g)16 b Fm(2)e Fo(W)817 1317 y Fk(2)p Fi(;)p Fk(2)864 1335 y Fp(\([0)p Fo(;)8 b Fp(1]\))13 b(:)1061 1304 y Fo(g)1086 1286 y Fj(0)1098 1304 y Fp(\(1\))h(=)g Fo(\024)1254 1311 y Fk(1)1274 1304 y Fp(\()p Fo(z)r Fp(\))p Fo(g)r Fp(\(1\))1061 1364 y Fo(g)1086 1346 y Fj(0)1098 1364 y Fp(\(0\))g(=)g Fm(\000)p Fo(\024)1293 1371 y Fk(0)1313 1364 y Fp(\()p Fo(z)r Fp(\))p Fo(g)r Fp(\(0\))p Fo(;)1497 1264 y Fl(\033)1543 1335 y Fo(;)269 1471 y Fp(\()p Fo(H)t Fp(\()p Fo(z)r Fp(\))p Fo(g)r Fp(\)\()p Fo(x)p Fp(\))42 b(:=)f Fm(\000)694 1451 y Fi(d)712 1440 y Ff(2)p 684 1459 56 2 v 684 1488 a Fi(dx)722 1479 y Ff(2)744 1471 y Fo(g)r Fp(\()p Fo(x)p Fp(\))11 b(+)g Fo(V)h Fp(\()p Fo(x)p Fp(\))p Fo(g)r Fp(\()p Fo(x)p Fp(\))p Fo(;)56 b(g)16 b Fm(2)e Fz(dom)o Fp(\()p Fo(H)t Fp(\()p Fo(z)r Fp(\)\))p Fo(;)1736 1388 y Fz(\(1.13\))89 1584 y Fo(m)p Fp(\()p Fo(x)p Fp(\))f Fm(\021)h Fp(1)p Fo(=)p Fp(2)p Fz(,)j(with)433 1703 y Fo(\024)461 1710 y Fk(1)481 1703 y Fp(\()p Fo(z)r Fp(\))c(:=)h Fo(i)640 1665 y Fm(p)p 681 1665 25 2 v 681 1703 a Fo(z)50 b Fz(and)g Fo(\024)910 1710 y Fk(0)930 1703 y Fp(\()p Fo(z)r Fp(\))13 b(:=)h Fo(i)1089 1659 y Fl(p)p 1138 1659 144 2 v 1138 1703 a Fo(z)f Fm(\000)e Fo(V)1252 1710 y Fj(\000)1282 1703 y Fo(;)56 b(z)16 b Fm(2)p 1438 1663 66 2 v 14 w Fn(C)1471 1710 y Fk(+)1504 1703 y Fo(;)218 b Fz(\(1.14\))89 1817 y(where)17 b(the)g(cut)h(of)f(the)g(square)h(ro)q(ot)g(is)g(along)g Fp([0)p Fo(;)8 b Fm(1)p Fp(\))17 b Fz(and)h Fm(=)p Fz(m)n Fp(\()1322 1781 y Fm(p)p 1364 1781 25 2 v 36 x Fo(z)r Fp(\))e Fm(\025)f Fp(0)j Fz(for)g Fo(z)f Fm(2)f Fn(C)1718 1824 y Fk(+)1751 1817 y Fz(.)25 b(Let)89 1877 y(us)17 b(consider)f(the)g(self-adjoin)o(t)g(Sc)o(hr\366)q(dinger)g(op)q (erator)h Fo(K)t Fz(,)391 1987 y(dom)o Fp(\()p Fo(K)t Fp(\))44 b(:=)d Fo(W)756 1969 y Fk(2)p Fi(;)p Fk(2)803 1987 y Fp(\()p Fn(R)p Fp(\))p Fo(;)391 2076 y Fp(\()p Fo(K)t(f)5 b Fp(\)\()p Fo(x)p Fp(\))41 b(:=)g Fm(\000)757 2057 y Fi(d)775 2045 y Ff(2)p 747 2065 56 2 v 747 2094 a Fi(dx)785 2084 y Ff(2)807 2076 y Fo(f)5 b Fp(\()p Fo(x)p Fp(\))11 b(+)g Fo(V)990 2083 y Fi(K)r(L)1049 2076 y Fp(\()p Fo(x)p Fp(\))p Fo(f)5 b Fp(\()p Fo(x)p Fp(\))p Fo(;)56 b(f)20 b Fm(2)14 b Fz(dom)o Fp(\()p Fo(K)t Fp(\))p Fo(;)1736 2032 y Fz(\(1.15\))89 2192 y(on)j(the)f(Hilb)q(ert)f(space)h Fh(K)e Fp(=)g Fo(L)671 2174 y Fk(2)691 2192 y Fp(\()p Fn(R)p Fp(\))f Fz(whose)k(p)q(oten)o(tial)f Fo(V)1159 2199 y Fi(K)r(L)1231 2192 y Fm(2)e Fo(C)t Fp(\()p Fn(R)p Fp(\))f Fz(lo)q(oks)k(lik)o(e)635 2365 y Fo(V)663 2372 y Fi(K)r(L)735 2365 y Fp(=)787 2263 y Fl(8)787 2308 y(<)787 2398 y(:)852 2305 y Fo(V)880 2312 y Fj(\000)959 2305 y Fp(:)48 b Fo(x)13 b Fm(2)h Fn(R)1148 2312 y Fj(\000)852 2365 y Fo(V)79 b Fp(:)48 b Fo(x)13 b Fm(2)h Fp([0)p Fo(;)8 b Fp(1])852 2425 y(0)83 b(:)48 b Fo(x)13 b Fm(2)h Fp(\(1)p Fo(;)8 b Fp(+)p Fm(1)p Fp(\))1736 2365 y Fz(\(1.16\))89 2535 y(where)16 b Fo(V)258 2542 y Fj(\000)302 2535 y Fo(>)e Fp(0)p Fz(,)672 2600 y Fo(V)d Fp(\(0\))j(=)g Fo(V)867 2607 y Fj(\000)946 2600 y Fz(and)49 b Fo(V)11 b Fp(\(1\))j(=)g(0)p Fo(:)458 b Fz(\(1.17\))89 2692 y(Op)q(erators)17 b(of)g(this)f(t)o(yp)q (e)g(w)o(ere)f(in)o(v)o(estigate)g(in)h([7].)k(It)c(turns)h(out)g(that) 521 2806 y Fo(P)559 2785 y Fe(K)552 2818 y(H)584 2806 y Fp(\()p Fo(K)e Fm(\000)c Fo(z)r Fp(\))753 2785 y Fj(\000)p Fk(1)800 2806 y Fm(j)p Fh(H)j Fp(=)g(\()p Fo(H)t Fp(\()p Fo(z)r Fp(\))d Fm(\000)g Fo(z)r Fp(\))1147 2785 y Fj(\000)p Fk(1)1194 2806 y Fo(;)56 b(z)16 b Fm(2)e Fn(C)1383 2813 y Fk(+)1415 2806 y Fo(:)307 b Fz(\(1.18\))p eop %%Page: 5 7 5 6 bop 89 23 a Fs(Density)18 b(and)g(current)h(of)f(a)g(dissip)l(a)l (tive)g(Schr)o(\366dinger)h(opera)l(tor)211 b Fz(5)89 147 y(whic)o(h)18 b(means)f(that)h(the)g(op)q(erator)h Fo(K)j Fz(can)d(b)q(e)f(regarded)g(as)h(a)f(self-adjoin)o(t)g(dilation) g(of)g(the)89 208 y(family)c Fm(f)p Fo(H)t Fp(\()p Fo(z)r Fp(\))p Fm(g)395 221 y Fi(z)q Fj(2)p 436 193 50 2 v Fg(C)457 225 y Ff(+)488 208 y Fz(.)21 b(Moreo)o(v)o(er,)15 b(it)g(holds)772 292 y Fl(_)746 400 y Fi(z)q Fj(2)p Fg(C)s Fj(n)p Fg(R)853 340 y Fp(\()p Fo(K)g Fm(\000)c Fo(z)r Fp(\))1022 319 y Fj(\000)p Fk(1)1069 340 y Fh(H)j Fp(=)g Fh(K)532 b Fz(\(1.19\))89 507 y(whic)o(h)15 b(means)g(that)h Fo(K)j Fz(is)d(a)g(minim)o(al)d(self-adjoin)o(t)i(dilation)g(and,)h(hence,)f (that)h Fo(K)j Fz(is)d(unique)89 567 y(up)22 b(to)f(a)h(certain)e (isomorphism.)34 b(Th)o(us,)22 b(if)e(w)o(e)h(w)o(an)o(t)g(to)h (compute)d(quan)o(tities)h(whic)o(h)h(are)89 627 y(related)13 b(to)h(the)g(subspace)g Fh(H)p Fz(,)f(for)h(instance,)g(carrier)e (densities)h(on)h(the)g(in)o(terv)m(al)e Fp([0)p Fo(;)c Fp(1])p Fz(,)13 b(w)o(e)h(can)89 687 y(do)h(this)f(using)g(either)f (the)h(self-adjoin)o(t)g(op)q(erator)h Fo(K)j Fz(or)d(the)f(family)e Fm(f)p Fo(H)t Fp(\()p Fo(\025)p Fp(\))p Fm(g)1540 694 y Fi(\025)p Fj(2)p Fg(R)1625 687 y Fz(of)i(maximal)89 747 y(dissipativ)o(e)h(op)q(erators.)22 b(Moreo)o(v)o(er,)15 b(if)g(w)o(e)h(are)g(only)g(in)o(terested)f(in)g(quan)o(tities)g(with)h (resp)q(ect)89 808 y(to)k(a)g(small)e(energy)h(in)o(terv)m(al)f(around) j(the)e(energy)g Fo(\025)1108 815 y Fk(0)1148 808 y Fm(2)g Fn(R)e Fz(\(in)i(the)g(limit)e(only)i(quan)o(tities)89 868 y(for)g(the)f(energy)h Fo(\025)439 875 y Fk(0)459 868 y Fz(\),)g(then)f(it)g(is)h(enough)g(to)g(consider)f(the)h(maximal) c(dissipativ)o(e)j(op)q(erator)89 928 y Fo(H)t Fp(\()p Fo(\025)180 935 y Fk(0)201 928 y Fp(\))e Fz(whic)o(h)f(leads)h(to)h(a)g (mo)q(del)e(describ)q(ed)g(at)i(the)f(b)q(eginning.)148 1012 y(The)f(pap)q(er)h(is)f(organized)h(as)g(follo)o(ws.)k(In)15 b(Section)g(2)h(w)o(e)e(rigorously)i(de\034ne)f(Sc)o(hr\366)q(dinger-) 89 1072 y(t)o(yp)q(e)j(op)q(erators)h(and)g(brie\035y)e(recall)g(their) h(prop)q(erties,)g(cf.)26 b([11].)h(Moreo)o(v)o(er,)17 b(w)o(e)g(in)o(tro)q(duce)89 1132 y(the)h(imp)q(ortan)o(t)g(notion)h (of)f(the)g(c)o(haracteristic)f(function)h Fp(\002)1238 1139 y Fi(H)1272 1132 y Fp(\()p Fo(z)r Fp(\))g Fz(and)h(brie\035y)e (describ)q(e)h(the)89 1192 y(minimal)8 b(self-adjoin)o(t)j(dilation)f Fo(K)16 b Fz(of)11 b Fo(H)t Fz(.)20 b(F)l(or)11 b(further)g(purp)q (oses)h(w)o(e)f(indicate)f(its)h(generalized)89 1253 y(eigenfunction)k(expansion.)21 b(Essen)o(tially)l(,)14 b(w)o(e)i(follo)o(w)f(here)g(the)g(pap)q(er)i([12].)k(In)15 b(Section)g(3)h(w)o(e)89 1313 y(in)o(tro)q(duce)22 b(generalized)f (steady)h(states)g(and)h(de\034ne)f(carrier)f(densities.)38 b(In)21 b(Section)h(4)g(w)o(e)89 1373 y(de\034ne)e(the)g(curren)o(t)f (densit)o(y)h(and)g(compute)f(it)h(in)g(terms)e(of)j(the)f(c)o (haracteristic)f(function.)89 1433 y(In)g(particular,)f(it)g(turns)h (out)g(that)h(if)e(the)g(steady)h(state)g(is)f(giv)o(en)g(b)o(y)g Fo(\045)g Fp(=)g Fo(f)5 b Fp(\()p Fo(K)t Fp(\))p Fz(,)20 b(then)e(the)89 1493 y(curren)o(t)k(densit)o(y)g(is)h(zero.)42 b(In)22 b(Section)h(5)g(w)o(e)g(commen)o(t)d(the)j(results,)h(in)e (particular,)i(w)o(e)89 1554 y(clarify)18 b(the)i(relation)f(to)h(the)f (Lax-Phillips)g(scattering)g(theory)h(and)g(v)o(erify)e(the)h(con)o (tin)o(uit)o(y)89 1614 y(equation.)35 b(Finally)l(,)21 b(w)o(e)f(giv)o(e)g(a)h(rigorous)h(setting)f(of)g(so-called)g (dissipativ)o(e)f(Sc)o(hr\366)q(dinger-)89 1674 y(P)o(oisson)d(systems) e(whic)o(h)h(ha)o(v)o(e)g(the)g(adv)m(an)o(tage)h(that)g(their)e (curren)o(t)h(densities)f(are)i(di\033eren)o(t)89 1734 y(from)e(zero)h(for)h(suitable)f(c)o(hosen)g(steady)g(states.)89 1932 y Fx(2)77 b(Sc)n(hr\366)r(dinger-t)n(yp)r(e)26 b(op)r(erators)89 2067 y Fz(F)l(ollo)o(wing)15 b(the)f(prop)q(osal)i(of)g([9])e(w)o(e)g (consider)h(the)g(non-selfadjoin)o(t)g(Sc)o(hr\366)q(dinger-t)o(yp)q(e) f(op)q(er-)89 2127 y(ator)j Fo(H)k Fz(on)16 b(the)g(Hilb)q(ert)f(space) i Fh(H)c Fp(=)h Fo(L)837 2109 y Fk(2)857 2127 y Fp(\([)p Fo(a;)8 b(b)p Fp(]\))14 b Fz(giv)o(en)i(b)o(y)g(the)g(domain)f (\(1.12\))i(and)592 2242 y Fp(\()p Fo(H)t(g)r Fp(\)\()p Fo(x)p Fp(\))d(=)g(\()p Fo(l)q Fp(\()p Fo(g)r Fp(\)\)\()p Fo(x)p Fp(\))p Fo(;)56 b(g)16 b Fm(2)e Fz(dom)o Fp(\()p Fo(H)t Fp(\))p Fo(;)403 b Fz(\(2.1\))89 2357 y(where)505 2434 y Fp(\()p Fo(l)q Fp(\()p Fo(g)r Fp(\)\)\()p Fo(x)p Fp(\))14 b(:=)f Fm(\000)811 2400 y Fp(1)p 811 2423 25 2 v 811 2468 a(2)859 2400 y Fo(d)p 845 2423 54 2 v 845 2468 a(dx)950 2400 y Fp(1)p 908 2423 109 2 v 908 2468 a Fo(m)p Fp(\()p Fo(x)p Fp(\))1040 2400 y Fo(d)p 1026 2423 54 2 v 1026 2468 a(dx)1084 2434 y(g)r Fp(\()p Fo(x)p Fp(\))e(+)g Fo(V)h Fp(\()p Fo(x)p Fp(\))p Fo(g)r Fp(\()p Fo(x)p Fp(\))p Fo(;)315 b Fz(\(2.2\))89 2558 y(and)12 b Fo(V)25 b Fm(2)14 b Fo(L)312 2540 y Fk(2)332 2558 y Fp(\([)p Fo(a;)8 b(b)p Fp(]\))h Fz(is)i(a)h(real)f(p)q(oten)o(tial,)g (the)g(e\033ectiv)o(e)e(mass)i Fo(m)p Fp(\()p Fo(x)p Fp(\))i Fo(>)h Fp(0)d Fz(ob)q(eys)h Fo(m)p Fp(\()p Fo(x)p Fp(\))q(+)1761 2539 y Fk(1)p 1730 2547 79 2 v 1730 2575 a Fi(m)p Fk(\()p Fi(x)p Fk(\))1828 2558 y Fm(2)89 2625 y Fo(L)122 2607 y Fj(1)160 2625 y Fp(\([)p Fo(a;)c(b)p Fp(]\))15 b Fz(and)i Fo(\024)433 2632 y Fi(a)454 2625 y Fo(;)8 b(\024)504 2632 y Fi(b)537 2625 y Fm(2)15 b Fn(C)618 2632 y Fk(+)650 2625 y Fz(.)24 b(The)17 b(op)q(erator)h Fo(H)j Fz(is)c(maximal)d(dissipativ)o(e)h(and)j(completely)89 2685 y(non-selfadjoin)o(t,)g(see)f([11].)25 b(The)17 b(sp)q(ectrum)g(of)g Fo(H)22 b Fz(consists)c(of)g(isolated)f(eigen)o(v) m(alues)g(in)g(the)89 2746 y(lo)o(w)o(er)j(half-plane)g(with)h(the)f (only)g(accum)o(ulation)f(p)q(oin)o(t)h(at)h(in\034nit)o(y)l(.)33 b(Since)19 b(the)i(op)q(erator)89 2806 y Fo(H)h Fz(is)17 b(completely)d(non-selfadjoin)o(t)j(there)g(do)g(not)h(exist)e(real)h (eigen)o(v)m(alues.)23 b(T)l(o)17 b(analyze)g(the)p eop %%Page: 6 8 6 7 bop 89 23 a Fz(6)716 b Fs(H.-Ch.)25 b(Kaiser,)18 b(H.)h(Neidhardt,)g(J.)g(Rehber)o(g)89 147 y Fz(op)q(erator)f Fo(H)i Fz(it)c(is)g(useful)g(to)g(in)o(tro)q(duce)g(the)g(elemen)o (tary)d(solutions)k Fo(v)1421 154 y Fi(a)1441 147 y Fp(\()p Fo(x;)8 b(z)r Fp(\))16 b Fz(and)h Fo(v)1689 154 y Fi(b)1706 147 y Fp(\()p Fo(x;)8 b(z)r Fp(\))p Fz(,)258 274 y Fo(l)q Fp(\()p Fo(v)317 281 y Fi(a)337 274 y Fp(\()p Fo(x;)g(z)r Fp(\)\))i Fm(\000)h Fo(z)r(v)578 281 y Fi(a)598 274 y Fp(\()p Fo(x;)d(z)r Fp(\))14 b(=)f(0)p Fo(;)57 b(v)895 281 y Fi(a)916 274 y Fp(\()p Fo(a;)8 b(z)r Fp(\))13 b(=)h(1)p Fo(;)1245 240 y Fp(1)p 1192 263 131 2 v 1192 308 a(2)p Fo(m)p Fp(\()p Fo(a)p Fp(\))1327 274 y Fo(v)1353 253 y Fj(0)1351 286 y Fi(a)1372 274 y Fp(\()p Fo(a;)8 b(z)r Fp(\))13 b(=)h Fm(\000)p Fo(\024)1615 281 y Fi(a)1635 274 y Fo(;)112 b Fz(\(2.3\))258 403 y Fo(l)q Fp(\()p Fo(v)317 410 y Fi(b)333 403 y Fp(\()p Fo(x;)8 b(z)r Fp(\)\))j Fm(\000)f Fo(z)r(v)574 410 y Fi(b)591 403 y Fp(\()p Fo(x;)e(z)r Fp(\))13 b(=)h(0)p Fo(;)57 b(v)888 410 y Fi(b)905 403 y Fp(\()p Fo(b;)8 b(z)r Fp(\))13 b(=)h(1)p Fo(;)1227 369 y Fp(1)p 1176 392 126 2 v 1176 437 a(2)p Fo(m)p Fp(\()p Fo(b)p Fp(\))1307 403 y Fo(v)1333 383 y Fj(0)1331 415 y Fi(b)1347 403 y Fp(\()p Fo(b;)8 b(z)r Fp(\))14 b(=)f Fo(\024)1546 410 y Fi(b)1564 403 y Fo(;)183 b Fz(\(2.4\))89 535 y Fo(x)16 b Fm(2)g Fp([)p Fo(a;)8 b(b)p Fp(])p Fz(,)16 b Fo(z)h Fm(2)g Fn(C)9 b Fz(,)20 b(whic)o(h)d(alw)o(a)o(ys)g(exist.)24 b(The)18 b(W)l(ronskian)f Fo(W)7 b Fp(\()p Fo(z)r Fp(\))18 b Fz(of)f Fo(v)1461 542 y Fi(a)1482 535 y Fp(\()p Fo(x;)8 b(z)r Fp(\))17 b Fz(and)h Fo(v)1732 542 y Fi(b)1748 535 y Fp(\()p Fo(x;)8 b(z)r Fp(\))89 596 y Fz(is)16 b(de\034ned)g(b)o(y)395 720 y Fo(W)7 b Fp(\()p Fo(z)r Fp(\))13 b(=)h Fo(v)600 727 y Fi(a)620 720 y Fp(\()p Fo(x;)8 b(z)r Fp(\))792 686 y(1)p 738 708 133 2 v 738 754 a(2)p Fo(m)p Fp(\()p Fo(x)p Fp(\))875 720 y Fo(v)901 699 y Fj(0)899 732 y Fi(b)916 720 y Fp(\()p Fo(x;)g(z)r Fp(\))i Fm(\000)h Fo(v)1113 727 y Fi(b)1130 720 y Fp(\()p Fo(x;)d(z)r Fp(\))1302 686 y(1)p 1248 708 V 1248 754 a(2)p Fo(m)p Fp(\()p Fo(x)p Fp(\))1385 720 y Fo(v)1411 699 y Fj(0)1409 732 y Fi(a)1429 720 y Fp(\()p Fo(x;)g(z)r Fp(\))p Fo(:)205 b Fz(\(2.5\))89 852 y(W)l(e)15 b(note)h(that)g(the)g(W)l(roskian)f(do)q(es)i(not)f(dep) q(end)g(on)g Fo(x)p Fz(.)k(Similarly)l(,)13 b(the)i(functions)g Fo(v)1710 859 y Fj(\003)p Fi(a)1748 852 y Fp(\()p Fo(x;)8 b(z)r Fp(\))89 912 y Fz(and)17 b Fo(v)208 919 y Fj(\003)p Fi(b)242 912 y Fp(\()p Fo(x;)8 b(z)r Fp(\))p Fz(,)377 1017 y Fo(v)401 1024 y Fj(\003)p Fi(a)439 1017 y Fp(\()p Fo(x;)g(z)r Fp(\))14 b(:=)p 631 973 157 2 v 13 w Fo(v)655 1024 y Fi(a)675 1017 y Fp(\()p Fo(x;)p 744 989 25 2 v 8 w(z)r Fp(\))48 b Fz(and)i Fo(v)988 1024 y Fj(\003)p Fi(b)1022 1017 y Fp(\()p Fo(x;)8 b(z)r Fp(\))13 b(:=)p 1214 973 154 2 v 14 w Fo(v)1238 1024 y Fi(b)1254 1017 y Fp(\()p Fo(x;)p 1323 989 25 2 v 8 w(z)r Fp(\))p Fo(;)56 b(z)16 b Fm(2)e Fn(C)9 b Fo(;)191 b Fz(\(2.6\))89 1121 y Fo(x)14 b Fm(2)g Fp([)p Fo(a;)8 b(b)p Fp(])p Fz(,)14 b Fo(z)i Fm(2)e Fn(C)9 b Fz(,)18 b(are)f(elemen)o(tary)c(solutions)k (of)223 1248 y Fo(l)q Fp(\()p Fo(v)282 1255 y Fj(\003)p Fi(a)319 1248 y Fp(\()p Fo(x;)8 b(z)r Fp(\)\))j Fm(\000)f Fo(z)r(v)560 1255 y Fj(\003)p Fi(a)598 1248 y Fp(\()p Fo(x;)e(z)r Fp(\))14 b(=)f(0)p Fo(;)57 b(v)895 1255 y Fj(\003)p Fi(a)933 1248 y Fp(\()p Fo(a;)8 b(z)r Fp(\))14 b(=)f(1)p Fo(;)1262 1214 y Fp(1)p 1209 1236 131 2 v 1209 1282 a(2)p Fo(m)p Fp(\()p Fo(a)p Fp(\))1345 1248 y Fo(v)1371 1227 y Fj(0)1369 1260 y(\003)p Fi(a)1407 1248 y Fp(\()p Fo(a;)8 b(z)r Fp(\))13 b(=)h Fm(\000)p 1622 1220 49 2 v Fo(\024)1650 1255 y Fi(a)1670 1248 y Fo(;)77 b Fz(\(2.7\))223 1400 y Fo(l)q Fp(\()p Fo(v)282 1407 y Fj(\003)p Fi(b)316 1400 y Fp(\()p Fo(x;)8 b(z)r Fp(\)\))i Fm(\000)h Fo(z)r(v)557 1407 y Fj(\003)p Fi(b)591 1400 y Fp(\()p Fo(x;)d(z)r Fp(\))13 b(=)h(0)p Fo(;)57 b(v)888 1407 y Fj(\003)p Fi(b)923 1400 y Fp(\()p Fo(b;)8 b(z)r Fp(\))13 b(=)h(1)p Fo(;)1244 1367 y Fp(1)p 1193 1389 126 2 v 1193 1435 a(2)p Fo(m)p Fp(\()p Fo(b)p Fp(\))1324 1400 y Fo(v)1350 1380 y Fj(0)1348 1413 y(\003)p Fi(b)1383 1400 y Fp(\()p Fo(b;)8 b(z)r Fp(\))13 b(=)p 1554 1373 46 2 v 14 w Fo(\024)1582 1407 y Fi(b)1599 1400 y Fo(;)148 b Fz(\(2.8\))89 1533 y Fo(x)22 b Fm(2)h Fp([)p Fo(a;)8 b(b)p Fp(])p Fz(.)35 b(The)21 b(W)l(ronskian)h(of)g Fp(\()p Fo(v)798 1540 y Fj(\003)p Fi(a)836 1533 y Fp(\()p Fo(x;)8 b(z)r Fp(\))21 b Fz(and)h Fo(v)1094 1540 y Fj(\003)p Fi(b)1128 1533 y Fp(\()p Fo(x;)8 b(z)r Fp(\))21 b Fz(is)g(denoted)g(b)o(y)g Fo(W)1624 1540 y Fj(\003)1644 1533 y Fp(\()p Fo(z)r Fp(\))g Fz(and)h(is)89 1593 y(also)16 b(indep)q(enden)o(t)e(of)h Fo(x)p Fz(.)21 b(By)14 b(these)g(elemen)o(tary)e(solutions)k(one)f(gets)g(for)h(the)e (resolv)o(en)o(ts)g(the)89 1653 y(represen)o(tations)203 1758 y Fp(\(\()p Fo(H)h Fm(\000)c Fo(z)r Fp(\))390 1737 y Fj(\000)p Fk(1)437 1758 y Fo(f)5 b Fp(\)\()p Fo(x)p Fp(\))14 b(=)1158 b Fz(\(2.9\))404 1864 y Fm(\000)448 1831 y Fo(v)472 1838 y Fi(b)489 1831 y Fp(\()p Fo(x;)8 b(z)r Fp(\))p 448 1853 154 2 v 467 1899 a Fo(W)f Fp(\()p Fo(z)r Fp(\))614 1797 y Fl(Z)664 1810 y Fi(x)642 1909 y(a)694 1864 y Fo(dy)k(v)778 1871 y Fi(a)798 1864 y Fp(\()p Fo(y)r(;)d(z)r Fp(\))p Fo(f)d Fp(\()p Fo(y)r Fp(\))10 b Fm(\000)1067 1831 y Fo(v)1091 1838 y Fi(a)1111 1831 y Fp(\()p Fo(x;)e(z)r Fp(\))p 1067 1853 157 2 v 1087 1899 a Fo(W)f Fp(\()p Fo(z)r Fp(\))1237 1797 y Fl(Z)1287 1810 y Fi(b)1265 1909 y(x)1312 1864 y Fo(dy)j(v)1395 1871 y Fi(b)1412 1864 y Fp(\()p Fo(y)r(;)e(z)r Fp(\))p Fo(f)d Fp(\()p Fo(y)r Fp(\))p Fo(;)89 1999 y Fz(for)17 b Fo(z)e Fm(2)f Fo(\045)p Fp(\()p Fo(H)t Fp(\))p Fz(,)j Fo(f)i Fm(2)14 b Fo(L)510 1981 y Fk(2)530 1999 y Fp(\([)p Fo(a;)8 b(b)p Fp(]\))14 b Fz(and)168 2103 y Fp(\(\()p Fo(H)250 2083 y Fj(\003)281 2103 y Fm(\000)d Fo(z)r Fp(\))375 2083 y Fj(\000)p Fk(1)422 2103 y Fo(f)5 b Fp(\)\()p Fo(x)p Fp(\))13 b(=)1149 b Fz(\(2.10\))369 2210 y Fm(\000)413 2177 y Fo(v)437 2184 y Fj(\003)p Fi(b)471 2177 y Fp(\()p Fo(x;)8 b(z)r Fp(\))p 413 2199 171 2 v 434 2244 a Fo(W)480 2251 y Fj(\003)499 2244 y Fp(\()p Fo(z)r Fp(\))597 2142 y Fl(Z)646 2156 y Fi(x)624 2255 y(a)677 2210 y Fo(dy)i(v)760 2217 y Fj(\003)p Fi(a)798 2210 y Fp(\()p Fo(y)r(;)e(z)r Fp(\))p Fo(f)d Fp(\()p Fo(y)r Fp(\))10 b Fm(\000)1067 2177 y Fo(v)1091 2184 y Fj(\003)p Fi(a)1129 2177 y Fp(\()p Fo(x;)e(z)r Fp(\))p 1067 2199 175 2 v 1090 2244 a Fo(W)1136 2251 y Fj(\003)1156 2244 y Fp(\()p Fo(z)r Fp(\))1255 2142 y Fl(Z)1304 2156 y Fi(b)1282 2255 y(x)1330 2210 y Fo(dy)i(v)1413 2217 y Fj(\003)p Fi(b)1448 2210 y Fp(\()p Fo(y)r(;)e(z)r Fp(\))p Fo(f)d Fp(\()p Fo(y)r Fp(\))p Fo(;)89 2345 y Fz(for)17 b Fo(z)e Fm(2)f Fo(\045)p Fp(\()p Fo(H)337 2327 y Fj(\003)358 2345 y Fp(\))i Fz(and)h Fo(f)i Fm(2)14 b Fo(L)611 2327 y Fk(2)631 2345 y Fp(\([)p Fo(a;)8 b(b)p Fp(]\))p Fz(,)14 b(see)i([12].)148 2429 y(Since)e Fo(H)19 b Fz(is)14 b(completely)e(non-selfadjoin)o(t)j(the)f(maximal)e (dissipativ)o(e)h(op)q(erator)j Fo(H)j Fz(can)c(b)q(e)89 2489 y(completely)j(c)o(haracterized)g(b)o(y)i(its)g(c)o(haracteristic) f(function)h Fp(\002)1320 2496 y Fi(H)1354 2489 y Fp(\()p Fo(z)r Fp(\))p Fz(,)h Fo(z)i Fm(2)e Fo(\045)p Fp(\()p Fo(H)t Fp(\))14 b Fm(\\)g Fo(\045)p Fp(\()p Fo(H)1808 2471 y Fj(\003)1828 2489 y Fp(\))p Fz(,)89 2549 y(cf.)21 b([8].)f(The)15 b(de\034nition)g(of)g(the)g(c)o(haracteristic)f (function)g(relies)g(on)i(the)f(so-called)g(b)q(oundary)89 2609 y(op)q(erators)20 b Fo(T)7 b Fp(\()p Fo(z)r Fp(\))16 b(:)h Fh(H)g Fm(\000)-9 b(!)17 b Fn(C)636 2591 y Fk(2)659 2609 y Fz(,)h Fo(z)h Fm(2)e Fo(\045)p Fp(\()p Fo(H)t Fp(\))i Fz(and)g Fo(T)1035 2616 y Fj(\003)1054 2609 y Fp(\()p Fo(z)r Fp(\))e(:)f Fh(H)h Fm(\000)-8 b(!)17 b Fn(C)1348 2591 y Fk(2)1371 2609 y Fz(,)h Fo(z)h Fm(2)e Fo(\045)p Fp(\()p Fo(H)1583 2591 y Fj(\003)1603 2609 y Fp(\))p Fz(,)h(whic)o(h)g(are)89 2669 y(de\034ned)e(in)g([12].)21 b(W)l(riting)440 2797 y Fo(\024)468 2804 y Fi(a)502 2797 y Fp(=)14 b Fo(q)576 2804 y Fi(a)608 2797 y Fp(+)666 2763 y Fo(i)p 662 2785 25 2 v 662 2831 a Fp(2)691 2797 y Fo(\013)722 2776 y Fk(2)722 2809 y Fi(a)792 2797 y Fz(and)49 b Fo(\024)947 2804 y Fi(b)978 2797 y Fp(=)14 b Fo(q)1052 2804 y Fi(b)1080 2797 y Fp(+)1138 2763 y Fo(i)p 1134 2785 V 1134 2831 a Fp(2)1163 2797 y Fo(\013)1194 2776 y Fk(2)1194 2809 y Fi(b)1214 2797 y Fo(;)57 b(\013)1316 2804 y Fi(a)1337 2797 y Fo(;)8 b(\013)1390 2804 y Fi(b)1421 2797 y Fo(>)13 b Fp(0)p Fo(;)226 b Fz(\(2.11\))p eop %%Page: 7 9 7 8 bop 89 23 a Fs(Density)18 b(and)g(current)h(of)f(a)g(dissip)l(a)l (tive)g(Schr)o(\366dinger)h(opera)l(tor)211 b Fz(7)89 147 y(the)16 b(b)q(oundary)i(op)q(erators)f(are)g(de\034ned)e(b)o(y)596 285 y Fo(T)7 b Fp(\()p Fo(z)r Fp(\))p Fo(f)18 b Fp(:=)803 215 y Fl(\022)884 255 y Fo(\013)915 262 y Fi(b)932 255 y Fp(\(\()p Fo(H)d Fm(\000)c Fo(z)r Fp(\))1119 237 y Fj(\000)p Fk(1)1166 255 y Fo(f)5 b Fp(\)\()p Fo(b)p Fp(\))860 315 y Fm(\000)p Fo(\013)930 322 y Fi(a)951 315 y Fp(\(\()p Fo(H)15 b Fm(\000)c Fo(z)r Fp(\))1138 297 y Fj(\000)p Fk(1)1185 315 y Fo(f)5 b Fp(\)\()p Fo(a)p Fp(\))1317 215 y Fl(\023)1736 285 y Fz(\(2.12\))89 423 y(and)569 501 y Fo(T)598 508 y Fj(\003)617 501 y Fp(\()p Fo(z)r Fp(\))p Fo(f)19 b Fp(:=)788 431 y Fl(\022)869 471 y Fo(\013)900 478 y Fi(b)918 471 y Fp(\(\()p Fo(H)1000 453 y Fj(\003)1031 471 y Fm(\000)10 b Fo(z)r Fp(\))1124 453 y Fj(\000)p Fk(1)1171 471 y Fo(f)5 b Fp(\)\()p Fo(b)p Fp(\))846 531 y Fm(\000)p Fo(\013)916 538 y Fi(a)936 531 y Fp(\(\()p Fo(H)1018 513 y Fj(\003)1049 531 y Fm(\000)11 b Fo(z)r Fp(\))1143 513 y Fj(\000)p Fk(1)1190 531 y Fo(f)5 b Fp(\)\()p Fo(a)p Fp(\))1323 431 y Fl(\023)1368 501 y Fo(;)354 b Fz(\(2.13\))89 621 y Fo(f)19 b Fm(2)14 b Fo(L)212 603 y Fk(2)232 621 y Fp(\([)p Fo(a;)8 b(b)p Fp(]\))p Fz(.)20 b(Using)c(the)g(resolv)o(en)o(t)f(represen)o(tations)g(\(2.9\))i(and)g (\(2.10\))g(w)o(e)e(obtain)518 776 y Fo(T)7 b Fp(\()p Fo(z)r Fp(\))p Fo(f)19 b Fp(=)762 743 y(1)p 716 765 116 2 v 716 811 a Fo(W)7 b Fp(\()p Fo(z)r Fp(\))845 691 y Fl( )905 746 y Fm(\000)p Fo(\013)975 753 y Fi(b)1001 706 y Fl(R)1034 719 y Fi(b)1024 763 y(a)1059 746 y Fo(dy)16 b(v)1148 753 y Fi(a)1169 746 y Fp(\()p Fo(y)r(;)8 b(z)r Fp(\))p Fo(f)d Fp(\()p Fo(y)r Fp(\))925 814 y Fo(\013)956 821 y Fi(a)985 774 y Fl(R)1018 787 y Fi(b)1008 831 y(a)1044 814 y Fo(dy)15 b(v)1132 821 y Fi(b)1149 814 y Fp(\()p Fo(y)r(;)8 b(z)r Fp(\))p Fo(f)d Fp(\()p Fo(y)r Fp(\))1392 691 y Fl(!)1736 776 y Fz(\(2.14\))89 929 y(and)486 1021 y Fo(T)515 1028 y Fj(\003)534 1021 y Fp(\()p Fo(z)r Fp(\))p Fo(f)19 b Fp(=)749 988 y(1)p 697 1010 129 2 v 697 1056 a Fo(W)743 1063 y Fj(\003)762 1056 y Fp(\()p Fo(z)r Fp(\))838 936 y Fl( )899 991 y Fm(\000)p Fo(\013)969 998 y Fi(b)994 951 y Fl(R)1027 964 y Fi(b)1017 1008 y(a)1053 991 y Fo(dy)c(v)1141 998 y Fj(\003)p Fi(a)1179 991 y Fp(\()p Fo(y)r(;)8 b(z)r Fp(\))p Fo(f)d Fp(\()p Fo(y)r Fp(\))920 1059 y Fo(\013)951 1066 y Fi(b)976 1019 y Fl(R)1010 1032 y Fi(b)1000 1076 y(a)1035 1059 y Fo(dy)16 b(v)1124 1066 y Fj(\003)p Fi(b)1158 1059 y Fp(\()p Fo(y)r(;)8 b(z)r Fp(\))p Fo(f)d Fp(\()p Fo(y)r Fp(\))1403 936 y Fl(!)1451 1021 y Fo(;)271 b Fz(\(2.15\))89 1156 y Fo(f)19 b Fm(2)14 b Fo(L)212 1138 y Fk(2)232 1156 y Fp(\([)p Fo(a;)8 b(b)p Fp(]\))p Fz(.)20 b(The)c(adjoin)o(t)g(op)q (erators)i(are)e(giv)o(en)f(b)o(y)420 1290 y Fp(\()p Fo(T)7 b Fp(\()p Fo(z)r Fp(\))538 1269 y Fj(\003)557 1290 y Fo(\030)r Fp(\))i(\()p Fo(x)p Fp(\))41 b(=)845 1256 y(1)p 799 1278 116 2 v 799 1288 V 799 1332 a Fo(W)7 b Fp(\()p Fo(z)r Fp(\))928 1235 y Fl(\020)958 1290 y Fm(\000)p Fo(\013)1028 1297 y Fi(b)p 1045 1247 157 2 v 1045 1290 a Fo(v)1069 1297 y Fi(a)1089 1290 y Fp(\()p Fo(x;)h(z)r Fp(\))p Fo(;)g(\013)1255 1297 y Fi(a)p 1275 1247 154 2 v 1275 1290 a Fo(v)1299 1297 y Fi(b)1316 1290 y Fp(\()p Fo(x;)g(z)r Fp(\))1428 1235 y Fl(\021)1466 1290 y Fo(\030)249 b Fz(\(2.16\))715 1427 y Fp(=)851 1393 y(1)p 799 1415 129 2 v 799 1461 a Fo(W)845 1468 y Fj(\003)865 1461 y Fp(\()p 884 1433 25 2 v Fo(z)r Fp(\))941 1427 y(\()p Fm(\000)p Fo(\013)1030 1434 y Fi(b)1047 1427 y Fo(v)1071 1434 y Fj(\003)p Fi(a)1109 1427 y Fp(\()p Fo(x;)p 1178 1399 V 8 w(z)q Fp(\))p Fo(;)8 b(\013)1274 1434 y Fi(a)1295 1427 y Fo(v)1319 1434 y Fj(\003)p Fi(b)1353 1427 y Fp(\()p Fo(x;)p 1422 1399 V 8 w(z)r Fp(\)\))g Fo(\030)r(;)89 1563 y Fz(and)409 1684 y Fp(\()p Fo(T)457 1691 y Fj(\003)476 1684 y Fp(\()p Fo(z)r Fp(\))539 1664 y Fj(\003)559 1684 y Fo(\030)r Fp(\))h(\()p Fo(x)p Fp(\))41 b(=)854 1651 y(1)p 802 1673 129 2 v 802 1683 V 802 1726 a Fo(W)848 1733 y Fj(\003)867 1726 y Fp(\()p Fo(z)r Fp(\))943 1629 y Fl(\020)973 1684 y Fm(\000)p Fo(\013)1043 1691 y Fi(b)p 1060 1641 175 2 v 1060 1684 a Fo(v)1084 1691 y Fj(\003)p Fi(a)1122 1684 y Fp(\()p Fo(x;)8 b(z)r Fp(\))p Fo(;)g(\013)1288 1691 y Fi(a)p 1308 1641 171 2 v 1308 1684 a Fo(v)1332 1691 y Fj(\003)p Fi(b)1367 1684 y Fp(\()p Fo(x;)g(z)r Fp(\))1479 1629 y Fl(\021)1517 1684 y Fo(\030)198 b Fz(\(2.17\))717 1821 y Fp(=)847 1787 y(1)p 802 1810 116 2 v 802 1855 a Fo(W)7 b Fp(\()p 874 1828 25 2 v Fo(z)q Fp(\))931 1821 y(\()o Fm(\000)p Fo(\013)1019 1828 y Fi(b)1036 1821 y Fo(v)1060 1828 y Fi(a)1081 1821 y Fp(\()p Fo(x;)p 1150 1794 V 8 w(z)q Fp(\))p Fo(;)h(\013)1246 1828 y Fi(a)1267 1821 y Fo(v)1291 1828 y Fi(b)1308 1821 y Fp(\()p Fo(x;)p 1377 1794 V 8 w(z)q Fp(\)\))g Fo(\030)r(;)89 1958 y Fz(where)786 2036 y Fo(\030)16 b Fp(=)875 1966 y Fl(\022)934 2005 y Fo(\030)957 1987 y Fi(b)932 2066 y Fo(\030)955 2048 y Fi(a)998 1966 y Fl(\023)1048 2036 y Fm(2)e Fn(C)1128 2015 y Fk(2)1151 2036 y Fo(:)571 b Fz(\(2.18\))89 2151 y(The)17 b(c)o(haracteristic)f(function)g Fp(\002)720 2158 y Fi(H)771 2151 y Fz(of)h(the)g(maximal)d(dissipativ)o(e)h(op)q (erator)k Fo(H)i Fz(is)c(a)g(t)o(w)o(o-b)o(y-)89 2211 y(t)o(w)o(o)f(matrix-v)m(alued)f(function)h(whic)o(h)f(satis\034es)i (the)f(relation)513 2323 y Fp(\002)551 2330 y Fi(H)584 2323 y Fp(\()p Fo(z)r Fp(\))p Fo(T)7 b Fp(\()p Fo(z)r Fp(\))p Fo(f)18 b Fp(=)c Fo(T)869 2330 y Fj(\003)888 2323 y Fp(\()p Fo(z)r Fp(\))p Fo(f)s(;)56 b(z)16 b Fm(2)e Fo(\045)p Fp(\()p Fo(H)t Fp(\))e Fm(\\)f Fo(\045)p Fp(\()p Fo(H)1385 2302 y Fj(\003)1405 2323 y Fp(\))p Fo(;)298 b Fz(\(2.19\))89 2434 y Fo(f)20 b Fm(2)15 b Fo(L)214 2416 y Fk(2)234 2434 y Fp(\([)p Fo(a;)8 b(b)p Fp(]\))p Fz(.)21 b(It)16 b(dep)q(ends)h(meromorphically)c(on)k Fo(z)g Fm(2)e Fo(\045)p Fp(\()p Fo(H)t Fp(\))c Fm(\\)h Fo(\045)p Fp(\()p Fo(H)1426 2416 y Fj(\003)1446 2434 y Fp(\))17 b Fz(and)g(is)f(con)o(tractiv)o(e)89 2494 y(in)g Fn(C)179 2501 y Fj(\000)212 2494 y Fz(,)f(i.e.)677 2556 y Fm(k)p Fp(\002)740 2563 y Fi(H)774 2556 y Fp(\()p Fo(z)r Fp(\))p Fm(k)e(\024)h Fp(1)49 b Fz(for)g Fo(z)16 b Fm(2)e Fn(C)1227 2563 y Fj(\000)1259 2556 y Fo(:)463 b Fz(\(2.20\))89 2645 y(Using)16 b(the)g(elemen)o(tary)d(solutions)k (the)f(c)o(haracteristic)f(function)h Fp(\002)1376 2652 y Fi(H)1426 2645 y Fz(tak)o(es)f(the)h(form)422 2783 y Fp(\002)460 2790 y Fi(H)494 2783 y Fp(\()p Fo(z)r Fp(\))d(=)h Fo(I)644 2791 y Fg(C)665 2782 y Ff(2)698 2783 y Fp(+)d Fo(i)821 2749 y Fp(1)p 769 2772 129 2 v 769 2817 a Fo(W)815 2824 y Fj(\003)834 2817 y Fp(\()p Fo(z)r Fp(\))911 2713 y Fl(\022)968 2753 y Fo(\013)999 2734 y Fk(2)999 2765 y Fi(b)1019 2753 y Fo(v)1043 2760 y Fj(\003)p Fi(a)1081 2753 y Fp(\()p Fo(b;)d(z)r Fp(\))82 b Fm(\000)p Fo(\013)1339 2760 y Fi(b)1356 2753 y Fo(\013)1387 2760 y Fi(a)1008 2813 y Fm(\000)p Fo(\013)1078 2820 y Fi(b)1095 2813 y Fo(\013)1126 2820 y Fi(a)1228 2813 y Fo(\013)1259 2795 y Fk(2)1259 2825 y Fi(a)1280 2813 y Fo(v)1304 2820 y Fj(\003)p Fi(b)1338 2813 y Fp(\()p Fo(a;)8 b(z)r Fp(\))1469 2713 y Fl(\023)1514 2783 y Fo(:)208 b Fz(\(2.21\))p eop %%Page: 8 10 8 9 bop 89 23 a Fz(8)716 b Fs(H.-Ch.)25 b(Kaiser,)18 b(H.)h(Neidhardt,)g(J.)g(Rehber)o(g)89 147 y Fz(for)e Fo(z)e Fm(2)f Fo(\045)p Fp(\()p Fo(H)t Fp(\))e Fm(\\)f Fo(\045)p Fp(\()p Fo(H)500 129 y Fj(\003)520 147 y Fp(\))p Fz(,)16 b(cf.)21 b([12])148 231 y(Since)16 b Fo(H)22 b Fz(is)17 b(a)g(maximal)d(dissipativ)o(e)i(op)q(erator)i(there)f(is)g (a)g(larger)g(Hilb)q(ert)f(space)h Fh(K)f Fm(\023)f Fh(H)89 291 y Fz(and)i(a)g(self-adjoin)o(t)e(op)q(erator)j Fo(K)i Fz(on)d Fh(K)g Fz(suc)o(h)e(that)513 396 y Fo(P)551 375 y Fe(K)544 408 y(H)577 396 y Fp(\()p Fo(K)g Fm(\000)c Fo(z)r Fp(\))746 375 y Fj(\000)p Fk(1)793 396 y Fm(j)p Fh(H)i Fp(=)h(\()p Fo(H)h Fm(\000)c Fo(z)r Fp(\))1076 375 y Fj(\000)p Fk(1)1123 396 y Fo(;)57 b Fm(=)p Fz(m)n Fp(\()p Fo(z)r Fp(\))14 b Fo(>)g Fp(0)p Fo(;)299 b Fz(\(2.22\))89 500 y(see)18 b([8].)27 b(The)18 b(op)q(erator)h Fo(K)j Fz(is)c(called)f(a)i(self-adjoin)o(t)f(dilation)g(of)g(the)g(maximal)d (dissipativ)o(e)89 560 y(op)q(erator)j Fo(H)t Fz(.)j(Ob)o(viously)l(,) 15 b(from)g(the)h(condition)g(\(2.22\))h(one)f(gets)503 664 y Fo(P)541 644 y Fe(K)534 677 y(H)567 664 y Fp(\()p Fo(K)f Fm(\000)c Fo(z)r Fp(\))736 644 y Fj(\000)p Fk(1)783 664 y Fm(j)p Fh(H)i Fp(=)h(\()p Fo(H)961 644 y Fj(\003)992 664 y Fm(\000)d Fo(z)r Fp(\))1086 644 y Fj(\000)p Fk(1)1133 664 y Fo(;)57 b Fm(=)p Fz(m)n Fp(\()p Fo(z)r Fp(\))14 b Fo(<)g Fp(0)p Fo(:)289 b Fz(\(2.23\))89 769 y Fo(K)21 b Fz(is)16 b(called)f(a)h(minimal)d(self-adjoin)o(t)j(dilation)g(of)g Fo(H)21 b Fz(if)15 b(the)h(condition)772 831 y Fl(_)746 939 y Fi(z)q Fj(2)p Fg(C)s Fj(n)p Fg(R)853 879 y Fp(\()p Fo(K)f Fm(\000)c Fo(z)r Fp(\))1022 858 y Fj(\000)p Fk(1)1069 879 y Fh(H)j Fp(=)g Fh(K)532 b Fz(\(2.24\))89 1035 y(is)12 b(satis\034ed.)20 b(Minimal)9 b(self-adjoin)o(t)i(dilations)h(of)g (maximal)d(dissipativ)o(e)h(op)q(erators)j(are)f(deter-)89 1095 y(mined)j(up)i(to)h(a)f(certain)f(isomorphism,)f(in)h(particular,) h(all)f(minimal)d(self-adjoin)o(t)k(dilations)89 1156 y(are)f(unitarily)g(equiv)m(alen)o(t.)148 1239 y(In)j(the)g(presen)o(t) f(case)h(the)g(minim)o(al)d(self-adjoin)o(t)j(dilation)f(of)i(the)e (maximal)e(dissipativ)o(e)89 1300 y(op)q(erator)j Fo(H)j Fz(can)c(b)q(e)g(constructed)f(in)h(an)g(explicit)e(manner.)24 b(F)l(ollo)o(wing)17 b([12)q(])g(w)o(e)g(in)o(tro)q(duce)89 1360 y(the)f(larger)g(Hilb)q(ert)f(space)772 1420 y Fh(K)f Fp(=)g Fm(D)909 1427 y Fj(\000)950 1420 y Fm(\010)d Fh(H)g Fm(\010)g(D)1135 1427 y Fk(+)1164 1420 y Fo(;)558 b Fz(\(2.25\))89 1505 y(where)16 b Fm(D)268 1512 y Fj(\006)312 1505 y Fp(:=)d Fo(L)410 1487 y Fk(2)430 1505 y Fp(\()p Fn(R)488 1512 y Fj(\006)515 1505 y Fo(;)8 b Fn(C)569 1487 y Fk(2)592 1505 y Fp(\))p Fz(.)21 b(In)o(tro)q(ducing)16 b(the)g(graph)1138 1492 y Fp(^)1133 1505 y(\012)p Fz(,)p 539 1666 333 4 v 705 1644 a Fn(R)744 1651 y Fj(\000)p 871 1666 V 1037 1644 a Fn(R)1076 1651 y Fk(+)p 539 1832 V 705 1810 a Fn(R)744 1817 y Fj(\000)p 871 1832 V 1037 1810 a Fn(R)1076 1817 y Fk(+)p 870 1831 4 167 v 892 1756 a Fp([)p Fo(a;)8 b(b)p Fp(])89 1992 y Fz(one)17 b(can)f(write)g(the)g(Hilb)q(ert)f (space)h Fh(K)g Fz(as)h Fo(L)917 1974 y Fk(2)937 1992 y Fp(\()961 1980 y(^)956 1992 y(\012\))p Fz(.)22 b(F)l(urthermore,)13 b(w)o(e)j(de\034ne)788 2097 y Fo(~)-22 b(g)16 b Fp(:=)d Fo(g)917 2104 y Fj(\000)958 2097 y Fm(\010)e Fo(g)i Fm(\010)e Fo(g)1117 2104 y Fk(+)1147 2097 y Fo(;)575 b Fz(\(2.26\))89 2201 y(where)416 2278 y Fo(g)439 2285 y Fj(\000)469 2278 y Fp(\()p Fo(x)p Fp(\))14 b(:=)614 2207 y Fl(\022)672 2247 y Fo(g)697 2229 y Fi(b)695 2260 y Fj(\000)724 2247 y Fp(\()p Fo(x)p Fp(\))672 2307 y Fo(g)697 2289 y Fi(a)695 2320 y Fj(\000)724 2307 y Fp(\()p Fo(x)p Fp(\))811 2207 y Fl(\023)953 2278 y Fz(and)50 b Fo(g)1104 2285 y Fk(+)1134 2278 y Fp(\()p Fo(x)p Fp(\))13 b(:=)1278 2207 y Fl(\022)1336 2247 y Fo(g)1361 2229 y Fi(b)1359 2260 y Fk(+)1389 2247 y Fp(\()p Fo(x)p Fp(\))1336 2307 y Fo(g)1361 2289 y Fi(a)1359 2320 y Fk(+)1389 2307 y Fp(\()p Fo(x)p Fp(\))1475 2207 y Fl(\023)1520 2278 y Fo(;)202 b Fz(\(2.27\))89 2395 y(for)17 b Fo(x)c Fm(2)h Fn(R)291 2402 y Fj(\000)334 2395 y Fz(and)j Fo(x)c Fm(2)h Fn(R)556 2402 y Fk(+)583 2395 y Fz(,)i(resp)q(ectiv)o(ely)l(.)i(Let)f(the)f(matrices)e Fo(K)1305 2377 y Fi(a)1301 2408 y Fj(\006)1347 2395 y Fz(and)j Fo(K)1487 2377 y Fi(b)1483 2408 y Fj(\006)1529 2395 y Fz(giv)o(en)f(b)o(y)384 2532 y Fo(K)429 2511 y Fi(a)425 2544 y Fj(\000)469 2532 y Fp(:=)553 2498 y(1)p 539 2520 52 2 v 539 2566 a Fo(\013)570 2573 y Fi(a)604 2462 y Fl(\022)662 2501 y Fp(0)54 b(0)662 2562 y(1)42 b Fo(\024)756 2569 y Fi(a)797 2462 y Fl(\023)940 2532 y Fz(and)98 b Fo(K)1161 2511 y Fi(a)1157 2544 y Fk(+)1201 2532 y Fp(:=)1285 2498 y(1)p 1271 2520 V 1271 2566 a Fo(\013)1302 2573 y Fi(a)1336 2462 y Fl(\022)1394 2501 y Fp(0)54 b(0)1394 2562 y(1)p 1460 2534 49 2 v 42 w Fo(\024)1488 2569 y Fi(a)1529 2462 y Fl(\023)1736 2532 y Fz(\(2.28\))89 2662 y(as)17 b(w)o(ell)e(as)342 2783 y Fo(K)387 2762 y Fi(b)383 2795 y Fj(\000)427 2783 y Fp(:=)509 2749 y(1)p 497 2772 V 497 2817 a Fo(\013)528 2824 y Fi(b)558 2713 y Fl(\022)616 2753 y Fp(1)42 b Fm(\000)p Fo(\024)749 2760 y Fi(b)616 2813 y Fp(0)72 b(0)786 2713 y Fl(\023)929 2783 y Fz(and)98 b Fo(K)1150 2762 y Fi(b)1146 2795 y Fk(+)1190 2783 y Fp(:=)1272 2749 y(1)p 1260 2772 V 1260 2817 a Fo(\013)1291 2824 y Fi(b)1322 2713 y Fl(\022)1379 2753 y Fp(1)42 b Fm(\000)p 1484 2725 46 2 v Fo(\024)1512 2760 y Fi(b)1379 2813 y Fp(0)72 b(0)1550 2713 y Fl(\023)1595 2783 y Fo(:)127 b Fz(\(2.29\))p eop %%Page: 9 11 9 10 bop 89 23 a Fs(Density)18 b(and)g(current)h(of)f(a)g(dissip)l(a)l (tive)g(Schr)o(\366dinger)h(opera)l(tor)211 b Fz(9)89 147 y(Using)16 b(these)g(notations)h(the)f(self-adjoin)o(t)g(dilation)g Fo(K)k Fz(is)c(de\034ned)g(b)o(y)462 320 y(dom)o Fp(\()p Fo(K)t Fp(\))e(:=)716 188 y Fl(8)716 233 y(>)716 248 y(>)716 263 y(<)716 353 y(>)716 368 y(>)716 382 y(:)758 320 y Fo(~)-21 b(g)16 b Fm(2)e Fh(K)g Fp(:)942 230 y Fo(g)965 237 y Fj(\006)1009 230 y Fm(2)g Fo(W)1109 211 y Fk(1)p Fi(;)p Fk(2)1156 230 y Fp(\()p Fn(R)1214 237 y Fj(\006)1240 230 y Fo(;)8 b Fn(C)1295 211 y Fk(2)1318 230 y Fp(\))p Fo(;)942 290 y(g)r(;)1001 270 y Fk(1)p 994 278 32 2 v 994 307 a Fi(m)1030 290 y Fo(g)1055 272 y Fj(0)1081 290 y Fm(2)14 b Fo(W)1181 272 y Fk(1)p Fi(;)p Fk(2)1228 290 y Fp(\([)p Fo(a;)8 b(b)p Fp(]\))p Fo(;)942 350 y(K)987 332 y Fi(a)983 362 y Fj(\000)1013 350 y Fo(g)1036 357 y Fi(a)1068 350 y Fp(+)j Fo(K)1162 332 y Fi(b)1158 362 y Fj(\000)1188 350 y Fo(g)1211 357 y Fi(b)1242 350 y Fp(=)j Fo(g)1317 357 y Fj(\000)1347 350 y Fp(\(0\))p Fo(;)942 410 y(K)987 392 y Fi(a)983 422 y Fk(+)1013 410 y Fo(g)1036 417 y Fi(a)1068 410 y Fp(+)d Fo(K)1162 392 y Fi(b)1158 422 y Fk(+)1188 410 y Fo(g)1211 417 y Fi(b)1242 410 y Fp(=)j Fo(g)1317 417 y Fk(+)1347 410 y Fp(\(0\))1444 188 y Fl(9)1444 233 y(>)1444 248 y(>)1444 263 y(=)1444 353 y(>)1444 368 y(>)1444 382 y(;)1736 320 y Fz(\(2.30\))89 493 y(and)432 565 y Fo(K)q(~)-21 b(g)16 b Fp(:=)d Fm(\000)p Fo(i)655 532 y(d)p 642 554 54 2 v 642 599 a(dx)700 565 y(g)723 572 y Fj(\000)763 565 y Fm(\010)e Fo(l)q Fp(\()p Fo(g)r Fp(\))g Fm(\010)g(\000)p Fo(i)1027 532 y(d)p 1014 554 V 1014 599 a(dx)1071 565 y(g)1094 572 y Fk(+)1124 565 y Fo(;)103 b(~)-21 b(g)16 b Fm(2)e Fz(dom)o Fp(\()p Fo(K)t Fp(\))p Fo(;)217 b Fz(\(2.31\))89 655 y(where)448 735 y Fo(g)471 742 y Fi(a)506 735 y Fp(=)558 665 y Fl(\022)659 681 y Fk(1)p 620 690 96 2 v 620 718 a(2)p Fi(m)p Fk(\()p Fi(a)p Fk(\))720 701 y Fo(g)745 683 y Fj(0)757 701 y Fp(\()p Fo(a)p Fp(\))674 769 y Fo(g)r Fp(\()p Fo(a)p Fp(\))841 665 y Fl(\023)935 735 y Fz(and)49 b Fo(g)1085 742 y Fi(b)1117 735 y Fp(=)1168 665 y Fl(\022)1268 681 y Fk(1)p 1231 690 92 2 v 1231 718 a(2)p Fi(m)p Fk(\()p Fi(b)p Fk(\))1327 701 y Fo(g)1352 683 y Fj(0)1364 701 y Fp(\()p Fo(b)p Fp(\))1282 769 y Fo(g)r Fp(\()p Fo(b)p Fp(\))1443 665 y Fl(\023)1488 735 y Fo(:)234 b Fz(\(2.32\))89 842 y(With)16 b(resp)q(ect)g(to)h(a)f(graph)h(picture)f(the)g(op)q (erator)h Fo(K)j Fz(lo)q(oks)d(lik)o(e)p 282 1048 416 4 v 697 1054 a Fp(\))282 971 y Fo(\013)313 978 y Fi(b)331 971 y Fo(g)356 953 y Fi(b)354 984 y Fj(\000)383 971 y Fp(\(0\))e(=)553 952 y Fk(1)p 516 960 92 2 v 516 988 a(2)p Fi(m)p Fk(\()p Fi(b)p Fk(\))613 971 y Fo(g)638 953 y Fj(0)649 971 y Fp(\()p Fo(b)p Fp(\))c Fm(\000)g Fo(\024)797 978 y Fi(b)814 971 y Fo(g)r Fp(\()p Fo(b)p Fp(\))365 1108 y Fm(\000)p Fo(i)436 1089 y Fi(d)p 426 1097 39 2 v 426 1125 a(dx)469 1108 y Fo(g)494 1090 y Fi(b)492 1121 y Fj(\000)p 1154 1048 416 4 v 1146 1054 a Fz(\()1071 952 y Fk(1)p 1035 960 92 2 v 1035 988 a(2)p Fi(m)p Fk(\()p Fi(b)p Fk(\))1131 971 y Fo(g)1156 953 y Fj(0)1168 971 y Fp(\()p Fo(b)p Fp(\))f Fm(\000)p 1287 944 46 2 v 11 w Fo(\024)1315 978 y Fi(b)1332 971 y Fo(g)r Fp(\()p Fo(b)p Fp(\))k(=)g Fo(\013)1513 978 y Fi(b)1530 971 y Fo(g)1555 953 y Fi(b)1553 984 y Fk(+)1583 971 y Fp(\(0\))1362 1108 y Fm(\000)p Fo(i)1432 1089 y Fi(d)p 1423 1097 39 2 v 1423 1125 a(dx)1465 1108 y Fo(g)1490 1090 y Fi(b)1488 1121 y Fk(+)p 282 1297 416 4 v 697 1303 a Fp(\))262 1386 y Fo(\013)293 1393 y Fi(a)313 1386 y Fo(g)338 1368 y Fi(a)336 1399 y Fj(\000)366 1386 y Fp(\(0\))g(=)538 1367 y Fk(1)p 499 1375 96 2 v 499 1404 a(2)p Fi(m)p Fk(\()p Fi(a)p Fk(\))599 1386 y Fo(g)624 1368 y Fj(0)636 1386 y Fp(\()p Fo(a)p Fp(\))d(+)g Fo(\024)788 1393 y Fi(a)809 1386 y Fo(g)r Fp(\()p Fo(a)p Fp(\))365 1254 y Fm(\000)p Fo(i)436 1234 y Fi(d)p 426 1242 39 2 v 426 1271 a(dx)469 1254 y Fo(g)494 1235 y Fi(a)492 1266 y Fj(\000)p 1154 1297 416 4 v 1146 1303 a Fp(\()1073 1367 y Fk(1)p 1035 1375 96 2 v 1035 1404 a(2)p Fi(m)p Fk(\()p Fi(a)p Fk(\))1135 1386 y Fo(g)1160 1368 y Fj(0)1171 1386 y Fp(\()p Fo(a)p Fp(\))g(+)p 1295 1359 49 2 v 11 w Fo(\024)1323 1393 y Fi(a)1344 1386 y Fo(g)r Fp(\()p Fo(a)p Fp(\))i(=)h Fo(\013)1529 1393 y Fi(a)1550 1386 y Fo(g)1575 1368 y Fi(a)1573 1399 y Fk(+)1603 1386 y Fp(\(0\))1362 1254 y Fm(\000)p Fo(i)1432 1234 y Fi(d)p 1423 1242 39 2 v 1423 1271 a(dx)1465 1254 y Fo(g)1490 1235 y Fi(a)1488 1266 y Fk(+)p 945 1295 4 250 v 988 1171 a Fo(l)q Fp(\()p Fo(g)r Fp(\))89 1525 y Fz(The)f(self-adjoin)o(t)f(op)q(erator)i Fo(K)j Fz(is)12 b(absolutely)h(con)o(tin)o(uous)f(and)i(its)e(sp)q(ectrum)f(coincides)h (with)89 1585 y(the)k(real)g(axis,)g(i.e.)k Fo(\033)r Fp(\()p Fo(K)t Fp(\))13 b(=)h Fn(R)p Fz(.)k(The)e(m)o(ultiplic)o(it)o (y)d(of)j(its)g(sp)q(ectrum)f(is)h(t)o(w)o(o.)21 b(The)16 b(resolv)o(en)o(t)89 1645 y(of)h Fo(K)j Fz(admits)15 b(the)h(represen)o(tation)293 1731 y Fp(\()p Fo(K)f Fm(\000)c Fo(z)r Fp(\))462 1711 y Fj(\000)p Fk(1)517 1731 y Fp(\()p Fo(f)560 1738 y Fj(\000)601 1731 y Fm(\010)g Fo(f)16 b Fm(\010)11 b Fo(f)765 1738 y Fk(+)794 1731 y Fp(\))j(=)871 b Fz(\(2.33\))278 1837 y Fo(i)303 1769 y Fl(Z)353 1782 y Fi(x)331 1881 y Fj(\0001)404 1837 y Fo(dy)16 b(e)492 1816 y Fi(i)p Fk(\()p Fi(x)p Fj(\000)p Fi(y)q Fk(\))p Fi(z)617 1837 y Fo(f)641 1844 y Fj(\000)670 1837 y Fp(\()p Fo(y)r Fp(\))25 b Fm(\010)f Fp(\()p Fo(H)16 b Fm(\000)11 b Fo(z)r Fp(\))991 1816 y Fj(\000)p Fk(1)1038 1837 y Fo(f)16 b Fp(+)11 b Fo(iT)1173 1844 y Fj(\003)1192 1837 y Fp(\()p 1211 1809 25 2 v Fo(z)r Fp(\))1255 1816 y Fj(\003)1283 1769 y Fl(Z)1333 1782 y Fk(0)1310 1881 y Fj(\0001)1384 1837 y Fo(dy)k(e)1471 1816 y Fj(\000)p Fi(iy)q(z)1549 1837 y Fo(f)1573 1844 y Fj(\000)1602 1837 y Fp(\()p Fo(y)r Fp(\))25 b Fm(\010)278 1978 y Fo(i)303 1910 y Fl(Z)353 1923 y Fi(x)331 2023 y Fk(0)383 1978 y Fo(dy)16 b(e)471 1957 y Fi(i)p Fk(\()p Fi(x)p Fj(\000)p Fi(y)q Fk(\))p Fi(z)596 1978 y Fo(f)620 1985 y Fk(+)650 1978 y Fp(\()p Fo(y)r Fp(\))10 b(+)h Fo(ie)813 1957 y Fi(iz)q(x)864 1978 y Fo(T)c Fp(\()p Fo(z)r Fp(\))p Fo(f)16 b Fp(+)11 b Fo(i)p Fp(\002)1107 1985 y Fi(H)1140 1978 y Fp(\()p 1159 1950 V Fo(z)r Fp(\))1203 1957 y Fj(\003)1231 1910 y Fl(Z)1281 1923 y Fk(0)1259 2023 y Fj(\0001)1332 1978 y Fo(dy)k(e)1419 1957 y Fi(i)p Fk(\()p Fi(x)p Fj(\000)p Fi(y)q Fk(\))p Fi(z)1544 1978 y Fo(f)1568 1985 y Fj(\000)1598 1978 y Fp(\()p Fo(y)r Fp(\))89 2094 y Fz(for)i Fm(=)p Fz(m)n Fp(\()p Fo(z)r Fp(\))d Fo(>)g Fp(0)i Fz(and)296 2180 y Fp(\()p Fo(K)f Fm(\000)c Fo(z)r Fp(\))465 2160 y Fj(\000)p Fk(1)520 2180 y Fp(\()p Fo(f)563 2187 y Fj(\000)604 2180 y Fm(\010)g Fo(f)16 b Fm(\010)11 b Fo(f)768 2187 y Fk(+)798 2180 y Fp(\))i(=)868 b Fz(\(2.34\))282 2285 y Fm(\000)p Fo(i)346 2217 y Fl(Z)395 2231 y Fk(0)373 2330 y Fi(x)423 2285 y Fo(dy)16 b(e)511 2265 y Fi(i)p Fk(\()p Fi(x)p Fj(\000)p Fi(y)q Fk(\))p Fi(z)636 2285 y Fo(f)660 2292 y Fj(\000)689 2285 y Fp(\()p Fo(y)r Fp(\))11 b Fm(\000)g Fo(ie)854 2265 y Fi(iz)q(x)905 2285 y Fo(T)934 2292 y Fj(\003)953 2285 y Fp(\()p Fo(z)r Fp(\))p Fo(f)16 b Fm(\000)11 b Fo(i)p Fp(\002)1161 2292 y Fi(H)1194 2285 y Fp(\()p Fo(z)r Fp(\))1265 2217 y Fl(Z)1315 2231 y Fj(1)1293 2330 y Fk(0)1361 2285 y Fo(dy)16 b(e)1449 2265 y Fi(i)p Fk(\()p Fi(x)p Fj(\000)p Fi(y)q Fk(\))p Fi(z)1573 2285 y Fo(f)1597 2292 y Fk(+)1627 2285 y Fp(\()p Fo(y)r Fp(\))11 b Fm(\010)282 2416 y Fp(\()p Fo(H)345 2395 y Fj(\003)376 2416 y Fm(\000)f Fo(z)r Fp(\))469 2395 y Fj(\000)p Fk(1)516 2416 y Fo(f)17 b Fm(\000)11 b Fo(iT)c Fp(\()p 679 2388 V Fo(z)q Fp(\))722 2395 y Fj(\003)749 2348 y Fl(Z)799 2361 y Fj(1)777 2461 y Fk(0)845 2416 y Fo(dy)16 b(e)933 2395 y Fj(\000)p Fi(iy)q(z)1010 2416 y Fo(f)1034 2423 y Fk(+)1064 2416 y Fp(\()p Fo(y)r Fp(\))24 b Fm(\010)h(\000)p Fo(i)1280 2348 y Fl(Z)1329 2361 y Fj(1)1307 2461 y Fi(x)1375 2416 y Fo(dy)16 b(e)1463 2395 y Fi(i)p Fk(\()p Fi(x)p Fj(\000)p Fi(y)q Fk(\))p Fi(z)1588 2416 y Fo(f)1612 2423 y Fk(+)1641 2416 y Fp(\()p Fo(y)r Fp(\))89 2540 y Fz(for)g Fm(=)p Fz(m)n Fp(\()p Fo(z)r Fp(\))e Fo(<)g Fp(0)p Fz(.)21 b(The)15 b(generalized)f(eigenfunctions)1101 2527 y Fo(~)1095 2540 y( )1129 2522 y Fj(\000)1158 2540 y Fp(\()p Fm(\001)p Fo(;)8 b(\025;)g(\034)e Fp(\))p Fz(,)14 b Fo(\025)h Fm(2)f Fn(R)p Fz(,)e Fo(\034)19 b Fp(=)14 b Fo(a;)8 b(b)p Fz(,)14 b(of)i Fo(K)j Fz(are)89 2600 y(giv)o(en)d(b)o(y)261 2673 y Fo(~)255 2686 y( )289 2666 y Fj(\000)318 2686 y Fp(\()p Fo(x;)8 b(\025;)g(\034)e Fp(\))14 b(:=)f Fo( )596 2666 y Fj(\000)594 2699 y(\000)625 2686 y Fp(\()p Fo(x;)8 b(\025;)g(\034)e Fp(\))11 b Fm(\010)f Fo( )884 2666 y Fj(\000)913 2686 y Fp(\()p Fo(x;)e(\025;)g(\034)e Fp(\))11 b Fm(\010)g Fo( )1173 2666 y Fj(\000)1171 2699 y Fk(+)1202 2686 y Fp(\()p Fo(x;)d(\025;)g(\034)e Fp(\))13 b(=)318 b Fz(\(2.35\))497 2751 y Fp(1)p 461 2773 96 2 v 461 2783 a Fm(p)p 503 2783 54 2 v 41 x Fp(2)p Fo(\031)562 2784 y(e)585 2764 y Fi(ix\025)639 2784 y Fo(e)662 2791 y Fi(\034)694 2784 y Fm(\010)784 2751 y Fp(1)p 749 2773 96 2 v 749 2783 a Fm(p)p 790 2783 54 2 v 790 2824 a Fp(2)p Fo(\031)849 2784 y Fp(\()p Fo(T)897 2791 y Fj(\003)917 2784 y Fp(\()p Fo(\025)p Fp(\))983 2764 y Fj(\003)1003 2784 y Fo(e)1026 2791 y Fi(\034)1047 2784 y Fp(\)\()p Fo(x)p Fp(\))10 b Fm(\010)1233 2751 y Fp(1)p 1197 2773 96 2 v 1197 2783 a Fm(p)p 1239 2783 54 2 v 41 x Fp(2)p Fo(\031)1298 2784 y(e)1321 2764 y Fi(ix\025)1375 2784 y Fp(\002)1413 2791 y Fi(H)1446 2784 y Fp(\()p Fo(\025)p Fp(\))1512 2764 y Fj(\003)1532 2784 y Fo(e)1555 2791 y Fi(\034)p eop %%Page: 10 12 10 11 bop 89 23 a Fz(10)692 b Fs(H.-Ch.)25 b(Kaiser,)18 b(H.)h(Neidhardt,)g(J.)g(Rehber)o(g)89 147 y Fz(where)612 224 y Fo(e)635 231 y Fi(b)665 224 y Fp(:=)731 154 y Fl(\022)788 194 y Fp(1)788 254 y(0)833 154 y Fl(\023)927 224 y Fz(and)49 b Fo(e)1077 231 y Fi(a)1112 224 y Fp(:=)1177 154 y Fl(\022)1234 194 y Fp(0)1234 254 y(1)1280 154 y Fl(\023)1325 224 y Fo(:)397 b Fz(\(2.36\))89 335 y(The)16 b(functions)h(are)f(m)o(utually) e(orthogonal,)j(i.e.)j(one)d(has)489 397 y Fl(\020)525 439 y Fo(~)519 452 y( )553 431 y Fj(\000)582 452 y Fp(\()p Fm(\001)p Fo(;)8 b(\025;)g(\034)e Fp(\))p Fo(;)760 439 y(~)755 452 y( )789 431 y Fj(\000)817 452 y Fp(\()p Fm(\001)p Fo(;)i(\025)900 431 y Fj(0)912 452 y Fo(;)g(\034)961 431 y Fj(0)972 452 y Fp(\))991 397 y Fl(\021)1021 486 y Fi(L)1045 477 y Ff(2)1062 486 y Fk(\()1080 478 y(^)1076 486 y(\012\))1131 452 y Fp(=)14 b Fo(\016)r Fp(\()p Fo(\025)d Fm(\000)g Fo(\025)1343 431 y Fj(0)1355 452 y Fp(\))p Fo(\016)1396 459 y Fi(\034)t(\034)1436 450 y Fd(0)1448 452 y Fo(;)274 b Fz(\(2.37\))89 581 y(in)14 b(the)f(sense)h(of)g (distribution)g(for)g Fo(\025;)8 b(\025)815 563 y Fj(0)841 581 y Fm(2)14 b Fn(R)p Fz(,)d Fo(\034)s(;)d(\034)1025 563 y Fj(0)1050 581 y Fp(=)14 b Fo(a;)8 b(b)p Fz(.)19 b(Moreo)o(v)o(er,)13 b(elemen)o(ts)e(of)j(the)g(form)693 640 y Fl(Z)721 753 y Fg(R)755 708 y Fo(d\025)840 661 y Fl(X)831 767 y Fi(\034)t Fk(=)p Fi(a;b)936 695 y Fo(~)930 708 y( )964 687 y Fj(\000)993 708 y Fp(\()p Fm(\001)p Fo(;)8 b(\025;)g(\034)e Fp(\))r(^)-26 b Fo(g)1169 687 y Fi(\034)1190 708 y Fp(\()p Fo(\025)p Fp(\))480 b Fz(\(2.38\))89 865 y(where)13 b Fp(^)-26 b Fo(g)250 847 y Fi(\034)272 865 y Fp(\()p Fm(\001)p Fp(\))p Fz(,)11 b Fo(\034)20 b Fp(=)14 b Fo(a;)8 b(b)p Fz(,)j(are)g(smo)q(oth)g(functions)g(with)g (compact)f(supp)q(ort,)j(are)e(dense)g(in)g Fh(K)p Fz(.)20 b(W)l(e)89 930 y(note)e(that)f(the)g(generalized)f(eigenfunctions)969 917 y Fo(~)963 930 y( )997 912 y Fj(\000)1026 930 y Fp(\()p Fm(\001)p Fo(;)8 b(\025;)g(\034)e Fp(\))17 b Fz(are)g(usually)g(called) f(the)h(incoming)89 991 y(eigenfunctions.)45 b(Using)24 b(the)g(incoming)e(eigenfunctions)i(one)g(de\034nes)g(a)h (transformation)89 1051 y Fp(\010)124 1058 y Fj(\000)168 1051 y Fp(:)13 b Fh(K)h Fm(\000)-8 b(!)341 1038 y Fp(^)336 1051 y Fh(K)15 b Fp(=)e Fo(L)468 1033 y Fk(2)488 1051 y Fp(\()p Fn(R)p Fo(;)8 b Fn(C)598 1033 y Fk(2)621 1051 y Fp(\))635 1185 y(\(\010)689 1192 y Fj(\000)717 1185 y Fo(~)-22 b(g)r Fp(\)\()p Fo(\025)p Fp(\))14 b(=:)h(^)-25 b Fo(g)r Fp(\()p Fo(\025)p Fp(\))14 b(=)1066 1115 y Fl(\022)1127 1155 y Fp(^)-26 b Fo(g)1150 1137 y Fi(b)1167 1155 y Fp(\()p Fo(\025)p Fp(\))1125 1215 y(^)g Fo(g)1148 1197 y Fi(a)1169 1215 y Fp(\()p Fo(\025)p Fp(\))1256 1115 y Fl(\023)1301 1185 y Fo(;)421 b Fz(\(2.39\))89 1317 y(where)485 1389 y Fp(^)-26 b Fo(g)508 1369 y Fi(\034)530 1389 y Fp(\()p Fo(\025)p Fp(\))14 b(:=)675 1321 y Fl(Z)707 1426 y Fk(^)703 1434 y(\012)739 1389 y Fo(dx)814 1334 y Fl(\020)842 1389 y Fo(~)-22 b(g)r Fp(\()p Fo(x)p Fp(\))p Fo(;)963 1376 y(~)957 1389 y( )991 1369 y Fj(\000)1020 1389 y Fp(\()p Fo(x;)8 b(\025;)g(\034)e Fp(\))1185 1334 y Fl(\021)1222 1389 y Fo(;)57 b(\034)19 b Fp(=)14 b Fo(a;)8 b(b:)268 b Fz(\(2.40\))89 1498 y Fp(\010)124 1505 y Fj(\000)167 1498 y Fz(is)13 b(unitary)h(and)g(called)e(the)h(incoming)f(F)l(ourier) g(transformation.)21 b(The)13 b(in)o(v)o(erse)e(incoming)89 1558 y(F)l(ourier)16 b(transformation)g Fp(\010)625 1537 y Fj(\000)p Fk(1)625 1569 y Fj(\000)689 1558 y Fz(is)g(giv)o(en)f(b)o (y)361 1690 y Fp(\(\010)415 1669 y Fj(\000)p Fk(1)415 1702 y Fj(\000)464 1690 y Fp(^)-26 b Fo(g)r Fp(\)\()p Fo(x)p Fp(\))14 b(=)637 1622 y Fl(Z)665 1735 y Fg(R)699 1690 y Fo(d\025)785 1642 y Fl(X)775 1748 y Fi(\034)t Fk(=)p Fi(a;b)880 1676 y Fo(~)874 1690 y( )908 1669 y Fj(\000)937 1690 y Fp(\()p Fo(x;)8 b(\025;)g(\034)e Fp(\))r(^)-26 b Fo(g)1127 1669 y Fi(\034)1148 1690 y Fp(\()p Fo(\025)p Fp(\))p Fo(;)59 b Fp(^)-26 b Fo(g)16 b Fm(2)e Fo(L)1404 1669 y Fk(2)1424 1690 y Fp(\()p Fn(R)p Fo(;)8 b Fn(C)1534 1669 y Fk(2)1557 1690 y Fp(\))p Fo(:)146 b Fz(\(2.41\))89 1844 y(W)l(e)16 b(note)h(that)820 1904 y Fp(\010)855 1911 y Fj(\000)885 1904 y Fo(K)t Fp(\010)965 1883 y Fj(\000)p Fk(1)965 1916 y Fj(\000)1026 1904 y Fp(=)d Fo(M)611 b Fz(\(2.42\))89 1996 y(where)16 b Fo(M)22 b Fz(is)16 b(the)g(m)o (ultipli)o(cation)e(op)q(erator)j(b)o(y)f(the)g(indep)q(enden)o(t)f(v)m (ariable)h Fo(\025)h Fz(on)1669 1983 y Fp(^)1664 1996 y Fh(K)q Fz(,)e(i.e.)422 2098 y(dom)o Fp(\()p Fo(M)5 b Fp(\))42 b(:=)f Fm(f)r Fp(^)-26 b Fo(g)16 b Fm(2)e Fo(L)883 2080 y Fk(2)903 2098 y Fp(\()p Fn(R)p Fo(;)8 b Fn(C)1013 2080 y Fk(2)1036 2098 y Fp(\))13 b(:)h Fo(\025)r Fp(^)-26 b Fo(g)r Fp(\()p Fo(\025)p Fp(\))15 b Fm(2)f Fo(L)1310 2080 y Fk(2)1330 2098 y Fp(\()p Fn(R)o Fo(;)8 b Fn(C)1439 2080 y Fk(2)1462 2098 y Fp(\))p Fm(g)g Fo(;)422 2182 y Fp(\()p Fo(M)f Fp(^)-26 b Fo(g)s Fp(\)\()p Fo(\025)p Fp(\))42 b(:=)f Fo(\025)r Fp(^)-26 b Fo(g)s Fp(\()p Fo(\025)p Fp(\))p Fo(;)59 b Fp(^)-26 b Fo(g)16 b Fm(2)e Fz(dom)o Fp(\()p Fo(M)5 b Fp(\))p Fo(:)1736 2140 y Fz(\(2.43\))89 2287 y(The)15 b(represen)o(tation)f(\(2.43\))i(induced)e(b)o(y)g Fp(\010)926 2294 y Fj(\000)970 2287 y Fz(is)h(called)f(the)g(incoming)g (sp)q(ectral)g(represen)o(ta-)89 2347 y(tion)i(of)h Fo(K)t Fz(.)148 2431 y(Finally)l(,)12 b(w)o(e)g(note)g(that)h(eac)o(h)f(b)q (ounded)i(self-adjoin)o(t)e(op)q(erator)i Fo(G)f Fz(on)g Fh(K)p Fz(,)g(whic)o(h)f(comm)o(utes)89 2491 y(with)23 b Fo(K)t Fz(,)i(corresp)q(onds)g(to)e(a)h(measurable)e(family)f Fm(f)p Fo(G)p Fp(\()p Fo(\025)p Fp(\))p Fm(g)1252 2498 y Fi(\025)p Fj(2)p Fg(R)1346 2491 y Fz(of)j(t)o(w)o(o-b)o(y-t)o(w)o(o)f (matrices,)89 2551 y(whic)o(h)17 b(are)h(uniformly)e(b)q(ounded,)i (i.e.,)e Fo(G)p Fp(\()p Fm(\001)p Fp(\))h Fm(2)g Fo(L)1035 2533 y Fj(1)1072 2551 y Fp(\()p Fn(R)p Fo(;)8 b Fm(B)q Fp(\()p Fn(C)1235 2533 y Fk(2)1258 2551 y Fp(\)\))p Fz(,)18 b(suc)o(h)f(that)h(the)g(m)o(ultiplic)o(a-)89 2611 y(tion)e(op)q (erator)397 2599 y Fp(^)386 2611 y Fo(G)h Fz(on)f Fo(L)541 2593 y Fk(2)561 2611 y Fp(\()p Fn(R)p Fo(;)8 b Fn(C)671 2593 y Fk(2)694 2611 y Fp(\))16 b Fz(de\034ned)g(b)o(y)391 2721 y(dom)o Fp(\()513 2708 y(^)502 2721 y Fo(G)q Fp(\))41 b(:=)g Fm(f)r Fp(^)-26 b Fo(g)16 b Fm(2)e Fo(L)838 2702 y Fk(2)858 2721 y Fp(\()p Fn(R)p Fo(;)7 b Fn(C)968 2702 y Fk(2)990 2721 y Fp(\))14 b(:)g Fo(G)p Fp(\()p Fo(\025)p Fp(\))r(^)-26 b Fo(g)s Fp(\()p Fo(\025)p Fp(\))14 b Fm(2)g Fo(L)1341 2702 y Fk(2)1361 2721 y Fp(\()p Fn(R)p Fo(;)7 b Fn(C)1471 2702 y Fk(2)1493 2721 y Fp(\))p Fm(g)h Fo(;)391 2813 y Fp(\()421 2800 y(^)410 2813 y Fo(G)r Fp(^)-26 b Fo(g)s Fp(\)\()p Fo(\025)p Fp(\))42 b(:=)f Fo(G)p Fp(\()p Fo(\025)p Fp(\))r(^)-26 b Fo(g)s Fp(\()p Fo(\025)p Fp(\))p Fo(;)59 b Fp(^)-26 b Fo(g)16 b Fm(2)e Fz(dom)o Fp(\()1191 2797 y(^)1158 2813 y Fo(G)p Fp(\()p Fm(\001)p Fp(\)\))1736 2764 y Fz(\(2.44\))p eop %%Page: 11 13 11 12 bop 89 23 a Fs(Density)18 b(and)g(current)h(of)f(a)g(dissip)l(a)l (tive)g(Schr)o(\366dinger)h(opera)l(tor)187 b Fz(11)89 147 y(is)16 b(unitarily)f(equiv)m(alen)o(t)g(to)i Fo(G)p Fz(,)f(i.e.)823 260 y Fp(\010)858 267 y Fj(\000)888 260 y Fo(G)p Fp(\010)961 240 y Fj(\000)p Fk(1)961 273 y Fj(\000)1023 260 y Fp(=)1086 248 y(^)1075 260 y Fo(G:)609 b Fz(\(2.45\))89 380 y(Indeed,)23 b(if)f Fo(G)h Fz(comm)o(utes)c(with)k Fo(K)t Fz(,)g(then)945 367 y Fp(^)934 380 y Fo(G)g Fz(comm)o(ute)o(s)d (with)i Fo(M)5 b Fz(.)41 b(Applying)21 b(Theorem)89 440 y(VI)q(I.2.3)12 b(of)h([3])g(one)g(imm)o(ediatel)o(y)d(gets)j(that)938 428 y Fp(^)927 440 y Fo(G)h Fz(is)e(a)i(m)o(ultipli)o(cation)c(op)q (erator)k(of)f(t)o(yp)q(e)g(\(2.44\).)89 500 y(The)j(represen)o(tation) g(\(2.44\))h(is)f(called)f(the)h(incoming)f(sp)q(ectral)h(represen)o (tation)f(of)i Fo(G)p Fz(.)89 695 y Fx(3)77 b(Carrier)26 b(densit)n(y)89 829 y Fz(In)16 b(the)g(follo)o(wing)g(w)o(e)f(call)h (an)g(op)q(erator)i Fo(\045)c Fp(:)f Fh(K)h Fm(\000)-8 b(!)13 b Fh(K)k Fz(a)f(densit)o(y)g(matrix)e(if)i Fo(\045)g Fz(is)g(a)g(b)q(ounded,)89 889 y(non-negativ)o(e)i(op)q(erator.)26 b(The)18 b(op)q(erator)g Fo(\045)g Fz(is)g(called)e(a)i(steady)g(state) g(if)f Fo(\045)g Fz(comm)o(utes)e(with)89 950 y Fo(K)t Fz(.)26 b(Ob)o(viously)l(,)17 b(a)h(steady)g(state)g(do)q(es)h(not)f(c) o(hange)g(in)g(time.)24 b(If)17 b Fo(\045)h Fz(is)g(a)g(steady)g (state,)g(then)89 1010 y(there)j(is)h(a)g(measurable)e(matrix-v)m (alued)g(function)i Fo(\045)p Fp(\()p Fm(\001)p Fp(\))h Fm(2)h Fo(L)1284 992 y Fj(1)1321 1010 y Fp(\()p Fn(R)p Fo(;)8 b Fm(B)q Fp(\()p Fn(C)1484 992 y Fk(2)1507 1010 y Fp(\)\))21 b Fz(suc)o(h)h(that)g(the)89 1070 y(m)o(ultiplic)o(ation) 16 b(op)q(erator)25 b Fp(^)-29 b Fo(\045)19 b Fz(on)g Fo(L)750 1052 y Fk(2)770 1070 y Fp(\()p Fn(R)p Fo(;)8 b Fn(C)880 1052 y Fk(2)903 1070 y Fp(\))18 b Fz(generated)h(b)o(y)f Fo(\045)p Fp(\()p Fm(\001)p Fp(\))h Fz(is)f(unitarily)f(equiv)m(alen)o (t)g(to)89 1130 y Fo(\045)p Fz(,)f(i.e.)837 1193 y Fo(\045)e Fp(=)f(\010)962 1173 y Fj(\000)p Fk(1)962 1206 y Fj(\000)1014 1193 y Fp(^)-28 b Fo(\045)p Fp(\010)1070 1200 y Fj(\000)1100 1193 y Fo(;)647 b Fz(\(3.1\))89 1284 y(see)16 b(ab)q(o)o(v)o(e.)21 b(Ob)o(viously)l(,)15 b(the)h(measurable)f(function)h Fo(\045)p Fp(\()p Fm(\001)p Fp(\))g Fz(tak)o(es)g(the)g(form)689 1426 y Fo(\045)p Fp(\()p Fo(\025)p Fp(\))e(=)846 1355 y Fl(\022)905 1395 y Fo(\045)930 1377 y Fi(bb)963 1395 y Fp(\()p Fo(\025)p Fp(\))45 b Fo(\045)1099 1377 y Fi(ba)1135 1395 y Fp(\()p Fo(\025)p Fp(\))904 1455 y Fo(\045)929 1437 y Fi(ab)965 1455 y Fp(\()p Fo(\025)p Fp(\))d Fo(\045)1098 1437 y Fi(aa)1137 1455 y Fp(\()p Fo(\025)p Fp(\))1224 1355 y Fl(\023)1761 1426 y Fz(\(3.2\))89 1568 y(where)19 b Fo(\045)258 1549 y Fi(\034)t(\027)299 1568 y Fp(\()p Fm(\001)p Fp(\))f Fm(2)h Fo(L)454 1549 y Fj(1)491 1568 y Fp(\()p Fn(R)p Fp(\))p Fz(,)d Fo(\034)s(;)8 b(\027)21 b Fp(=)e Fo(a;)8 b(b)p Fz(.)28 b(Since)18 b Fo(\045)g Fm(\025)g Fp(0)i Fz(one)f(gets)g(that)g Fo(\045)p Fp(\()p Fo(\025)p Fp(\))g Fm(\025)f Fp(0)h Fz(a.e.)29 b(\(with)89 1628 y(resp)q(ect)16 b(to)h(the)f(Leb)q(esgue)h(measure\).)89 1771 y Fq(De\034nition)j(3.1)k Fz(A)12 b(b)q(ounded)i(self-adjoin)o(t)f (op)q(erator)h Fo(A)e Fz(on)i Fh(K)f Fz(is)g(called)e(an)j(observ)m (able.)20 b(W)l(e)89 1831 y(sa)o(y)c(the)g(observ)m(able)h Fo(A)135 1960 y Fz(\(i\))24 b(is)11 b(admissible)f(with)h(resp)q(ect)g (to)h Fo(\045)f Fz(if)g Fo(\045A)g Fz(is)g(a)h(n)o(uclear)e(op)q (erator)j(on)f Fh(K)p Fz(,)g(i.e.)18 b Fo(\045A)c Fm(2)g(L)1756 1967 y Fk(1)1776 1960 y Fp(\()p Fh(K)p Fp(\))p Fz(,)122 2067 y(\(ii\))23 b(is)15 b(admissible)f(with)h(resp)q(ect)g(to)g Fo(K)20 b Fz(if)14 b Fo(E)969 2074 y Fi(K)1004 2067 y Fp(\(\001\))p Fo(A)f Fm(2)h(L)1214 2074 y Fk(1)1234 2067 y Fp(\()p Fh(K)p Fp(\))h Fz(for)h(eac)o(h)f(b)q(ounded)h(in)o(terv)m (al)211 2127 y Fp(\001)e Fm(\022)f Fn(R)g Fz(where)j Fo(E)547 2134 y Fi(K)582 2127 y Fp(\()p Fm(\001)p Fp(\))g Fz(denotes)g(the)g(sp)q(ectral)g(measure)f(of)i Fo(K)t Fz(.)89 2271 y(If)d(the)g(observ)m(able)h Fo(A)f Fz(is)g(admissible)f (with)h(resp)q(ect)g(to)h Fo(\045)p Fz(,)f(then)h(its)f(exp)q(ectation) g(v)m(alue)g Fn(E)1763 2278 y Fi(\045)1786 2271 y Fp(\()p Fo(A)p Fp(\))89 2331 y Fz(with)i(resp)q(ect)g(to)h(the)f(densit)o(y)f (matrix)f Fo(\045)j Fz(is)f(de\034ned)g(b)o(y)796 2444 y Fn(E)826 2451 y Fi(\045)849 2444 y Fp(\()p Fo(A)p Fp(\))e(:=)f Fz(tr)p Fp(\()p Fo(\045A)p Fp(\))p Fo(:)606 b Fz(\(3.3\))89 2556 y(In)19 b(the)f(follo)o(wing)h(w)o(e)f(sho)o(w)h(that)g(the)g (admissibilit)o(y)c(of)k Fo(A)g Fz(with)f(resp)q(ect)h(to)g Fo(K)k Fz(leads)18 b(to)h(a)89 2617 y(certain)d(lo)q(calization)f(in)h (the)g(incoming)f(sp)q(ectral)h(represen)o(tation:)89 2746 y Fq(Prop)r(osition)k(3.2)k Fc(Assume)f Fo(m)15 b Fp(+)798 2726 y Fk(1)p 791 2734 32 2 v 791 2763 a Fi(m)852 2746 y Fm(2)24 b Fo(L)942 2728 y Fj(1)980 2746 y Fp(\([)p Fo(a;)8 b(b)p Fp(]\))p Fc(,)22 b Fo(V)36 b Fm(2)24 b Fo(L)1306 2728 y Fk(2)1326 2746 y Fp(\([)p Fo(a;)8 b(b)p Fp(]\))21 b Fc(and)j Fo(\024)1611 2753 y Fi(a)1632 2746 y Fo(;)8 b(\024)1682 2753 y Fi(b)1723 2746 y Fm(2)24 b Fn(C)1813 2753 y Fk(+)1846 2746 y Fc(.)89 2806 y(If)19 b(the)h(observable)i Fo(A)d Fc(is)g(admissible)i(with)e(r)n(esp)n(e)n (ct)g(to)h Fo(K)t Fc(,)g(then)h(ther)n(e)e(exists)h(a)g(me)n(asur)n (able)p eop %%Page: 12 14 12 13 bop 89 23 a Fz(12)692 b Fs(H.-Ch.)25 b(Kaiser,)18 b(H.)h(Neidhardt,)g(J.)g(Rehber)o(g)89 147 y Fc(matrix-value)n(d)14 b(function)h Fo(A)p Fp(\()p Fm(\001)p Fp(\))e Fm(2)h Fo(L)759 129 y Fk(1)759 160 y Fi(loc)805 147 y Fp(\()p Fn(R)p Fo(;)7 b Fm(B)r Fp(\()p Fn(C)968 129 y Fk(2)990 147 y Fp(\)\))p Fc(,)14 b(such)g(that)f Fo(A)p Fp(\()p Fo(\025)p Fp(\))h(=)g Fo(A)p Fp(\()p Fo(\025)p Fp(\))1531 129 y Fj(\003)1564 147 y Fc(for)e(a.e.)22 b Fo(\025)14 b Fm(2)g Fn(R)89 208 y Fc(and)575 282 y Fz(tr)o Fp(\()p Fo(\045AE)729 289 y Fi(K)763 282 y Fp(\(\001\)\))g(=)927 214 y Fl(Z)954 327 y Fk(\001)994 282 y Fo(d\025)h Fz(tr)1100 290 y Fg(C)1121 281 y Ff(2)1143 282 y Fp(\()p Fo(\045)p Fp(\()p Fo(\025)p Fp(\))p Fo(A)p Fp(\()p Fo(\025)p Fp(\)\))386 b Fz(\(3.4\))89 395 y Fc(for)21 b(any)h(b)n(ounde)n(d)g(Bor)n(el)f(set) i Fp(\001)e Fm(\022)h Fn(R)c Fc(and)k(any)g(ste)n(ady)f(state)h Fo(\045)g Fc(of)g Fo(K)t Fc(.)35 b(The)22 b(me)n(asur)n(able)89 455 y(function)d Fo(A)p Fp(\()p Fm(\001)p Fp(\))e Fc(is)g(uniquely)i (de\034ne)n(d)f(up)g(to)f(a)h(Bor)n(el)f(set)h(of)f(L)n(eb)n(esgue)h (me)n(asur)n(e)f(zer)n(o.)148 539 y(If)k(the)g(observable)i Fo(A)e Fc(is)g(in)g(addition)g(admissible)g(with)h(r)n(esp)n(e)n(ct)e (to)h(the)h(ste)n(ady)e(state)i Fo(\045)p Fc(,)89 599 y(then)c Fz(tr)235 608 y Fg(C)256 598 y Ff(2)279 599 y Fp(\()p Fo(\045)p Fp(\()p Fm(\001)p Fp(\))p Fo(A)p Fp(\()p Fm(\001)p Fp(\)\))13 b Fm(2)h Fo(L)576 581 y Fk(1)596 599 y Fp(\()p Fn(R)p Fp(\))g Fc(and)k(the)g(r)n(epr)n (esentation)652 736 y Fz(tr)p Fp(\()p Fo(\045A)p Fp(\))13 b(=)855 668 y Fl(Z)883 781 y Fg(R)917 736 y Fo(d\025)i Fz(tr)1023 744 y Fg(C)1043 735 y Ff(2)1066 736 y Fp(\()p Fo(\045)p Fp(\()p Fo(\025)p Fp(\))p Fo(A)p Fp(\()p Fo(\025)p Fp(\)\))463 b Fz(\(3.5\))89 872 y Fc(holds.)89 1000 y Fq(Pro)r(of.)23 b Fz(Notice)16 b(that)h(the)f(sp)q(ectral)h(measure)e Fo(E)1018 1007 y Fi(K)1053 1000 y Fp(\()p Fm(\001)p Fp(\))h Fz(of)h Fo(K)k Fz(is)c(absolutely)f(con)o(tin)o(uous)h(with)89 1060 y(resp)q(ect)f(to)h(Leb)q(esgue)g(measure.)i(Hence,)c(the)h(set)g (function)703 1173 y Fo(\026)732 1180 y Fi(\045;A)789 1173 y Fp(\(\001\))d(:=)h Fz(tr)p Fp(\()p Fo(\045AE)1102 1180 y Fi(K)1136 1173 y Fp(\(\001\)\))p Fo(;)513 b Fz(\(3.6\))89 1285 y(where)22 b Fp(\001)g Fz(is)g(b)q(ounded)h(Borel)e(set)i(of)f Fn(R)p Fz(,)e(is)i(in)g(fact)h(a)f(Leb)q(esgue)h(absolutely)f(con)o (tin)o(uous)89 1345 y(measure.)e(Let)390 1320 y Fi(d\026)429 1326 y Fb(\032;A)p 390 1334 90 2 v 416 1363 a Fi(d\025)501 1345 y Fz(denote)d(its)f(Radon-Nik)o(o)q(dym)f(deriv)m(ativ)o(e)f(and)j (de\034ne)808 1487 y Fo(T)837 1494 y Fi(A)865 1487 y Fp(\()t(^)-28 b Fo(\045)p Fp(\))14 b(:=)1012 1454 y Fo(d\026)1066 1461 y Fi(\045;A)p 1012 1476 112 2 v 1041 1521 a Fo(d\025)1128 1487 y(:)619 b Fz(\(3.7\))89 1617 y(Then)17 b Fo(T)246 1624 y Fi(A)274 1617 y Fp(\()p Fm(\001)p Fp(\))g Fz(maps)g Fo(L)505 1599 y Fj(1)542 1617 y Fp(\()p Fn(R)p Fo(;)8 b Fm(B)q Fp(\()p Fn(C)705 1599 y Fk(2)728 1617 y Fp(\)\))17 b Fz(con)o(tin)o(uously)f(in)o(to)h Fo(L)1200 1599 y Fk(1)1200 1630 y Fi(loc)1246 1617 y Fp(\()p Fn(R)p Fp(\))d Fz(\(the)j(latter)f(in)h(its)g(canonic)89 1677 y(F)l(rec)o(het)e(top)q (ology\))i(and,)g(additionally)l(,)e(one)h(has)444 1815 y(tr)p Fp(\()p Fo(\045AE)599 1822 y Fi(K)633 1815 y Fp(\(\001\)\))d(=) 796 1747 y Fl(Z)824 1860 y Fk(\001)864 1815 y Fo(d\025)936 1781 y(d\026)990 1788 y Fi(\045;A)p 936 1803 V 965 1849 a Fo(d\025)1052 1815 y Fp(\()p Fo(\025)p Fp(\))h(=)1184 1747 y Fl(Z)1212 1860 y Fk(\001)1252 1815 y Fo(d\025)g(T)1348 1822 y Fi(A)1376 1815 y Fp(\()t(^)-28 b Fo(\045)q Fp(\)\()p Fo(\025)p Fp(\))255 b Fz(\(3.8\))89 1953 y(for)13 b(an)o(y)f(b)q (ounded)h(Borel)f(set)g Fp(\001)h Fm(\032)h Fn(R)p Fz(.)j(It)12 b(is)g(not)h(hard)g(to)f(see)g(that)h(\(3.8\))g(implies)c(the)k (equalit)o(y)760 2066 y Fo(T)789 2073 y Fi(A)817 2066 y Fp(\()p Fo(\037)867 2073 y Fk(\001)903 2066 y Fp(^)-29 b Fo(\045)p Fp(\))14 b(=)g Fo(\037)1039 2073 y Fk(\001)1070 2066 y Fo(T)1099 2073 y Fi(A)1127 2066 y Fp(\()t(^)-28 b Fo(\045)p Fp(\))571 b Fz(\(3.9\))89 2178 y(for)17 b(an)o(y)f(b)q (ounded)h(Borel)e(set)h Fp(\001)e Fm(\022)f Fn(R)h Fz(where)i Fo(\037)992 2185 y Fk(\001)1039 2178 y Fz(denotes)g(its)g(indicator)g (function.)21 b(W)l(e)16 b(put)161 2317 y Fo(e)184 2324 y Fi(bb)230 2317 y Fp(:=)295 2247 y Fl(\022)353 2286 y Fp(1)41 b(0)353 2346 y(0)g(0)464 2247 y Fl(\023)509 2317 y Fo(;)21 b(e)567 2324 y Fi(ba)617 2317 y Fp(:=)682 2247 y Fl(\022)739 2286 y Fp(0)42 b(1)739 2346 y(0)g(0)850 2247 y Fl(\023)895 2317 y Fo(;)22 b(e)954 2324 y Fi(ab)1003 2317 y Fp(:=)1069 2247 y Fl(\022)1126 2286 y Fp(0)42 b(0)1126 2346 y(1)g(0)1237 2247 y Fl(\023)1282 2317 y Fo(;)22 b(e)1341 2324 y Fi(aa)1394 2317 y Fp(:=)1459 2247 y Fl(\022)1517 2286 y Fp(0)41 b(0)1517 2346 y(0)g(1)1628 2247 y Fl(\023)1736 2317 y Fz(\(3.10\))89 2458 y(and)17 b(de\034ne)f(for)g(an)o(y)g Fo(h)e Fm(2)g Fo(L)613 2440 y Fj(1)651 2458 y Fp(\()p Fn(R)p Fp(\))599 2571 y Fo(A)636 2578 y Fi(ij)666 2571 y Fp(\()p Fo(h)p Fp(\))g(:=)f Fo(T)840 2578 y Fi(A)868 2571 y Fp(\()p Fo(h)e Fm(\001)g Fo(e)974 2578 y Fi(j)r(i)1004 2571 y Fp(\))p Fo(;)105 b(i;)8 b(j)17 b Fp(=)c Fo(a;)8 b(b:)384 b Fz(\(3.11\))89 2683 y(Clearly)l(,)21 b(eac)o(h)g(of)g(the)g(mappings)g Fo(A)800 2690 y Fi(ij)829 2683 y Fp(\()p Fm(\001)p Fp(\))g Fz(then)g(maps)g Fo(L)1184 2665 y Fj(1)1221 2683 y Fp(\()p Fn(R)p Fp(\))d Fz(con)o(tin)o(uously)j (in)o(to)f Fo(L)1741 2665 y Fk(1)1741 2696 y Fi(loc)1787 2683 y Fp(\()p Fn(R)p Fp(\))89 2743 y Fz(and)d(\(3.11\))g(implies)747 2806 y Fo(A)784 2813 y Fi(ij)814 2806 y Fp(\()p Fo(\037)864 2813 y Fk(\001)895 2806 y Fo(h)p Fp(\))d(=)g Fo(\037)1039 2813 y Fk(\001)1070 2806 y Fo(A)1107 2813 y Fi(ij)1137 2806 y Fp(\()p Fo(h)p Fp(\))533 b Fz(\(3.12\))p eop %%Page: 13 15 13 14 bop 89 23 a Fs(Density)18 b(and)g(current)h(of)f(a)g(dissip)l(a)l (tive)g(Schr)o(\366dinger)h(opera)l(tor)187 b Fz(13)89 147 y(for)17 b(an)o(y)f Fo(h)e Fm(2)g Fo(L)378 129 y Fj(1)415 147 y Fp(\()p Fn(R)p Fp(\))f Fz(and)k(an)o(y)f(Borel)g(set)g Fp(\001)p Fz(.)21 b(T)l(aking)16 b(in)g(particular)g Fo(h)e Fm(\021)g Fp(1)p Fz(,)i(this)g(yields)756 233 y Fo(A)793 240 y Fi(ij)823 233 y Fp(\()p Fo(\037)873 240 y Fk(\001)904 233 y Fp(\))e(=)g Fo(\037)1020 240 y Fk(\001)1051 233 y Fo(A)1088 240 y Fi(ij)1118 233 y Fp(\(1\))p Fo(:)542 b Fz(\(3.13\))89 319 y(Ob)o(viously)l(,)17 b(b)o(y)h(the)g(linearit)o(y)f(of)h(the)h(mappings)e Fo(A)1083 326 y Fi(ij)1113 319 y Fp(\()p Fm(\001)p Fp(\))p Fz(,)h(this)h(last)f(equation)g(remains)f(true)89 379 y(if)22 b Fo(\037)171 386 y Fk(\001)225 379 y Fz(is)h(there)f(replaced) g(b)o(y)g(an)o(y)h(\(\034nite\))f(linear)g(com)o(bination)f(of)i (indicator)f(functions.)89 439 y(Because)11 b(the)h(set)g(of)g (\034nite)f(linear)h(com)o(binations)e(of)i(indicator)g(functions)g(is) g(dense)f(in)h Fo(L)1736 421 y Fj(1)1773 439 y Fp(\()p Fn(R)p Fp(\))p Fz(,)89 499 y(one)17 b(gets)f(for)h(all)e Fo(h)f Fm(2)g Fo(L)543 481 y Fj(1)581 499 y Fp(\()p Fn(R)p Fp(\))f Fz(the)j(equation)772 585 y Fo(A)809 592 y Fi(ij)839 585 y Fp(\()p Fo(h)p Fp(\))e(=)g Fo(h)d Fm(\001)g Fo(A)1072 592 y Fi(ij)1102 585 y Fp(\(1\))p Fo(:)558 b Fz(\(3.14\))89 671 y(Since)471 731 y Fo(\045)p Fp(\()p Fm(\001)p Fp(\))14 b(=)f Fo(\045)638 711 y Fi(bb)671 731 y Fp(\()p Fm(\001)p Fp(\))p Fo(e)746 738 y Fi(bb)789 731 y Fp(+)e Fo(\045)863 711 y Fi(ba)899 731 y Fp(\()p Fm(\001)p Fp(\))p Fo(e)974 738 y Fi(ba)1020 731 y Fp(+)g Fo(\045)1094 711 y Fi(ab)1130 731 y Fp(\()p Fm(\001)p Fp(\))p Fo(e)1205 738 y Fi(ab)1251 731 y Fp(+)g Fo(\045)1325 711 y Fi(aa)1365 731 y Fp(\()p Fm(\001)p Fp(\))p Fo(e)1440 738 y Fi(aa)1736 731 y Fz(\(3.15\))89 808 y(one)17 b(gets)251 915 y(tr)p Fp(\()p Fo(\045AE)406 922 y Fi(K)440 915 y Fp(\(\001\)\))d(=)603 847 y Fl(Z)631 960 y Fk(\001)671 915 y Fo(d\025)h(T)768 922 y Fi(A)796 915 y Fp(\()t(^)-28 b Fo(\045)p Fp(\)\()p Fo(\025)p Fp(\))14 b(=)759 b Fz(\(3.16\))471 997 y Fl(X)452 1103 y Fi(i;j)r Fk(=)p Fi(a;b)570 977 y Fl(Z)598 1089 y Fk(\001)637 1044 y Fo(d\025)15 b(A)742 1051 y Fi(ij)772 1044 y Fp(\()p Fo(\045)816 1024 y Fi(ij)846 1044 y Fp(\)\()p Fo(\025)p Fp(\))f(=)1016 997 y Fl(X)997 1103 y Fi(i;j)r Fk(=)p Fi(a;b)1115 977 y Fl(Z)1142 1089 y Fk(\001)1182 1044 y Fo(d\025)h(\045)1275 1024 y Fi(ij)1305 1044 y Fp(\()p Fo(\025)p Fp(\))p Fo(A)1408 1051 y Fi(j)r(i)1438 1044 y Fp(\(1\)\()p Fo(\025)p Fp(\))p Fo(:)89 1179 y Fz(Setting)625 1265 y Fo(A)p Fp(\()p Fm(\001)p Fp(\))e(:=)792 1195 y Fl(\022)851 1235 y Fo(A)888 1242 y Fi(bb)920 1235 y Fp(\(1\)\()p Fm(\001)p Fp(\))46 b Fo(A)1117 1242 y Fi(ba)1152 1235 y Fp(\(1\)\()p Fm(\001)p Fp(\))850 1295 y Fo(A)887 1302 y Fi(ab)922 1295 y Fp(\(1\)\()p Fm(\001)p Fp(\))c Fo(A)1115 1302 y Fi(aa)1154 1295 y Fp(\(1\)\()p Fm(\001)p Fp(\))1289 1195 y Fl(\023)1736 1265 y Fz(\(3.17\))89 1371 y(w)o(e)17 b(\034nally)f(obtain)h(\(3.4\))h(for)f(an)o(y)g(b)q(ounded)h(Borel)e (set)g Fp(\001)f Fm(\032)g Fn(R)f Fz(and)j(an)o(y)g(steady)g(state)g Fo(\045)h Fz(of)89 1431 y Fo(K)t Fz(.)148 1515 y(Assume)e(that)450 1502 y Fp(~)437 1515 y Fo(A)p Fp(\()p Fm(\001)p Fp(\))h Fz(ob)q(eys)h(also)g(the)f(conditions)g(of)h(the)f(prop)q(osition.)26 b(Setting)18 b Fo(G)p Fp(\()p Fo(\025)p Fp(\))e(:=)89 1575 y Fo(A)p Fp(\()p Fo(\025)p Fp(\))11 b Fm(\000)266 1563 y Fp(~)253 1575 y Fo(A)p Fp(\()p Fo(\025)p Fp(\))p Fz(,)16 b Fo(\025)e Fm(2)g Fn(R)p Fz(,)f(one)j(gets)h(that)738 1617 y Fl(Z)766 1730 y Fk(\001)806 1685 y Fz(tr)844 1693 y Fg(C)865 1684 y Ff(2)887 1685 y Fp(\()p Fo(\045)p Fp(\()p Fo(\025)p Fp(\))p Fo(G)p Fp(\()p Fo(\025)p Fp(\)\))e(=)f(0)525 b Fz(\(3.18\))89 1797 y(for)17 b(an)o(y)f(b)q(ounded)h(Borel)e(set)h Fp(\001)f Fz(and)i(an)o(y)f(steady)g(state)h Fo(\045)p Fz(.)k(Hence)15 b(tr)1419 1806 y Fg(C)1440 1796 y Ff(2)1462 1797 y Fp(\()p Fo(\045)p Fp(\()p Fo(\025)p Fp(\))p Fo(G)p Fp(\()p Fo(\025)p Fp(\)\))g(=)f(0)j Fz(for)89 1857 y(a.e.)26 b Fo(\025)18 b Fz(and)h(an)o(y)f(steady)g(state)g Fo(\045)g Fz(whic)o(h)f(imme)o(diately)d(yields)j Fo(G)p Fp(\()p Fo(\025)p Fp(\))h(=)e(0)i Fz(or)h Fo(A)p Fp(\()p Fo(\025)p Fp(\))d(=)1771 1845 y(~)1758 1857 y Fo(A)o Fp(\()p Fo(\025)p Fp(\))89 1918 y Fz(for)h(a.e.)j Fo(\025)15 b Fm(2)f Fn(R)p Fz(.)148 2001 y(If)26 b Fo(\045)f Fz(is)h(admissible)e(with)h(resp)q (ect)h(to)g Fo(A)p Fz(,)h(then)f Fm(j)p Fz(tr)p Fp(\()p Fo(\045AE)1296 2008 y Fi(K)1330 2001 y Fp(\(\001\))p Fm(j)j Fo(<)h Fm(k)p Fo(\045A)p Fm(k)1632 2008 y Fj(L)1656 2013 y Ff(1)1701 2001 y Fz(for)c(an)o(y)89 2062 y(Borel)d(set)h Fp(\001)j Fm(\022)g Fn(R)p Fz(.)42 b(By)24 b(\(3.4\))g(this)g(implies)d (that)k(tr)p Fp(\()p Fo(\045)p Fp(\()p Fm(\001)p Fp(\))p Fo(A)p Fp(\()p Fm(\001)p Fp(\)\))h Fm(2)i Fo(L)1500 2043 y Fk(1)1520 2062 y Fp(\()p Fn(R)p Fp(\))p Fz(.)41 b(Since)24 b(one)89 2122 y(has)17 b Fp(lim)244 2129 y Fk(\001)p Fj(")p Fg(R)325 2122 y Fz(tr)p Fp(\()p Fo(\045AE)480 2129 y Fi(K)514 2122 y Fp(\(\001\)\))d(=)f Fz(tr)p Fp(\()p Fo(\045A)p Fp(\))j Fz(w)o(e)g(obtain)h(from)e(\(3.4\))h(the)g(equalit)o (y)f(\(3.5\).)152 b Fa(\003)148 2206 y Fz(Prop)q(osition)21 b(3.2)f(sa)o(ys)g(that)g(the)f(a)o(v)o(eraging)h(pro)q(cedure)g(lo)q (calizes)e(with)i(resp)q(ect)f(to)h(the)89 2266 y(incoming)10 b(sp)q(ectral)i(represen)o(tation.)19 b(Indeed,)12 b(the)g(quan)o(tit)o (y)e(tr)p Fp(\()p Fo(\045)p Fp(\()p Fo(\025)p Fp(\))p Fo(A)p Fp(\()p Fo(\025)p Fp(\)\))j Fz(can)f(b)q(e)g(regarded)89 2326 y(as)17 b(the)f(lo)q(cal)h(a)o(v)o(erage)e(of)i(the)f(observ)m (able)h Fo(A)p Fp(\()p Fo(\025)p Fp(\))f Fz(with)g(resp)q(ect)g(to)h (the)f(densit)o(y)g(matrix)e Fo(\045)p Fp(\()p Fo(\025)p Fp(\))89 2386 y Fz(at)j(energy)f Fo(\025)e Fm(2)g Fn(R)p Fz(,)f(i.e.)672 2446 y Fn(E)703 2454 y Fi(\045)p Fk(\()p Fi(\025)p Fk(\))774 2446 y Fp(\()p Fo(A)p Fp(\()p Fo(\025)p Fp(\)\))g(:=)h Fz(tr)p Fp(\()p Fo(\045)p Fp(\()p Fo(\025)p Fp(\))p Fo(A)p Fp(\()p Fo(\025)p Fp(\)\))p Fo(:)458 b Fz(\(3.19\))89 2523 y(F)l(orm)o(ula)15 b(\(3.5\))h(has)h(the)f(meaning) f(that)h(the)g(total)g(a)o(v)o(erage)g Fn(E)1273 2530 y Fi(\045)1296 2523 y Fp(\()p Fo(A)p Fp(\))g Fz(is)g(the)g(sum)f(of)h (the)g(lo)q(cal)89 2583 y(a)o(v)o(erages)g Fn(E)313 2591 y Fi(\045)p Fk(\()p Fi(\025)p Fk(\))385 2583 y Fp(\()p Fo(A)p Fp(\()p Fo(\025)p Fp(\)\))p Fz(,)f(i.e.)686 2697 y Fn(E)716 2704 y Fi(\045)739 2697 y Fp(\()p Fo(A)p Fp(\))e(=)879 2629 y Fl(Z)907 2742 y Fg(R)941 2697 y Fo(d\025)i Fn(E)1039 2705 y Fi(\045)p Fk(\()p Fi(\025)p Fk(\))1110 2697 y Fp(\()p Fo(A)p Fp(\()p Fo(\025)p Fp(\)\))p Fo(:)471 b Fz(\(3.20\))89 2806 y(Prop)q(osition)18 b(3.2)e(giv)o(es)g(rise)f(to)i (the)f(follo)o(wing)p eop %%Page: 14 16 14 15 bop 89 23 a Fz(14)692 b Fs(H.-Ch.)25 b(Kaiser,)18 b(H.)h(Neidhardt,)g(J.)g(Rehber)o(g)89 147 y Fq(De\034nition)h(3.3)k Fz(Let)f Fo(A)h Fz(b)q(e)f(an)i(observ)m(able)e(whic)o(h)g(is)h (admissible)d(with)j(resp)q(ect)f(to)h Fo(K)t Fz(.)89 208 y(An)e(elemen)n(t)d Fo(A)p Fp(\()p Fm(\001)p Fp(\))k Fm(2)g Fo(L)561 189 y Fk(1)561 221 y Fi(loc)606 208 y Fp(\()p Fn(R)p Fo(;)8 b Fm(B)q Fp(\()p Fn(C)769 189 y Fk(2)792 208 y Fp(\)\))22 b Fz(is)f(called)g(a)h(lo)q(calizer)e(of)i Fo(A)g Fz(with)f(resp)q(ect)h(to)g Fo(K)j Fz(if)89 268 y Fo(A)p Fp(\()p Fo(\025)p Fp(\))192 250 y Fj(\003)226 268 y Fp(=)13 b Fo(A)p Fp(\()p Fo(\025)p Fp(\))k Fz(for)f(a.e.)21 b Fo(\025)14 b Fm(2)g Fn(R)g Fz(and)596 405 y(tr)p Fp(\()p Fo(\045AE)751 412 y Fi(K)785 405 y Fp(\(\001\)\))g(=)948 337 y Fl(Z)976 450 y Fk(\001)1016 405 y Fo(d\025)g Fz(tr)p Fp(\()p Fo(\045)p Fp(\()p Fo(\025)p Fp(\))p Fo(A)p Fp(\()p Fo(\025)p Fp(\)\))383 b Fz(\(3.21\))89 541 y(holds)17 b(for)f(an)o(y)g(b)q(ounded)h(Borel)f(subset)g Fp(\001)e Fm(\022)f Fn(R)h Fz(and)j(an)o(y)f(steady)g(state)g Fo(\045)p Fz(.)89 685 y(Prop)q(osition)j(3.2)g(sa)o(ys)f(that)g(an)g(observ)m (able)h(whic)o(h)e(is)g(admissible)f(with)i(resp)q(ect)g(to)g Fo(K)k Fz(has)89 745 y(alw)o(a)o(ys)16 b(a)h(unique)e(lo)q(calizer.)148 829 y(T)l(o)k(calculate)f(the)g(carrier)g(densit)o(y)g(w)o(e)g (consider)g(the)g(observ)m(able)h Fo(U)5 b Fp(\()p Fo(!)r Fp(\))p Fz(,)19 b Fo(!)h Fm(\022)e Fp(\012)p Fz(,)h(giv)o(en)89 889 y(b)o(y)471 953 y Fp(\()p Fo(U)5 b Fp(\()p Fo(!)r Fp(\))605 940 y Fo(~)598 953 y(f)19 b Fp(\)\()p Fo(x)p Fp(\))14 b(=)g(0)d Fm(\010)g Fo(\037)908 960 y Fi(!)933 953 y Fp(\()p Fo(x)p Fp(\))p Fo(f)5 b Fp(\()p Fo(x)p Fp(\))10 b Fm(\010)h Fp(0)p Fo(;)1256 940 y(~)1249 953 y(f)19 b Fm(2)c Fo(L)1373 932 y Fk(2)1392 953 y Fp(\()1416 940 y(^)1411 953 y(\012)q(\))p Fo(;)256 b Fz(\(3.22\))89 1043 y(for)19 b(an)o(y)f(Borel)f(subset)i Fo(!)h Fm(\022)d Fp(\012)p Fz(.)27 b(W)l(e)18 b(note)h(that)g(the)f(observ)m(able)g Fo(U)5 b Fp(\()p Fo(!)r Fp(\))19 b Fz(is)f(a)h(pro)s(jection)e(on)89 1104 y Fh(K)j Fz(with)f(ran)p Fp(\()p Fo(U)5 b Fp(\()p Fo(!)r Fp(\)\))20 b Fm(\022)e Fh(H)p Fz(.)31 b(Lo)q(osely)20 b(sp)q(eaking,)g(the)f(pro)s(jection)g(acts)g(in)g(fact)g(only)g(on)h (the)89 1164 y(subspace)d Fh(H)d Fm(\022)f Fh(K)p Fz(.)89 1293 y Fq(Lemma)18 b(3.4)24 b Fc(Assume)16 b Fo(m)9 b Fp(+)674 1274 y Fk(1)p 667 1282 32 2 v 667 1310 a Fi(m)717 1293 y Fm(2)14 b Fo(L)797 1275 y Fj(1)835 1293 y Fp(\([)p Fo(a;)8 b(b)p Fp(]\))p Fc(,)14 b Fo(V)25 b Fm(2)14 b Fo(L)1132 1275 y Fk(2)1152 1293 y Fp(\([)p Fo(a;)8 b(b)p Fp(]\))15 b Fc(and)h Fo(\024)1423 1300 y Fi(a)1444 1293 y Fo(;)8 b(\024)1494 1300 y Fi(b)1525 1293 y Fm(2)14 b Fn(C)1605 1300 y Fk(+)1638 1293 y Fc(.)22 b(Then)16 b(for)89 1353 y(any)j(Bor)n(el)f(set)h Fo(!)f Fm(\022)e Fp(\012)j Fc(the)g(observable)h Fo(U)5 b Fp(\()p Fo(!)r Fp(\))19 b Fc(is)f(admissible)h(with)g(r)n(esp)n(e)n(ct)f(to)h(the)g (minimal)89 1414 y(self-adjoint)h(dilation)f Fo(K)j Fc(of)c(the)h (maximal)g(dissip)n(ative)f(op)n(er)n(ator)f Fo(H)t Fc(.)25 b(If)18 b(the)h(ste)n(ady)f(state)h Fo(\045)89 1474 y Fc(satis\034es)f(the)g(c)n(ondition)597 1587 y Fo(C)635 1594 y Fk(^)-21 b Fi(\045)666 1587 y Fp(:=)14 b(sup)734 1628 y Fi(\025)p Fj(2)p Fg(R)813 1543 y Fm(p)p 855 1543 133 2 v 44 x Fo(\025)883 1572 y Fk(2)914 1587 y Fp(+)d(1)p Fm(k)p Fo(\045)p Fp(\()p Fo(\025)p Fp(\))p Fm(k)1128 1595 y Fj(B)q Fk(\()p Fg(C)1188 1586 y Ff(2)1208 1595 y Fk(\))1237 1587 y Fo(<)j Fm(1)p Fo(;)383 b Fz(\(3.23\))89 1729 y Fc(then)18 b(for)f(any)h(Bor)n(el)f(set)h Fo(!)e Fm(\022)e Fp(\012)j Fc(the)h(observable)i Fo(U)5 b Fp(\()p Fo(!)r Fp(\))17 b Fc(is)h(admissible)g(with)g(r)n(esp)n(e)n(ct)e(to)i Fo(\045)p Fc(.)89 1859 y Fq(Pro)r(of.)j Fz(The)c(relation)e(\(2.33\))i (implies)133 2004 y Fo(U)5 b Fp(\()p Fo(!)r Fp(\)\()p Fo(K)16 b Fm(\000)11 b Fo(z)r Fp(\))411 1984 y Fj(\000)p Fk(1)464 1991 y Fo(~)458 2004 y(f)19 b Fp(=)14 b(0)d Fm(\010)g Fo(\037)669 2011 y Fi(!)694 2004 y Fp(\()p Fo(H)k Fm(\000)c Fo(z)r Fp(\))862 1984 y Fj(\000)p Fk(1)909 2004 y Fo(f)16 b Fp(+)11 b Fo(i\037)1046 2011 y Fi(!)1071 2004 y Fo(T)1100 2011 y Fj(\003)1119 2004 y Fp(\()p 1138 1977 25 2 v Fo(z)r Fp(\))1182 1984 y Fj(\003)1210 1936 y Fl(Z)1260 1950 y Fk(0)1237 2049 y Fj(\0001)1311 2004 y Fo(dy)k(e)1398 1984 y Fj(\000)p Fi(iy)q(z)1476 2004 y Fo(f)1500 2011 y Fj(\000)1529 2004 y Fp(\()p Fo(y)r Fp(\))25 b Fm(\010)11 b Fp(0)44 b Fz(\(3.24\))89 2158 y(for)172 2145 y Fo(~)165 2158 y(f)23 b Fm(2)17 b Fh(K)h Fz(and)h Fo(z)g Fm(2)e Fn(C)535 2165 y Fk(+)567 2158 y Fz(.)27 b(By)17 b(Theorem)g(3.1)h(of)g([11)q(])f(one)i(gets)f(that)g Fp(\()p Fo(H)f Fm(\000)12 b Fo(z)r Fp(\))1597 2140 y Fj(\000)p Fk(1)1662 2158 y Fz(is)18 b(a)g(trace)89 2219 y(class)g(op)q(erator)h(for)e(eac)o(h)g Fo(z)h Fm(2)f Fn(C)713 2226 y Fk(+)745 2219 y Fz(.)25 b(Hence)17 b Fo(U)5 b Fp(\()p Fo(!)r Fp(\)\()p Fo(H)16 b Fm(\000)c Fo(z)r Fp(\))1209 2201 y Fj(\000)p Fk(1)1274 2219 y Fz(is)17 b(a)h(trace)f(class)h(op)q(erator)h(for)89 2279 y(eac)o(h)d Fo(!)g Fm(\022)d Fp(\012)p Fz(.)21 b(Since)15 b(the)h(op)q(erator)h Fo(T)803 2286 y Fj(\003)822 2279 y Fp(\()p 841 2251 V Fo(z)r Fp(\))885 2261 y Fj(\003)921 2279 y Fz(acts)f(from)f(the)g(t)o (w)o(o)h(dimensional)e(Hilb)q(ert)h(space)89 2339 y Fn(C)122 2321 y Fk(2)161 2339 y Fz(in)o(to)g Fh(H)g Fz(one)h(easily)f(gets)h (that)g(the)f(second)h(addend)g(of)g(\(3.24\))h(is)e(a)h(trace)f(class) h(op)q(erator,)89 2399 y(to)q(o.)29 b(Hence)17 b Fo(U)5 b Fp(\()p Fo(!)r Fp(\)\()p Fo(K)17 b Fm(\000)12 b Fo(z)r Fp(\))627 2381 y Fj(\000)p Fk(1)691 2399 y Fm(2)18 b(L)776 2406 y Fk(1)796 2399 y Fp(\()p Fh(K)p Fp(\))h Fz(for)f(eac)o(h)g(Borel) g(set)g Fo(!)h Fm(\022)f Fp(\012)g Fz(and)h Fo(z)g Fm(2)f Fn(C)1666 2406 y Fk(+)1698 2399 y Fz(.)28 b(Using)89 2459 y(the)16 b(represen)o(tation)375 2572 y Fo(U)5 b Fp(\()p Fo(!)r Fp(\))p Fo(E)519 2579 y Fi(K)554 2572 y Fp(\(\001\))13 b(=)h Fo(U)5 b Fp(\()p Fo(!)r Fp(\)\()p Fo(K)15 b Fm(\000)c Fo(z)r Fp(\))975 2552 y Fj(\000)p Fk(1)1022 2572 y Fp(\()p Fo(K)k Fm(\000)c Fo(z)r Fp(\))p Fo(E)1227 2579 y Fi(K)1261 2572 y Fp(\(\001\))p Fo(;)56 b(z)16 b Fm(2)e Fn(C)1529 2579 y Fk(+)1562 2572 y Fo(;)160 b Fz(\(3.25\))89 2685 y(w)o(e)13 b(\034nd)g(that)h Fo(U)5 b Fp(\()p Fo(!)r Fp(\))p Fo(E)499 2692 y Fi(K)534 2685 y Fp(\(\001\))12 b Fz(is)h(a)h(trace)f(class)g(op)q(erator)h(for)g(eac) o(h)e(Borel)g(sets)i Fo(!)i Fm(\022)d Fp(\012)h Fz(and)f(eac)o(h)89 2746 y(b)q(ounded)18 b(in)o(terv)m(al)d Fp(\001)f Fm(\022)h Fn(R)p Fz(.)k(Hence,)c(the)i(observ)m(able)f Fo(U)5 b Fp(\()p Fo(!)r Fp(\))18 b Fz(is)e(admissible)f(with)h(resp)q(ect)g(to) 89 2806 y Fo(K)21 b Fz(for)16 b(eac)o(h)g(Borel)f(set)h Fo(!)g Fm(\022)e Fp(\012)p Fz(.)p eop %%Page: 15 17 15 16 bop 89 23 a Fs(Density)18 b(and)g(current)h(of)f(a)g(dissip)l(a)l (tive)g(Schr)o(\366dinger)h(opera)l(tor)187 b Fz(15)148 147 y(Moreo)o(v)o(er,)15 b(taking)h(in)o(to)g(accoun)o(t)g(\(3.23\))h (and)g(the)f(represen)o(tation)620 245 y Fo(U)5 b Fp(\()p Fo(!)r Fp(\))j Fo(\045)15 b Fp(=)f Fo(U)5 b Fp(\()p Fo(!)r Fp(\)\()p Fo(K)15 b Fm(\000)c Fo(i)p Fp(\))1097 224 y Fj(\000)p Fk(1)1144 245 y Fp(\()p Fo(K)k Fm(\000)c Fo(i)p Fp(\))p Fo(\045)406 b Fz(\(3.26\))89 343 y(one)17 b(imme)o(diately)c (gets)k(that)g Fo(U)5 b Fp(\()p Fo(!)r Fp(\))j Fo(\045)17 b Fz(is)g(a)g(trace)f(class)h(op)q(erator)g(for)g(eac)o(h)f Fo(!)h Fm(\022)d Fp(\012)p Fz(,)j(b)q(ecause)89 403 y Fp(\()p Fo(K)8 b Fm(\000)t Fo(i)p Fp(\))p Fo(\045)k Fz(is)h(b)q (ounded.)21 b(Hence,)12 b Fo(\045)c(U)d Fp(\()p Fo(!)r Fp(\))13 b Fz(is)g(a)g(trace)f(class)h(op)q(erator)h(for)f(eac)o(h)f (Borel)g(set)h Fo(!)j Fm(\022)e Fp(\012)89 463 y Fz(whic)o(h)i(yields)f (that)h Fo(U)5 b Fp(\()p Fo(!)r Fp(\))j Fo(\045)18 b Fz(is)e(admissible)e(with)i(resp)q(ect)g(to)g Fo(\045)545 b Fa(\003)148 547 y Fz(Since)11 b Fo(U)5 b Fp(\()p Fo(!)r Fp(\))13 b Fz(is)f(admissible)d(with)j(resp)q(ect)g(to)g Fo(K)k Fz(for)d(an)o(y)e(Borel)h(set)f Fo(!)16 b Fm(\022)e Fp(\012)e Fz(b)o(y)g(Prop)q(osition)89 607 y(3.2)j(there)g(is)f(a)h (unique)f(lo)q(calizer)g Fo(U)5 b Fp(\()p Fo(!)r Fp(\)\()p Fm(\001)p Fp(\))14 b(:)f Fn(R)e Fm(\000)-8 b(!)13 b(B)r Fp(\()p Fn(C)1155 589 y Fk(2)1177 607 y Fp(\))p Fz(.)21 b(W)l(e)15 b(are)f(going)i(to)f(calculate)f(this)89 667 y(lo)q(calizer.)89 782 y Fq(Prop)r(osition)20 b(3.5)k Fc(Assume)f Fo(m)15 b Fp(+)798 762 y Fk(1)p 791 771 32 2 v 791 799 a Fi(m)852 782 y Fm(2)24 b Fo(L)942 764 y Fj(1)980 782 y Fp(\([)p Fo(a;)8 b(b)p Fp(]\))p Fc(,)22 b Fo(V)36 b Fm(2)24 b Fo(L)1306 764 y Fk(2)1326 782 y Fp(\([)p Fo(a;)8 b(b)p Fp(]\))21 b Fc(and)j Fo(\024)1611 789 y Fi(a)1632 782 y Fo(;)8 b(\024)1682 789 y Fi(b)1723 782 y Fm(2)24 b Fn(C)1813 789 y Fk(+)1846 782 y Fc(.)89 842 y(Then)f(for)e(any)h(Bor)n(el)h(set)f Fo(!)j Fm(\022)d Fp(\012)g Fc(the)h(lo)n(c)n(alizer)f(of)g Fo(U)5 b Fp(\()p Fo(!)r Fp(\)\()p Fm(\001)p Fp(\))22 b Fc(of)g(the)h(observable)g Fo(U)5 b Fp(\()p Fo(!)r Fp(\))23 b Fc(is)89 902 y(given)c(by)712 984 y Fo(U)5 b Fp(\()p Fo(!)r Fp(\)\()p Fo(\025)p Fp(\))14 b(=)952 916 y Fl(Z)980 1029 y Fi(!)1013 984 y Fo(dx)g(D)q Fp(\()p Fo(x;)8 b(\025)p Fp(\))499 b Fz(\(3.27\))89 1089 y Fc(wher)n(e)268 1209 y Fo(D)q Fp(\()p Fo(x;)8 b(\025)p Fp(\))15 b(:=)505 1138 y Fl(\022)652 1178 y Fm(j)p Fo( )700 1160 y Fj(\000)729 1178 y Fp(\()p Fo(x;)8 b(\025;)g(b)p Fp(\))p Fm(j)902 1160 y Fk(2)1052 1178 y Fo( )1086 1160 y Fj(\000)1115 1178 y Fp(\()p Fo(x;)g(\025;)g(a)p Fp(\))p 1279 1135 222 2 v Fo( )1313 1164 y Fj(\000)1341 1178 y Fp(\()p Fo(x;)g(\025;)g(b)p Fp(\))562 1243 y Fo( )596 1225 y Fj(\000)625 1243 y Fp(\()p Fo(x;)g(\025;)g(b)p Fp(\))p 784 1200 227 2 v Fo( )818 1229 y Fj(\000)847 1243 y Fp(\()p Fo(x;)g(\025;)g(a)p Fp(\))128 b Fm(j)p Fo( )1187 1225 y Fj(\000)1216 1243 y Fp(\()p Fo(x;)8 b(\025;)g(a)p Fp(\))p Fm(j)1394 1225 y Fk(2)1521 1138 y Fl(\023)1736 1209 y Fz(\(3.28\))89 1338 y Fo(x)14 b Fm(2)g Fp(\012)p Fc(,)j Fo(\025)e Fm(2)f Fn(R)p Fc(.)148 1421 y(If)j(the)h(ste)n(ady)f(state)h Fo(\045)g Fc(satis\034es)g(the)g (c)n(ondition)g(\(3.23\),)f(then)578 1543 y Fn(E)608 1550 y Fi(\045)631 1543 y Fp(\()p Fo(U)5 b Fp(\()p Fo(!)r Fp(\)\))14 b(=)843 1476 y Fl(Z)871 1588 y Fg(R)905 1543 y Fo(d\025)h Fz(tr)1011 1552 y Fg(C)1032 1542 y Ff(2)1054 1543 y Fp(\()p Fo(\045)p Fp(\()p Fo(\025)p Fp(\))f Fo(U)5 b Fp(\()p Fo(!)r Fp(\)\()p Fo(\025)p Fp(\)\))365 b Fz(\(3.29\))89 1664 y Fc(for)17 b(any)g(Bor)n(el)h(subset)g Fo(!)e Fm(\022)e Fp(\012)p Fc(.)89 1779 y Fq(Pro)r(of.)24 b Fz(By)17 b(Lemma)e(3.4)j (the)f(observ)m(able)g Fo(U)5 b Fp(\()p Fo(!)r Fp(\))18 b Fz(is)f(admissible)e(for)i(an)o(y)g(Borel)g(set)g Fo(!)h Fm(\022)d Fp(\012)p Fz(.)89 1839 y(By)k(Prop)q(osition)j(3.2)e(there)f (is)h(a)g(unique)g(lo)q(calizer)e Fo(U)5 b Fp(\()p Fo(!)r Fp(\)\()p Fm(\001)p Fp(\))21 b Fm(2)f(L)1368 1821 y Fk(1)1368 1852 y Fi(loc)1414 1839 y Fp(\()p Fn(R)p Fo(;)8 b Fm(B)q Fp(\()p Fn(C)1577 1821 y Fk(2)1600 1839 y Fp(\)\))20 b Fz(suc)o(h)f(that)89 1899 y(\(3.21\))13 b(is)e(satis\034ed)h(for)g (an)o(y)g(b)q(ounded)h(Borel)e(set)g Fp(\001)j Fm(\022)g Fn(R)8 b Fz(and)13 b(an)o(y)e(steady)h(state)g Fo(\045)p Fz(.)20 b(F)l(ollo)o(wing)89 1960 y(the)c(pro)q(of)h(of)g(Prop)q (osition)g(3.2)g(w)o(e)f(consider)g(the)g(measure)648 2057 y Fo(\026)677 2065 y Fi(\045;U)t Fk(\()p Fi(!)q Fk(\))785 2057 y Fp(\(\001\))e(=)g Fz(tr)p Fp(\()p Fo(\045)g(U)5 b Fp(\()p Fo(!)r Fp(\))p Fo(E)1170 2064 y Fi(K)1204 2057 y Fp(\(\001\)\))434 b Fz(\(3.30\))89 2162 y(for)22 b(an)o(y)f(b)q (ounded)i(Borel)d(set)i Fp(\001)g Fm(\022)h Fn(R)p Fz(.)34 b(W)l(e)21 b(set)1078 2149 y Fp(^)1069 2162 y Fo(U)6 b Fp(\()p Fo(!)r Fp(\))22 b(:=)h(\010)1310 2169 y Fj(\000)1339 2162 y Fo(U)5 b Fp(\()p Fo(!)r Fp(\)\010)1482 2141 y Fj(\000)p Fk(1)1482 2173 y Fj(\000)1552 2162 y Fz(and)1660 2149 y Fp(^)1652 2162 y Fo(U)1685 2169 y Fk(\001)1716 2162 y Fp(\()p Fo(!)r Fp(\))23 b(:=)89 2228 y Fo(E)125 2235 y Fi(M)165 2228 y Fp(\(\001\))252 2215 y(^)244 2228 y Fo(U)5 b(E)318 2235 y Fi(M)357 2228 y Fp(\(\001\))23 b Fz(where)f Fo(M)30 b Fp(=)25 b(\010)781 2235 y Fj(\000)811 2228 y Fo(K)t Fp(\010)891 2207 y Fj(\000)p Fk(1)891 2239 y Fj(\000)939 2228 y Fz(,)f(see)e(\(2.42\).)41 b(By)22 b(Lemma)f(3.4)i(the)g(op)q(erator)97 2281 y Fp(^)89 2293 y Fo(U)122 2300 y Fk(\001)154 2293 y Fp(\()p Fo(!)r Fp(\))16 b Fz(is)g(n)o(uclear.)k(Hence,)15 b(w)o(e)h(\034nd)703 2391 y Fo(\026)732 2399 y Fi(\045;U)t Fk(\()p Fi(!)q Fk(\))840 2391 y Fp(\(\001\))d(=)h Fz(tr)p Fp(\()t(^)-28 b Fo(\045)1088 2378 y Fp(^)1080 2391 y Fo(U)1113 2398 y Fk(\001)1145 2391 y Fp(\()p Fo(!)r Fp(\)\))p Fo(:)488 b Fz(\(3.31\))89 2496 y(Let)21 b(us)g(calculate)f(the)h(k)o(ernel)e(of) 762 2483 y Fp(^)753 2496 y Fo(U)786 2503 y Fk(\001)818 2496 y Fp(\()p Fo(!)r Fp(\))p Fz(.)35 b(T)l(o)22 b(his)f(end)f(w)o(e)h (consider)f(the)h(scalar)g(pro)q(duct)89 2556 y Fp(\()116 2543 y(^)108 2556 y Fo(U)141 2563 y Fk(\001)173 2556 y Fp(\()p Fo(!)r Fp(\))r(^)-26 b Fo(g)r(;)300 2543 y Fp(^)290 2556 y Fo(f)5 b Fp(\))p Fz(,)18 b Fp(^)-26 b Fo(g)r(;)426 2543 y Fp(^)415 2556 y Fo(f)19 b Fm(2)14 b Fo(L)538 2538 y Fk(2)558 2556 y Fp(\(\001)p Fo(;)8 b Fn(C)673 2538 y Fk(2)695 2556 y Fp(\))p Fz(.)21 b(Using)c(\(2.41\))f (one)h(has)158 2654 y Fp(\()186 2641 y(^)177 2654 y Fo(U)210 2661 y Fk(\001)242 2654 y Fp(\()p Fo(!)r Fp(\))r(^)-26 b Fo(g)r(;)370 2641 y Fp(^)359 2654 y Fo(f)5 b Fp(\))14 b(=)1277 b Fz(\(3.32\))359 2693 y Fl(Z)387 2806 y Fi(!)421 2761 y Fo(dx)496 2693 y Fl(Z)523 2806 y Fk(\001)563 2761 y Fo(d\025)648 2713 y Fl(X)639 2819 y Fi(\034)t Fk(=)p Fi(a;b)738 2761 y Fo( )772 2740 y Fj(\000)801 2761 y Fp(\()p Fo(x;)8 b(\025;)g(\034)e Fp(\))r(^)-26 b Fo(g)991 2740 y Fi(\034)1012 2761 y Fp(\()p Fo(\025)p Fp(\))p 1078 2687 582 2 v -68 x Fl(Z)1106 2806 y Fk(\001)1146 2761 y Fo(d\026)1231 2713 y Fl(X)1223 2819 y Fi(\030)q Fk(=)p Fi(a;b)1319 2761 y Fo( )1353 2746 y Fj(\000)1382 2761 y Fp(\()p Fo(x;)8 b(\026;)g(\030)r Fp(\))1555 2747 y(^)1544 2761 y Fo(f)1573 2746 y Fi(\030)1593 2761 y Fp(\()p Fo(\026)p Fp(\))p Fo(:)p eop %%Page: 16 18 16 17 bop 89 23 a Fz(16)692 b Fs(H.-Ch.)25 b(Kaiser,)18 b(H.)h(Neidhardt,)g(J.)g(Rehber)o(g)89 147 y Fz(Setting)369 229 y Fo(U)407 205 y Fi(\034)t(\030)402 243 y Fk(\001)446 229 y Fp(\()p Fo(!)r Fp(\)\()p Fo(\026;)8 b(\025)p Fp(\))14 b(:=)712 161 y Fl(Z)740 274 y Fi(!)774 229 y Fo(dx)g( )875 208 y Fj(\000)904 229 y Fp(\()p Fo(x;)8 b(\025;)g(\030)r Fp(\))p 1065 186 229 2 v Fo( )1099 215 y Fj(\000)1128 229 y Fp(\()p Fo(x;)g(\026;)g(\034)e Fp(\))p Fo(;)56 b(\034)s(;)8 b(\030)17 b Fp(=)c Fo(a;)8 b(b;)154 b Fz(\(3.33\))89 333 y(w)o(e)16 b(get)162 438 y Fp(\()189 425 y(^)181 438 y Fo(U)214 445 y Fk(\001)246 438 y Fp(\()p Fo(!)r Fp(\))r(^)-26 b Fo(g)r(;)373 425 y Fp(^)363 438 y Fo(f)5 b Fp(\))14 b(=)1273 b Fz(\(3.34\))245 477 y Fl(Z)273 590 y Fk(\001)313 545 y Fo(d\026)389 475 y Fl(\022)426 477 y(Z)454 590 y Fk(\001)494 545 y Fo(d\025)569 475 y Fl(\022)629 515 y Fo(U)667 497 y Fi(bb)662 527 y Fk(\001)699 515 y Fp(\()p Fo(!)r Fp(\)\()p Fo(\026;)8 b(\025)p Fp(\))46 b Fo(U)970 497 y Fi(ba)965 527 y Fk(\001)1006 515 y Fp(\()p Fo(!)r Fp(\)\()p Fo(\026;)8 b(\025)p Fp(\))627 575 y Fo(U)665 557 y Fi(ab)660 587 y Fk(\001)701 575 y Fp(\()p Fo(!)r Fp(\)\()p Fo(\026;)g(\025)p Fp(\))42 b Fo(U)968 557 y Fi(aa)963 587 y Fk(\001)1008 575 y Fp(\()p Fo(!)r Fp(\)\()p Fo(\026;)8 b(\025)p Fp(\))1217 475 y Fl(\023)g(\022)1323 515 y Fp(^)-26 b Fo(g)1346 497 y Fi(b)1363 515 y Fp(\()p Fo(\025)p Fp(\))1321 575 y(^)g Fo(g)1344 557 y Fi(a)1365 575 y Fp(\()p Fo(\025)p Fp(\))1452 475 y Fl(\023)1497 545 y Fo(;)1519 475 y Fl(\022)1589 502 y Fp(^)1578 515 y Fo(f)1607 497 y Fi(b)1625 515 y Fp(\()p Fo(\026)p Fp(\))1587 567 y(^)1576 580 y Fo(f)1605 562 y Fi(a)1627 580 y Fp(\()p Fo(\026)p Fp(\))1715 475 y Fl(\023)o(\023)89 681 y Fz(what)17 b(sho)o(ws)g(that)467 802 y Fo(U)500 809 y Fk(\001)532 802 y Fp(\()p Fo(!)r Fp(\)\()p Fo(\026;)8 b(\025)p Fp(\))14 b(:=)799 732 y Fl(\022)858 772 y Fo(U)896 754 y Fi(bb)891 784 y Fk(\001)929 772 y Fp(\()p Fo(!)r Fp(\)\()p Fo(\026;)8 b(\025)p Fp(\))45 b Fo(U)1199 754 y Fi(ba)1194 784 y Fk(\001)1236 772 y Fp(\()p Fo(!)r Fp(\)\()p Fo(\026;)8 b(\025)p Fp(\))856 832 y Fo(U)894 814 y Fi(ab)889 845 y Fk(\001)930 832 y Fp(\()p Fo(!)r Fp(\)\()p Fo(\026;)g(\025)p Fp(\))43 b Fo(U)1198 814 y Fi(aa)1193 845 y Fk(\001)1238 832 y Fp(\()p Fo(!)r Fp(\)\()p Fo(\026;)8 b(\025)p Fp(\))1446 732 y Fl(\023)1736 802 y Fz(\(3.35\))89 946 y(is)16 b(the)g(k)o(ernel)f (of)h(the)g(in)o(tegral)g(op)q(erator)887 933 y Fp(^)879 946 y Fo(U)912 953 y Fk(\001)944 946 y Fp(\()p Fo(!)r Fp(\))p Fz(.)21 b(Setting)239 1084 y Fo(D)q Fp(\()p Fo(x;)8 b(\026;)g(\025)p Fp(\))15 b(:=)527 1014 y Fl(\022)587 1054 y Fo( )621 1036 y Fj(\000)650 1054 y Fp(\()p Fo(x;)8 b(\025;)g(b)p Fp(\))p 809 1011 223 2 v Fo( )843 1040 y Fj(\000)872 1054 y Fp(\()p Fo(x;)g(\026;)g(b)p Fp(\))46 b Fo( )1112 1036 y Fj(\000)1141 1054 y Fp(\()p Fo(x;)8 b(\025;)g(a)p Fp(\))p 1305 1011 V Fo( )1339 1040 y Fj(\000)1367 1054 y Fp(\()p Fo(x;)g(\026;)g(b)p Fp(\))585 1119 y Fo( )619 1101 y Fj(\000)648 1119 y Fp(\()p Fo(x;)g(\025;)g(b)p Fp(\))p 807 1076 228 2 v Fo( )841 1105 y Fj(\000)869 1119 y Fp(\()p Fo(x;)g(\026;)g(a)p Fp(\))41 b Fo( )1109 1101 y Fj(\000)1138 1119 y Fp(\()p Fo(x;)8 b(\025;)g(a)p Fp(\))p 1302 1076 V Fo( )1336 1105 y Fj(\000)1365 1119 y Fp(\()p Fo(x;)g(\026;)g(a)p Fp(\))1550 1014 y Fl(\023)1736 1084 y Fz(\(3.36\))89 1220 y(w)o(e)16 b(obtain)h(the)f(represen)o (tation)532 1346 y Fo(U)565 1353 y Fk(\001)597 1346 y Fp(\()p Fo(!)r Fp(\)\()p Fo(\026;)8 b(\025)p Fp(\))14 b(=)850 1278 y Fl(Z)878 1391 y Fi(!)911 1346 y Fo(dx)g(D)q Fp(\()p Fo(x;)8 b(\026;)g(\025)p Fp(\))p Fo(;)23 b(\026;)8 b(\025)14 b Fm(2)h Fp(\001)p Fo(:)317 b Fz(\(3.37\))89 1477 y(Since)24 b Fo(U)258 1484 y Fk(\001)289 1477 y Fp(\()p Fo(!)r Fp(\)\()p Fo(\026;)8 b(\025)p Fp(\))29 b(=)f Fo(U)604 1484 y Fk(\001)633 1475 y Fd(0)647 1477 y Fp(\()p Fo(!)r Fp(\)\()p Fo(\026;)8 b(\025)p Fp(\))25 b Fz(for)g Fo(\026;)8 b(\025)28 b Fm(2)h Fp(\001)e Fm(\022)h Fp(\001)1287 1459 y Fj(0)1323 1477 y Fz(it)c(mak)o(es)f(sense)h(to)h (de\034ne)89 1537 y Fo(U)5 b Fp(\()p Fo(!)r Fp(\)\()p Fo(\026;)j(\025)p Fp(\))21 b(:=)f(lim)475 1544 y Fk(\001)p Fj(")p Fg(R)556 1537 y Fo(U)589 1544 y Fk(\001)621 1537 y Fp(\()p Fo(!)r Fp(\)\()p Fo(\026;)8 b(\025)p Fp(\))p Fz(,)21 b Fo(\026;)8 b(\025)20 b Fm(2)h Fn(R)p Fz(.)29 b(Hence)18 b Fo(U)1259 1544 y Fk(\001)1291 1537 y Fp(\()p Fo(!)r Fp(\)\()p Fo(\026;)8 b(\025)p Fp(\))21 b(=)f Fo(U)5 b Fp(\()p Fo(!)r Fp(\)\()p Fo(\026;)j(\025)p Fp(\))21 b Fz(for)89 1597 y Fo(\026;)8 b(\025)15 b Fm(2)f Fp(\001)i Fz(and)528 1679 y Fo(U)5 b Fp(\()p Fo(!)r Fp(\)\()p Fo(\026;)j(\025)p Fp(\))15 b(=)820 1611 y Fl(Z)847 1724 y Fi(!)881 1679 y Fo(dx)f(D)q Fp(\()p Fo(x;)8 b(\026;)g(\025)p Fp(\))p Fo(;)57 b(\026;)8 b(\025)15 b Fm(2)f Fp(\001)p Fo(:)313 b Fz(\(3.38\))89 1787 y(Since)16 b(the)f(k)o(ernel)g(dep)q(ends)i(con)o (tin)o(uously)e(on)i Fo(\026)f Fz(and)h Fo(\025)g Fz(one)f(gets)h(that) 525 1913 y(tr)p Fp(\()t(^)-28 b Fo(\045)629 1901 y Fp(^)621 1913 y Fo(U)654 1920 y Fk(\001)686 1913 y Fp(\()p Fo(!)r Fp(\)\))13 b(=)840 1845 y Fl(Z)868 1958 y Fk(\001)908 1913 y Fo(d\025)h Fz(tr)1013 1922 y Fg(C)1034 1912 y Ff(2)1057 1913 y Fp(\()p Fo(\045)p Fp(\()p Fo(\025)p Fp(\))g Fo(U)5 b Fp(\()p Fo(!)r Fp(\)\()p Fo(\025;)j(\025)p Fp(\)\))312 b Fz(\(3.39\))89 2044 y(for)22 b(an)o(y)f(b)q(ounded)h (Borel)f(set)g Fp(\001)h Fm(\022)g Fn(R)d Fz(and)j(an)o(y)f(steady)g (state)h Fo(\045)p Fz(.)37 b(F)l(rom)20 b(\(3.38\))i(w)o(e)f(\034nd)89 2104 y(that)e Fo(U)5 b Fp(\()p Fo(!)r Fp(\)\()p Fo(\025;)j(\025)p Fp(\))421 2086 y Fj(\003)459 2104 y Fp(=)17 b Fo(U)5 b Fp(\()p Fo(!)r Fp(\)\()p Fo(\025;)j(\025)p Fp(\))19 b Fz(for)g Fo(\025)e Fm(2)h Fn(R)p Fz(.)24 b(Since)17 b(the)h(eigenfunctions)f Fo( )1576 2086 y Fj(\000)1606 2104 y Fp(\()p Fo(x;)8 b(\025;)g(b)p Fp(\))17 b Fz(and)89 2164 y Fo( )123 2146 y Fj(\000)152 2164 y Fp(\()p Fo(x;)8 b(\025;)g(a)p Fp(\))19 b Fz(are)h(b)q(ounded)g(on)h(compact)d(sets)i (of)g Fo(x)f Fz(and)h Fo(\025)g Fz(w)o(e)g(obtain)g(that)g Fo(U)5 b Fp(\()p Fo(!)r Fp(\)\()p Fo(\025;)j(\025)p Fp(\))21 b Fm(2)89 2225 y Fo(L)122 2206 y Fk(1)122 2238 y Fi(loc)168 2225 y Fp(\()p Fn(R)p Fo(;)7 b Fm(B)r Fp(\()p Fn(C)331 2206 y Fk(2)353 2225 y Fp(\)\))p Fz(.)21 b(By)12 b(\(3.39\))i(the)f (matrix-v)m(alued)f(function)h Fo(U)5 b Fp(\()p Fo(!)r Fp(\)\()p Fo(\025)p Fp(\))15 b(:=)e Fo(U)5 b Fp(\()p Fo(!)r Fp(\)\()p Fo(\025;)j(\025)p Fp(\))p Fz(,)15 b Fo(\025)f Fm(2)g Fn(R)p Fz(,)89 2285 y(is)19 b(the)f(unique)g(lo)q (calizer)f(of)i(the)f(observ)m(able)h Fo(U)5 b Fp(\()p Fo(!)r Fp(\))p Fz(.)28 b(By)18 b(condition)g(\(3.23\))h(the)g(observ)m (able)89 2345 y Fo(U)5 b Fp(\()p Fo(!)r Fp(\))17 b Fz(is)f(admissible)e (with)i(resp)q(ect)g(to)h Fo(\045)p Fz(.)k(Applying)15 b(Prop)q(osition)j(3.2)e(w)o(e)g(v)o(erify)f(\(3.29\).)148 2429 y(It)22 b(remains)f(to)i(v)o(erify)d(\(3.27\))j(and)g(\(3.28\).)40 b(F)l(rom)21 b(\(3.38\))i(and)g(\(3.39\))g(w)o(e)f(obtain)h(the)89 2489 y(represen)o(tation)459 2615 y(tr)o Fp(\()t(^)-28 b Fo(\045)563 2602 y Fp(^)554 2615 y Fo(U)587 2622 y Fk(\001)619 2615 y Fp(\()p Fo(!)r Fp(\)\))14 b(=)774 2547 y Fl(Z)801 2660 y Fk(\001)841 2615 y Fo(d\025)917 2547 y Fl(Z)945 2660 y Fi(!)978 2615 y Fo(dx)g Fz(tr)1083 2623 y Fg(C)1104 2614 y Ff(2)1126 2615 y Fp(\()p Fo(\045)p Fp(\()p Fo(\025)p Fp(\))h Fo(D)q Fp(\()p Fo(x;)8 b(\025;)g(\025)p Fp(\)\))p Fo(:)245 b Fz(\(3.40\))89 2746 y(Setting)21 b Fo(D)q Fp(\()p Fo(x;)8 b(\025)p Fp(\))22 b(:=)f Fo(D)q Fp(\()p Fo(x;)8 b(\025;)g(\025)p Fp(\))p Fz(,)23 b Fo(x)e Fm(2)h Fp(\012)p Fz(.)35 b Fo(\025)22 b Fm(2)g Fn(R)p Fz(,)c(w)o(e)i(immediatel)o(y)e(obtain)j(\(3.27\))g(and)89 2806 y(\(3.28\).)1595 b Fa(\003)p eop %%Page: 17 19 17 18 bop 89 23 a Fs(Density)18 b(and)g(current)h(of)f(a)g(dissip)l(a)l (tive)g(Schr)o(\366dinger)h(opera)l(tor)187 b Fz(17)148 147 y(W)l(e)16 b(set)662 208 y Fo(u)693 215 y Fk(^)-21 b Fi(\045)710 208 y Fp(\()p Fo(x;)8 b(\025)p Fp(\))14 b(:=)f Fz(tr)943 216 y Fg(C)964 206 y Ff(2)987 208 y Fp(\()p Fo(\045)p Fp(\()p Fo(\025)p Fp(\))h Fo(D)q Fp(\()p Fo(x;)8 b(\025)p Fp(\)\))449 b Fz(\(3.41\))89 290 y(for)18 b Fo(x)c Fm(2)i Fp(\012)h Fz(and)h Fo(\025)e Fm(2)f Fn(R)p Fz(.)21 b(A)16 b(simple)f(computation)h(sho)o(ws)i(that)g(the)f(eigen)o (v)m(alues)f(of)h Fo(D)q Fp(\()p Fo(x;)8 b(\025)p Fp(\))89 351 y Fz(are)16 b(equal)g(to)g Fm(j)p Fo( )406 333 y Fj(\000)435 351 y Fp(\()p Fo(x;)8 b(\025;)g(b)p Fp(\))p Fm(j)608 333 y Fk(2)643 351 y Fz(and)17 b Fm(j)p Fo( )786 333 y Fj(\000)815 351 y Fp(\()p Fo(x;)8 b(\025;)g(a)p Fp(\))p Fm(j)993 333 y Fk(2)1028 351 y Fz(whic)o(h)15 b(sho)o(ws)i(that)f(the)g(matrix)e Fo(D)q Fp(\()p Fo(x;)8 b(\025)p Fp(\))17 b Fz(is)89 411 y(non-negativ)o(e)i(for)h(eac)o(h)f Fo(x)g Fm(2)g Fp(\012)g Fz(and)h Fo(\025)g Fm(2)f Fn(R)p Fz(.)27 b(Since)19 b(for)g(a.e.)30 b Fo(\025)20 b Fm(2)f Fn(R)e Fz(the)i(matrix)e Fo(\045)p Fp(\()p Fo(\025)p Fp(\))j Fz(is)89 471 y(non-negativ)o(e,)f(to)q(o)h(one)e(gets)h(that)g Fo(u)814 478 y Fk(^)-21 b Fi(\045)831 471 y Fp(\()p Fo(x;)8 b(\025)p Fp(\))18 b Fm(\025)g Fp(0)h Fz(for)g Fo(x)e Fm(2)h Fp(\012)h Fz(and)g(a.e.)28 b Fo(\025)19 b Fm(2)f Fn(R)p Fz(.)25 b(This)19 b(fact)89 531 y(can)e(also)f(b)q(e)h(v)o (eri\034ed)d(taking)j(in)o(to)f(accoun)o(t)g(the)g(represen)o(tation) 409 658 y Fo(u)440 665 y Fk(^)-21 b Fi(\045)457 658 y Fp(\()p Fo(x;)8 b(\025)p Fp(\))14 b(=)639 587 y Fl(\034)676 658 y Fo(\045)701 637 y Fi(t)716 658 y Fp(\()p Fo(\025)p Fp(\))790 587 y Fl(\022)851 627 y Fo( )885 609 y Fj(\000)914 627 y Fp(\()p Fo(x;)8 b(\025;)g(b)p Fp(\))848 687 y Fo( )882 669 y Fj(\000)911 687 y Fp(\()p Fo(x;)g(\025;)g(a)p Fp(\))1095 587 y Fl(\023)1140 658 y Fo(;)1162 587 y Fl(\022)1222 627 y Fo( )1256 609 y Fj(\000)1285 627 y Fp(\()p Fo(x;)g(\025;)g(b)p Fp(\))1220 687 y Fo( )1254 669 y Fj(\000)1283 687 y Fp(\()p Fo(x;)g(\025;)g(a)p Fp(\))1467 587 y Fl(\023)o(\035)1736 658 y Fz(\(3.42\))89 788 y(where)24 b Fo(\045)263 770 y Fi(t)278 788 y Fp(\()p Fo(\025)p Fp(\))h Fz(is)f(the)g(transp)q(osed) i(matrix)d(to)h(\(3.2\).)46 b(Moreo)o(v)o(er,)25 b(if)f(condition)g (\(3.23\))h(is)89 848 y(satis\034ed,)16 b(then)g(from)f(Prop)q(osition) j(\(3.5\))e(w)o(e)g(obtain)h(the)f(represen)o(tation)627 972 y Fn(E)658 979 y Fi(\045)681 972 y Fp(\()p Fo(U)5 b Fp(\()p Fo(!)r Fp(\)\))14 b(=)893 905 y Fl(Z)920 1017 y Fg(R)955 972 y Fo(d\025)1031 905 y Fl(Z)1058 1017 y Fi(!)1092 972 y Fo(dx)g(u)1190 979 y Fk(^)-21 b Fi(\045)1207 972 y Fp(\()p Fo(x;)8 b(\025)p Fp(\))413 b Fz(\(3.43\))89 1096 y(for)17 b(Borel)e(sets)h Fo(!)g Fm(\022)e Fp(\012)p Fz(.)21 b(T)l(aking)c(in)o(to)f(accoun)o(t)g(F)l(ubini's)f(theorem)g(w) o(e)h(\034nd)g(that)715 1217 y Fn(E)745 1224 y Fi(\045)768 1217 y Fp(\()p Fo(U)5 b Fp(\()p Fo(!)r Fp(\)\))14 b(=)980 1149 y Fl(Z)1008 1262 y Fi(!)1041 1217 y Fo(dx)g(u)1139 1224 y Fk(^)-21 b Fi(\045)1156 1217 y Fp(\()p Fo(x)p Fp(\))p Fo(:)500 b Fz(\(3.44\))89 1341 y(where)582 1412 y Fo(u)613 1419 y Fk(^)-21 b Fi(\045)630 1412 y Fp(\()p Fo(x)p Fp(\))14 b(:=)775 1345 y Fl(Z)803 1457 y Fg(R)837 1412 y Fo(d\025)h(u)936 1419 y Fk(^)-21 b Fi(\045)953 1412 y Fp(\()p Fo(x;)8 b(\025)p Fp(\))14 b Fm(\025)f Fp(0)p Fo(;)57 b(x)14 b Fm(2)g Fp(\012)p Fo(;)368 b Fz(\(3.45\))89 1524 y(and)19 b Fo(u)217 1531 y Fk(^)-21 b Fi(\045)251 1524 y Fm(2)17 b Fo(L)334 1506 y Fk(1)354 1524 y Fp(\(\012\))p Fz(.)27 b(The)18 b(represen)o(tation)g(\(3.44\))h(sho)o(ws)g(that)f Fn(E)1313 1531 y Fi(\045)1336 1524 y Fp(\()p Fo(U)5 b Fp(\()p Fm(\001)p Fp(\)\))19 b Fz(de\034nes)f(a)g(measure)89 1584 y(on)h Fp(\012)g Fz(whic)o(h)g(is)f(absolutely)h(con)o(tin)o(uous) f(with)h(resp)q(ect)f(to)h(the)g(Leb)q(esgue)g(measure.)28 b(Since)89 1644 y(the)15 b(exp)q(ectation)f(v)m(alue)g Fn(E)588 1651 y Fi(\045)611 1644 y Fp(\()p Fo(U)5 b Fp(\()p Fo(!)r Fp(\)\))15 b Fz(has)g(the)g(meaning)e(of)i(the)g(n)o(um)o(b)q (er)e(of)i(carriers)f(in)g Fo(!)i Fm(\022)e Fp(\012)89 1704 y Fz(its)f(Radon-Nik)o(o)q(dym)e(deriv)m(ativ)o(e)g(can)i(b)q(e)g (in)o(terpreted)e(as)i(the)f(carrier)g(densit)o(y)g(of)h(the)f(system) 89 1765 y(describ)q(ed)k(b)o(y)g Fo(K)k Fz(and)d(m)o(utatis)d(m)o (utandis)h(b)o(y)h Fo(H)t Fz(.)89 1891 y Fq(De\034nition)k(3.6)k Fz(Assume)15 b Fo(m)c Fp(+)743 1871 y Fk(1)p 736 1880 32 2 v 736 1908 a Fi(m)787 1891 y Fm(2)k Fo(L)868 1873 y Fj(1)905 1891 y Fp(\([)p Fo(a;)8 b(b)p Fp(]\))p Fz(,)15 b Fo(V)26 b Fm(2)15 b Fo(L)1204 1873 y Fk(2)1224 1891 y Fp(\([)p Fo(a;)8 b(b)p Fp(]\))15 b Fz(and)i Fo(\024)1497 1898 y Fi(a)1518 1891 y Fo(;)8 b(\024)1568 1898 y Fi(b)1600 1891 y Fm(2)15 b Fn(C)1681 1898 y Fk(+)1713 1891 y Fz(.)23 b(Then)89 1951 y(the)11 b(matrix)e Fo(D)q Fp(\()p Fo(x;)f(\025)p Fp(\))p Fz(,)13 b Fo(x)g Fm(2)h Fp(\012)p Fz(,)e(and)g(the)f(v)m(alue)f Fo(u)974 1958 y Fk(^)-21 b Fi(\045)991 1951 y Fp(\()p Fo(x;)8 b(\025)p Fp(\))j Fz(are)g(called)f(carrier)h(densit)o(y)f (observ)m(able)89 2011 y(and)20 b(carrier)e(densit)o(y)g(at)i Fo(x)e Fm(2)h Fp(\012)h Fz(and)f(at)h(energy)f Fo(\025)g Fm(2)g Fn(R)d Fz(of)j(the)g(system)f(describ)q(ed)g(b)o(y)h Fo(H)t Fz(,)89 2072 y(resp)q(ectiv)o(ely)l(.)89 2198 y(This)j(de\034nition)g(is)g(justi\034ed)g(b)o(y)f(the)h(fact)g(that)g (b)o(y)g(\(3.41\))h(the)f(carrier)f(densit)o(y)g Fo(u)1728 2205 y Fk(^)-21 b Fi(\045)1745 2198 y Fp(\()p Fo(x;)8 b(\025)p Fp(\))89 2258 y Fz(is)20 b(the)h(exp)q(ectation)f(v)m(alue)g (of)h(the)f(carrier)f(densit)o(y)h(observ)m(able)g Fo(D)q Fp(\()p Fo(x;)8 b(\025)p Fp(\))p Fz(,)22 b(i.e.)33 b Fo(u)1669 2265 y Fk(^)-21 b Fi(\045)1686 2258 y Fp(\()p Fo(x;)8 b(\025)p Fp(\))21 b(=)89 2318 y Fn(E)119 2326 y Fi(\045)p Fk(\()p Fi(\025)p Fk(\))191 2318 y Fp(\()p Fo(D)q Fp(\()p Fo(x;)8 b(\025)p Fp(\)\))15 b Fz(at)g Fo(x)e Fm(2)h Fp(\012)h Fz(and)g(at)g Fo(\025)f Fm(2)g Fn(R)p Fz(.)k(Moreo)o(v)o(er,)13 b(w)o(e)h(note)h(that)g(\(3.45\))g (can)g(b)q(e)f(written)89 2379 y(as)568 2451 y Fo(u)599 2458 y Fk(^)-21 b Fi(\045)616 2451 y Fp(\()p Fo(x)p Fp(\))13 b(=)747 2383 y Fl(Z)775 2496 y Fg(R)809 2451 y Fo(d\025)i Fn(E)907 2458 y Fi(\045)p Fk(\()p Fi(\025)p Fk(\))978 2451 y Fp(\()p Fo(D)q Fp(\()p Fo(x;)8 b(\025)p Fp(\)\))p Fo(;)58 b(x)13 b Fm(2)h Fp(\012)p Fo(;)354 b Fz(\(3.46\))89 2557 y(i.e,)12 b(the)g(carrier)f(densit)o(y)g(at)i Fo(x)g Fm(2)h Fp(\012)f Fz(is)f(the)g(sum)f(of)i(exp)q(ectation)f(v)m(alues)g (of)g(the)g(carrier)g(densit)o(y)89 2617 y(observ)m(able)k(at)h Fo(x)d Fm(2)g Fp(\012)i Fz(o)o(v)o(er)f(all)h(energies.)148 2701 y(On)h(the)e(Hilb)q(ert)g(space)i Fh(K)d Fp(=)g Fo(L)744 2683 y Fk(2)764 2701 y Fp(\()788 2688 y(^)783 2701 y(\012\))i Fz(w)o(e)g(consider)g(the)g(m)o(ultipli)o(cation)e(op)q (erator)j Fo(M)5 b Fp(\()p Fo(h)p Fp(\))p Fz(,)533 2806 y Fp(\()p Fo(M)g Fp(\()p Fo(h)p Fp(\))677 2793 y Fo(~)670 2806 y(f)h Fp(\)\()p Fo(x)p Fp(\))13 b(=)h(0)d Fm(\010)g Fo(h)p Fp(\()p Fo(x)p Fp(\))p Fo(f)5 b Fp(\()p Fo(x)p Fp(\))11 b Fm(\010)g Fp(0)p Fo(;)57 b(x)13 b Fm(2)h Fp(\012)p Fo(;)319 b Fz(\(3.47\))p eop %%Page: 18 20 18 19 bop 89 23 a Fz(18)692 b Fs(H.-Ch.)25 b(Kaiser,)18 b(H.)h(Neidhardt,)g(J.)g(Rehber)o(g)89 147 y Fz(for)14 b(real)f(functions)h Fo(h)g Fm(2)g Fo(L)583 129 y Fj(1)620 147 y Fp(\(\012\))p Fz(.)21 b(W)l(e)13 b(note)h(that)g Fo(M)5 b Fp(\()p Fo(\037)1120 154 y Fi(!)1145 147 y Fp(\))14 b(=)g Fo(U)5 b Fp(\()p Fo(!)r Fp(\))14 b Fz(for)g(an)o(y)f(Borel)g(set) g Fo(!)j Fm(\022)e Fp(\012)p Fz(.)89 208 y(In)i(particular,)g(one)g (has)h Fo(M)5 b Fp(\()p Fo(\037)667 215 y Fk(\012)694 208 y Fp(\))14 b(=)g Fo(U)5 b Fp(\(\012\))14 b(=)g Fo(P)994 189 y Fe(K)987 220 y(H)1019 208 y Fz(.)22 b(Ob)o(viously)l(,)14 b(the)i(represen)o(tation)564 320 y Fo(\045)e(M)5 b Fp(\()p Fo(h)p Fp(\))14 b(=)g Fo(\045)g(U)5 b Fp(\(\012\))14 b Fo(M)5 b Fp(\()p Fo(h)p Fp(\))p Fo(;)57 b(h)14 b Fm(2)g Fo(L)1262 300 y Fj(1)1300 320 y Fp(\(\012\))p Fo(;)349 b Fz(\(3.48\))89 433 y(is)23 b(v)m(alid.)39 b(Since)22 b(the)g(observ)m(able)h Fo(U)5 b Fp(\(\012\))23 b Fz(is)f(admissible)f (with)h(resp)q(ect)g(to)h Fo(\045)g Fz(the)f(pro)q(duct)89 493 y Fo(\045)14 b(U)5 b Fp(\(\012\))16 b Fz(is)f(a)g(n)o(uclear)f(op)q (erator)j(on)e Fh(K)h Fz(whic)o(h)e(yields)g(that)i Fo(\045M)5 b Fp(\()p Fo(h)p Fp(\))15 b Fz(is)g(a)h(n)o(uclear)e(op)q(erator)i(on) 89 553 y Fh(K)p Fz(,)g(i.e.)k(the)c(observ)m(able)h Fo(M)5 b Fp(\()p Fo(h)p Fp(\))16 b Fz(is)g(admissible)e(with)j(resp)q(ect)e (to)i Fo(\045)p Fz(.)89 682 y Fq(Prop)r(osition)j(3.7)k Fc(Assume)c Fo(m)12 b Fp(+)789 662 y Fk(1)p 782 670 32 2 v 782 699 a Fi(m)836 682 y Fm(2)17 b Fo(L)919 664 y Fj(1)957 682 y Fp(\([)p Fo(a;)8 b(b)p Fp(]\))p Fc(,)18 b Fo(V)29 b Fm(2)17 b Fo(L)1265 664 y Fk(2)1285 682 y Fp(\([)p Fo(a;)8 b(b)p Fp(]\))18 b Fc(and)h Fo(\024)1562 689 y Fi(a)1583 682 y Fo(;)8 b(\024)1633 689 y Fi(b)1668 682 y Fm(2)17 b Fn(C)1751 689 y Fk(+)1784 682 y Fc(.)28 b(If)89 742 y(the)19 b(ste)n(ady)f(state)i Fo(\045)e Fc(satis\034es)i(the)f(c)n(ondition)g(\(3.23\),)f(then)i(the)f(c)n (arrier)e(density)i Fo(u)1678 749 y Fk(^)-21 b Fi(\045)1714 742 y Fc(de\034ne)n(d)89 802 y(by)18 b(\(3.45\))f(is)g(a)g(non-ne)n (gative)j Fo(L)712 784 y Fk(1)732 802 y Fc(-function)f(such)f(that)664 939 y Fz(tr)p Fp(\()p Fo(\045M)5 b Fp(\()p Fo(h)p Fp(\)\))14 b(=)949 871 y Fl(Z)976 984 y Fk(\012)1012 939 y Fo(dx)g(u)1110 946 y Fk(^)-21 b Fi(\045)1127 939 y Fp(\()p Fo(x)p Fp(\))p Fo(h)p Fp(\()p Fo(x)p Fp(\))449 b Fz(\(3.49\))89 1078 y Fc(for)17 b(r)n(e)n(al)g(functions)h Fo(h)c Fm(2)g Fo(L)593 1059 y Fj(1)631 1078 y Fp(\(\012\))p Fc(.)22 b(In)c(p)n(articular,)f(one)h(has)469 1190 y Fm(k)p Fo(u)525 1197 y Fk(^)-21 b Fi(\045)542 1190 y Fm(k)567 1198 y Fi(L)591 1189 y Ff(1)608 1198 y Fk(\()p Fi(!)q Fk(\))675 1190 y Fp(=)14 b Fz(tr)o Fp(\()p Fo(\045)8 b(U)d Fp(\()p Fo(!)r Fp(\)\))15 b Fm(\024)f Fo(C)1049 1197 y Fk(^)-21 b Fi(\045)1066 1190 y Fm(k)p Fp(\()p Fo(K)15 b Fm(\000)c Fo(i)p Fp(\))1252 1169 y Fj(\000)p Fk(1)1298 1190 y Fo(P)1336 1169 y Fe(K)1329 1202 y(H)1362 1190 y Fm(k)1387 1198 y Fj(L)1411 1203 y Ff(1)1428 1198 y Fk(\()p Fe(K)p Fk(\))1736 1190 y Fz(\(3.50\))89 1303 y Fc(for)17 b(e)n(ach)h(Bor)n(el)f(set)h Fo(!)e Fm(\022)e Fp(\012)p Fc(.)89 1431 y Fq(Pro)r(of.)24 b Fz(Setting)17 b Fo(!)h Fp(=)d(\012)j Fz(w)o(e)e(obtain)i(from)e (\(3.44\))i(that)g Fo(u)1209 1438 y Fk(^)-21 b Fi(\045)1241 1431 y Fm(2)16 b Fo(L)1323 1413 y Fk(1)1343 1431 y Fp(\(\012\))p Fz(.)24 b(W)l(e)17 b(c)o(ho)q(ose)h Fo(h)d Fp(=)h Fo(\037)1823 1438 y Fi(!)1847 1431 y Fz(,)89 1491 y Fo(!)g Fm(\022)e Fp(\012)p Fz(.)21 b(By)16 b(\(3.44\))h(w)o(e)e(get)629 1628 y(tr)p Fp(\()p Fo(\045M)5 b Fp(\()p Fo(\037)813 1635 y Fi(!)838 1628 y Fp(\)\))14 b(=)942 1560 y Fl(Z)970 1673 y Fk(\012)1005 1628 y Fo(dx)g(u)1103 1635 y Fk(^)-21 b Fi(\045)1120 1628 y Fp(\()p Fo(x)p Fp(\))p Fo(\037)1217 1635 y Fi(!)1242 1628 y Fp(\()p Fo(x)p Fp(\))p Fo(:)414 b Fz(\(3.51\))89 1764 y(By)16 b(linearit)o(y)e(this)i(equation)g (extends)g(to)664 1898 y(tr)p Fp(\()p Fo(\045M)5 b Fp(\()p Fo(h)p Fp(\)\))14 b(=)949 1830 y Fl(Z)976 1943 y Fk(\012)1012 1898 y Fo(dx)g(u)1110 1905 y Fk(^)-21 b Fi(\045)1127 1898 y Fp(\()p Fo(x)p Fp(\))p Fo(h)p Fp(\()p Fo(x)p Fp(\))449 b Fz(\(3.52\))89 2039 y(where)19 b Fo(h)g Fz(is)g(an)h(arbitrary)f (step)g(function)g(on)g Fp(\012)p Fz(.)30 b(Since)18 b Fo(u)1223 2046 y Fk(^)-21 b Fi(\045)1260 2039 y Fz(is)18 b(from)g Fo(L)1462 2021 y Fk(1)1482 2039 y Fp(\(\012\))i Fz(and)f Fo(\045U)5 b Fp(\(\012\))20 b Fz(is)89 2099 y(a)c(n)o(uclear)f(op)q(erator)i(b)q(oth)g(sides)e(of)h(\(3.52\))g (admit)f(a)h(con)o(tin)o(uation)f(to)h Fo(L)1474 2081 y Fj(1)1512 2099 y Fz(-functions)g(whic)o(h)89 2159 y(v)o(eri\034es)f (\(3.49\).)148 2243 y(It)h(remains)f(to)h(sho)o(w)h(the)f(estimate)f (\(3.50\).)21 b(F)l(rom)15 b(\(3.44\))i(w)o(e)f(immedi)o(ately)d(get) 384 2312 y Fl(Z)412 2425 y Fk(\012)448 2380 y Fo(dx)g(u)542 2387 y Fi(g)562 2380 y Fp(\()p Fo(x)p Fp(\))h(=)g Fz(tr)o Fp(\()p Fo(\045U)5 b Fp(\(\012\)\))15 b(=)f Fz(tr)p Fp(\()p Fo(\045)p Fp(\()p Fo(K)h Fm(\000)c Fo(i)p Fp(\)\()p Fo(K)j Fm(\000)d Fo(i)p Fp(\))1375 2359 y Fj(\000)p Fk(1)1422 2380 y Fo(U)5 b Fp(\(\012\)\))p Fo(:)170 b Fz(\(3.53\))89 2521 y(Since)16 b Fo(U)5 b Fp(\(\012\)\()p Fo(K)15 b Fm(\000)c Fo(i)p Fp(\))489 2503 y Fj(\000)p Fk(1)550 2521 y Fm(2)j(L)631 2528 y Fk(1)651 2521 y Fp(\()p Fh(K)p Fp(\))i Fz(and)h Fo(\045)p Fp(\()p Fo(K)f Fm(\000)10 b Fo(i)p Fp(\))16 b Fz(is)g(a)h(b)q(ounded)g(op)q(erator)h(whic)o(h)d (norm)h(can)89 2581 y(b)q(e)h(estimated)d(b)o(y)i Fo(C)484 2588 y Fk(^)-21 b Fi(\045)517 2581 y Fz(w)o(e)16 b(obtain)418 2693 y(tr)p Fp(\()p Fo(\045)p Fp(\()p Fo(K)f Fm(\000)c Fo(i)p Fp(\)\()p Fo(K)k Fm(\000)10 b Fo(i)p Fp(\))821 2673 y Fj(\000)p Fk(1)868 2693 y Fo(U)5 b Fp(\()p Fo(!)r Fp(\)\))14 b Fm(\024)g Fo(C)1100 2700 y Fk(^)-21 b Fi(\045)1117 2693 y Fm(k)p Fp(\()p Fo(K)15 b Fm(\000)c Fo(i)p Fp(\))1303 2673 y Fj(\000)p Fk(1)1350 2693 y Fo(P)1388 2673 y Fe(K)1381 2706 y(H)1413 2693 y Fm(k)1438 2701 y Fj(L)1462 2706 y Ff(1)1479 2701 y Fk(\()p Fe(K)p Fk(\))1736 2693 y Fz(\(3.54\))89 2806 y(whic)o(h)16 b(v)o(eri\034es)f(\(3.50\).)1292 b Fa(\003)p eop %%Page: 19 21 19 20 bop 89 23 a Fs(Density)18 b(and)g(current)h(of)f(a)g(dissip)l(a)l (tive)g(Schr)o(\366dinger)h(opera)l(tor)187 b Fz(19)148 147 y(Ob)o(viously)l(,)15 b(the)h(relation)g(\(3.49\))h(tak)o(es)e(the) h(form)743 264 y(tr)o Fp(\()p Fo(\045M)5 b Fp(\()p Fo(h)p Fp(\)\))15 b(=)f Fm(h)p Fo(u)1078 271 y Fk(^)-21 b Fi(\045)1095 264 y Fo(;)8 b(h)p Fm(i)1164 280 y Fi(L)1188 270 y Ff(1)1736 264 y Fz(\(3.55\))89 380 y(where)16 b(b)o(y)g Fm(h\001)p Fo(;)8 b(h)p Fm(i)400 395 y Fi(L)424 385 y Ff(1)444 380 y Fz(,)15 b Fo(h)f Fm(2)g Fo(L)595 362 y Fj(1)633 380 y Fp(\(\012\))p Fz(,)i(w)o(e)g(denote)g(the)g(linear)f(functionals)h (on)h Fo(L)1533 362 y Fk(1)1553 380 y Fp(\(\012\))p Fz(.)148 464 y(W)l(e)g(conclude)g(this)g(section)g(with)g(some)f(considerations) i(whic)o(h)e(w)o(e)h(need)g(in)g(the)g(follo)o(w-)89 524 y(ing)23 b(section.)42 b(Since)22 b Fm(f)p Fo(\045)p Fp(\()p Fo(\025)p Fp(\))p Fm(g)655 531 y Fi(\025)p Fj(2)p Fg(R)749 524 y Fz(is)h(a)g(measurable)f(family)f(of)i(non-negativ)o(e)g (self-adjoin)o(t)89 584 y(op)q(erators)c(there)f(is)f(a)i(measurable)d (family)g Fm(f)p Fo(V)11 b Fp(\()p Fo(\025)p Fp(\))p Fm(g)1088 591 y Fi(\025)p Fj(2)p Fg(R)1176 584 y Fz(of)18 b(unitary)g(op)q(erators)h(on)g Fn(C)1727 566 y Fk(2)1767 584 y Fz(suc)o(h)89 644 y(that)580 728 y Fo(\045)p Fp(\()p Fo(\025)p Fp(\))14 b(=)g Fo(V)d Fp(\()p Fo(\025)p Fp(\))850 657 y Fl(\022)908 697 y Fo(\026)937 704 y Fi(b)955 697 y Fp(\()p Fo(\025)p Fp(\))88 b(0)952 757 y(0)f Fo(\026)1092 764 y Fi(a)1113 757 y Fp(\()p Fo(\025)p Fp(\))1200 657 y Fl(\023)1245 728 y Fo(V)11 b Fp(\()p Fo(\025)p Fp(\))1350 707 y Fj(\003)1736 728 y Fz(\(3.56\))89 851 y(for)j(a.e.)19 b Fo(\025)c Fm(2)f Fn(R)c Fz(where)j Fo(\026)560 858 y Fi(b)577 851 y Fp(\()p Fo(\025)p Fp(\))h Fz(and)g Fo(\026)778 858 y Fi(a)799 851 y Fp(\()p Fo(\025)p Fp(\))f Fz(are)h(the)f (non-negativ)o(e)g(eigen)o(v)m(alues)f(of)h Fo(\045)p Fp(\()p Fo(\025)p Fp(\))p Fz(.)21 b(F)l(rom)89 911 y(\(3.56\))c(w)o(e)f (get)g(that)557 1057 y Fo(\045)582 1036 y Fi(t)597 1057 y Fp(\()p Fo(\025)p Fp(\))f(=)e Fo(V)769 1036 y Fi(t)784 1057 y Fp(\()p Fo(\025)p Fp(\))850 1036 y Fj(\003)878 987 y Fl(\022)935 1026 y Fo(\026)964 1033 y Fi(b)982 1026 y Fp(\()p Fo(\025)p Fp(\))88 b(0)980 1087 y(0)e Fo(\026)1119 1094 y Fi(a)1140 1087 y Fp(\()p Fo(\025)p Fp(\))1227 987 y Fl(\023)1272 1057 y Fo(V)1311 1036 y Fi(t)1326 1057 y Fp(\()p Fo(\025)p Fp(\))344 b Fz(\(3.57\))89 1203 y(for)18 b(a.e.)23 b Fo(\025)16 b Fm(2)f Fn(R)f Fz(where)j Fo(V)588 1185 y Fi(t)603 1203 y Fp(\()p Fo(\025)p Fp(\))g Fz(is)g(the)g(transp)q(osed)i(matrix)c(to)i Fo(V)12 b Fp(\()p Fo(\025)p Fp(\))p Fz(.)24 b(Inserting)16 b(\(3.57\))i(in)o (to)89 1264 y(\(3.42\))f(w)o(e)f(obtain)h(the)f(expression)146 1380 y Fo(u)177 1387 y Fk(^)-21 b Fi(\045)194 1380 y Fp(\()p Fo(x;)8 b(\025)p Fp(\))14 b(=)1374 b Fz(\(3.58\))288 1413 y Fl(\034)q(\022)383 1453 y Fo(\026)412 1460 y Fi(b)430 1453 y Fp(\()p Fo(\025)p Fp(\))88 b(0)427 1513 y(0)f Fo(\026)567 1520 y Fi(a)588 1513 y Fp(\()p Fo(\025)p Fp(\))675 1413 y Fl(\023)720 1483 y Fo(V)759 1463 y Fi(t)774 1483 y Fp(\()p Fo(\025)p Fp(\))848 1413 y Fl(\022)908 1453 y Fo( )942 1435 y Fj(\000)971 1453 y Fp(\()p Fo(x;)8 b(\025;)g(b)p Fp(\))906 1513 y Fo( )940 1495 y Fj(\000)969 1513 y Fp(\()p Fo(x;)g(\025;)g(a)p Fp(\))1153 1413 y Fl(\023)1198 1483 y Fo(;)g(V)1259 1463 y Fi(t)1274 1483 y Fp(\()p Fo(\025)p Fp(\))1348 1413 y Fl(\022)1409 1453 y Fo(\036)1438 1435 y Fj(\000)1467 1453 y Fp(\()p Fo(x;)g(\025;)g(b)p Fp(\))1406 1513 y Fo(\036)1435 1495 y Fj(\000)1465 1513 y Fp(\()p Fo(x;)g(\025;)g(a)p Fp(\))1649 1413 y Fl(\023)o(\035)1731 1483 y Fo(:)89 1629 y Fz(Let)17 b(us)f(in)o(tro)q(duce)g(the)g(unit)g (v)o(ectors)f Fo(e)831 1636 y Fi(\034)852 1629 y Fp(\()p Fo(\025)p Fp(\))p Fz(,)696 1745 y Fo(e)719 1752 y Fi(\034)741 1745 y Fp(\()p Fo(\025)p Fp(\))f(=)g Fo(V)d Fp(\()p Fo(\025)p Fp(\))p Fo(e)1001 1752 y Fi(\034)1022 1745 y Fo(;)57 b(\034)20 b Fp(=)13 b Fo(b;)8 b(a)482 b Fz(\(3.59\))89 1861 y(whic)o(h)16 b(p)q(erform)f(an)i(orthonormal)f(basis)g(in)g Fn(C)969 1843 y Fk(2)992 1861 y Fz(.)21 b(W)l(e)16 b(set)155 1965 y Fo(~)149 1978 y( )q Fp(\()p Fo(x;)8 b(\025;)g(e)324 1985 y Fi(\034)345 1978 y Fp(\()p Fo(\025)p Fp(\)\))15 b(:=)e Fo( )542 1985 y Fj(\000)571 1978 y Fp(\()p Fo(x;)8 b(\025;)g(e)713 1985 y Fi(\034)734 1978 y Fp(\()p Fo(\025)p Fp(\)\))k Fm(\010)e Fo( )r Fp(\()p Fo(x;)e(\025;)g(e)1056 1985 y Fi(\034)1077 1978 y Fp(\()p Fo(\025)p Fp(\)\))j Fm(\010)g Fo( )1255 1985 y Fk(+)1284 1978 y Fp(\()p Fo(x;)d(\025;)g(e) 1426 1985 y Fi(\034)1447 1978 y Fp(\()p Fo(\025)p Fp(\)\))15 b(=)151 b Fz(\(3.60\))391 2039 y Fp(1)p 355 2061 96 2 v 355 2071 a Fm(p)p 397 2071 54 2 v 41 x Fp(2)p Fo(\031)455 2073 y(e)478 2052 y Fi(ix\025)533 2073 y Fo(e)556 2080 y Fi(\034)577 2073 y Fp(\()p Fo(\025)p Fp(\))11 b Fm(\010)744 2039 y Fp(1)p 709 2061 96 2 v 709 2071 a Fm(p)p 750 2071 54 2 v 750 2112 a Fp(2)p Fo(\031)809 2073 y Fp(\()p Fo(T)857 2080 y Fj(\003)877 2073 y Fp(\()p Fo(\025)p Fp(\))943 2052 y Fj(\003)963 2073 y Fo(e)986 2080 y Fi(\034)1007 2073 y Fp(\()p Fo(\025)p Fp(\)\)\()p Fo(x)p Fp(\))g Fm(\010)1259 2039 y Fp(1)p 1224 2061 96 2 v 1224 2071 a Fm(p)p 1265 2071 54 2 v 1265 2112 a Fp(2)p Fo(\031)1324 2073 y(e)1347 2052 y Fi(ix\025)1401 2073 y Fp(\002)1439 2080 y Fi(H)1473 2073 y Fp(\()p Fo(\025)p Fp(\))1539 2052 y Fj(\003)1559 2073 y Fo(e)1582 2080 y Fi(\034)1603 2073 y Fp(\()p Fo(\025)p Fp(\))p Fo(:)89 2228 y Fz(Ob)o(viously)l(,)j(the)h(system)f Fm(f)602 2215 y Fo(~)596 2228 y( )q Fp(\()p Fo(x;)8 b(\025;)g(e)771 2235 y Fi(\034)792 2228 y Fp(\()p Fo(\025)p Fp(\)\))p Fm(g)902 2235 y Fi(\034)t Fk(=)p Fi(b;a)1010 2228 y Fz(p)q(erforms)15 b(an)g(orthonormal)h(basis)f(of)h(gener-)89 2288 y(alized)f (eigenfunctions.)21 b(Moreo)o(v)o(er,)14 b(a)j(straigh)o(tforw)o(ard)g (computation)e(sho)o(ws)j(that)509 2431 y Fo(V)548 2410 y Fi(t)563 2431 y Fp(\()p Fo(\025)p Fp(\))637 2361 y Fl(\022)698 2400 y Fo( )732 2382 y Fj(\000)761 2400 y Fp(\()p Fo(x;)8 b(\025;)g(b)p Fp(\))695 2460 y Fo( )729 2442 y Fj(\000)758 2460 y Fp(\()p Fo(x;)g(\025;)g(a)p Fp(\))943 2361 y Fl(\023)993 2431 y Fp(=)1045 2361 y Fl(\022)1104 2400 y Fo( )r Fp(\()p Fo(x;)g(\025;)g(e)1280 2407 y Fi(b)1296 2400 y Fp(\()p Fo(\025)p Fp(\)\))1102 2460 y Fo( )r Fp(\()p Fo(x;)g(\025;)g(e)1278 2467 y Fi(a)1298 2460 y Fp(\()p Fo(\025)p Fp(\)\))1404 2361 y Fl(\023)1736 2431 y Fz(\(3.61\))89 2573 y(whic)o(h)16 b(leads)g(to)389 2689 y Fo(u)420 2696 y Fk(^)-21 b Fi(\045)437 2689 y Fp(\()p Fo(x;)8 b(\025)p Fp(\))14 b(=)f Fo(\026)647 2696 y Fi(b)665 2689 y Fp(\()p Fo(\025)p Fp(\))p Fm(j)p Fo( )r Fp(\()p Fo(x;)8 b(\025;)g(e)921 2696 y Fi(b)937 2689 y Fp(\()p Fo(\025)p Fp(\)\))p Fm(j)1036 2669 y Fk(2)1067 2689 y Fp(+)j Fo(\026)1145 2696 y Fi(a)1166 2689 y Fp(\()p Fo(\025)p Fp(\))p Fm(j)p Fo( )r Fp(\()p Fo(x;)d(\025;)g(e)1422 2696 y Fi(a)1442 2689 y Fp(\()p Fo(\025)p Fp(\)\))p Fm(j)1541 2669 y Fk(2)1736 2689 y Fz(\(3.62\))89 2806 y(for)17 b Fo(x)c Fm(2)h Fp(\012)j Fz(and)g Fo(\025)d Fm(2)g Fn(R)p Fz(.)p eop %%Page: 20 22 20 21 bop 89 23 a Fz(20)692 b Fs(H.-Ch.)25 b(Kaiser,)18 b(H.)h(Neidhardt,)g(J.)g(Rehber)o(g)89 155 y Fx(4)77 b(Curren)n(t)26 b(densit)n(y)89 289 y Fz(In)13 b(accordance)g(with)g (\(1.11\))h(the)e(curren)o(t)h(densit)o(y)f Fo(j)1074 296 y Fk(^)-21 b Fi(\045)1091 289 y Fp(\()p Fo(x;)8 b(\025)p Fp(\))13 b Fz(for)g(the)g(energy)g Fo(\025)h Fm(2)g Fn(R)c Fz(is)j(de\034ned)89 350 y(b)o(y)337 424 y Fo(j)360 431 y Fk(^)-21 b Fi(\045)377 424 y Fp(\()p Fo(x;)8 b(\025)p Fp(\))41 b(:=)g Fo(\026)656 431 y Fi(b)674 424 y Fp(\()p Fo(\025)p Fp(\))p Fm(=)p Fz(m)825 368 y Fl(\020)890 404 y Fk(1)p 860 412 79 2 v 860 441 a Fi(m)p Fk(\()p Fi(x)p Fk(\))943 424 y Fo( )r Fp(\()p Fo(x;)8 b(\025;)g(e)1119 431 y Fi(b)1136 424 y Fp(\()p Fo(\025)p Fp(\)\))1221 406 y Fj(0)p 1246 380 278 2 v 1246 424 a Fo( )r Fp(\()p Fo(x;)g(\025;)g(e)1422 431 y Fi(b)1439 424 y Fp(\()p Fo(\025)p Fp(\)\))1524 368 y Fl(\021)1562 424 y Fp(+)627 523 y Fo(\026)656 530 y Fi(a)678 523 y Fp(\()p Fo(\025)p Fp(\))p Fm(=)p Fz(m)829 467 y Fl(\020)894 503 y Fk(1)p 864 511 79 2 v 864 540 a Fi(m)p Fk(\()p Fi(x)p Fk(\))947 523 y Fo( )r Fp(\()p Fo(x;)g(\025;)g(e)1123 530 y Fi(a)1143 523 y Fp(\()p Fo(\025)p Fp(\)\))1228 505 y Fj(0)p 1254 479 282 2 v 1254 523 a Fo( )r Fp(\()p Fo(x;)g(\025;)g(e)1430 530 y Fi(a)1449 523 y Fp(\()p Fo(\025)p Fp(\)\))1535 467 y Fl(\021)1621 473 y Fo(;)126 b Fz(\(4.1\))89 627 y Fo(x)16 b Fm(2)h Fp(\012)p Fz(,)g(where)h(the)f(eigenfunctions)g Fo( )r Fp(\()p Fo(x;)8 b(\025;)g(e)973 634 y Fi(\034)994 627 y Fp(\()p Fo(\025)p Fp(\)\))p Fz(,)18 b Fo(\034)k Fp(=)16 b Fo(a;)8 b(b)p Fz(,)17 b(are)g(de\034ned)h(b)o(y)f(\(3.61\).) 26 b(W)l(e)89 687 y(are)15 b(going)i(to)e(relate)g(the)g(curren)o(t)f (densit)o(y)g(to)i(the)f(c)o(haracteristic)e(function)i(of)h(the)f (maximal)89 747 y(dissipativ)o(e)g(op)q(erator)j Fo(H)t Fz(.)89 877 y Fq(Prop)r(osition)i(4.1)k Fc(Assume)f Fo(m)15 b Fp(+)798 857 y Fk(1)p 791 865 32 2 v 791 894 a Fi(m)852 877 y Fm(2)24 b Fo(L)942 858 y Fj(1)980 877 y Fp(\([)p Fo(a;)8 b(b)p Fp(]\))p Fc(,)22 b Fo(V)36 b Fm(2)24 b Fo(L)1306 858 y Fk(2)1326 877 y Fp(\([)p Fo(a;)8 b(b)p Fp(]\))21 b Fc(and)j Fo(\024)1611 884 y Fi(a)1632 877 y Fo(;)8 b(\024)1682 884 y Fi(b)1723 877 y Fm(2)24 b Fn(C)1813 884 y Fk(+)1846 877 y Fc(.)89 937 y(F)l(urther,)17 b(let)i Fo(\045)e Fc(b)n(e)h(a)g(ste)n(ady)f(state.)23 b(Then)18 b(the)g(curr)n(ent)f(density)h Fo(j)1346 944 y Fk(^)-21 b Fi(\045)1363 937 y Fp(\()p Fo(x;)8 b(\025)p Fp(\))p Fc(,)18 b Fo(x)13 b Fm(2)h Fp(\012)p Fc(,)k Fo(\025)c Fm(2)h Fn(R)p Fc(,)f(is)89 997 y(indep)n(endent)19 b(fr)n(om)e Fo(x)g Fc(and)g(admits)g(the)h(r)n(epr)n(esentation)731 1110 y Fo(j)754 1117 y Fk(^)-21 b Fi(\045)771 1110 y Fp(\()p Fo(\025)p Fp(\))15 b(=)e Fz(tr)941 1118 y Fg(C)962 1109 y Ff(2)985 1110 y Fp(\()p Fo(\045)p Fp(\()p Fo(\025)p Fp(\))p Fo(C)t Fp(\()p Fo(\025)p Fp(\)\))542 b Fz(\(4.2\))89 1223 y Fc(wher)n(e)428 1296 y Fo(C)t Fp(\()p Fo(\025)p Fp(\))13 b(:=)h Fm(\000)678 1262 y Fp(1)p 656 1284 71 2 v 656 1330 a(2)p Fo(\031)r(i)745 1262 y(\013)776 1269 y Fi(a)797 1262 y Fo(\013)828 1269 y Fi(b)p 736 1284 120 2 v 736 1294 V 736 1337 a Fo(W)7 b Fp(\()p Fo(\025)p Fp(\))860 1296 y Fo(E)s Fp(\002)937 1303 y Fi(H)971 1296 y Fp(\()p Fo(\025)p Fp(\))1037 1275 y Fj(\003)1070 1296 y Fp(=)1150 1262 y(1)p 1127 1284 71 2 v 1127 1330 a(2)p Fo(\031)r(i)1217 1262 y(\013)1248 1269 y Fi(b)1265 1262 y Fo(\013)1296 1269 y Fi(a)p 1208 1284 120 2 v 1208 1330 a Fo(W)g Fp(\()p Fo(\025)p Fp(\))1332 1296 y(\002)1370 1303 y Fi(H)1403 1296 y Fp(\()p Fo(\025)p Fp(\))p Fo(E)s(;)239 b Fz(\(4.3\))89 1424 y Fo(\025)14 b Fm(2)h Fn(R)o Fc(.)20 b(Mor)n(e)n(over,)c(if)h Fz(tr)566 1432 y Fg(C)587 1423 y Ff(2)610 1424 y Fp(\()p Fo(\045)p Fp(\()p Fm(\001)p Fp(\)\))c Fm(2)h Fo(L)818 1406 y Fk(1)838 1424 y Fp(\()p Fn(R)p Fp(\))p Fc(,)h(then)j(the)g(total)g(curr)n(ent)f Fo(j)1439 1431 y Fk(^)-21 b Fi(\045)1457 1424 y Fc(,)790 1563 y Fo(j)813 1570 y Fk(^)g Fi(\045)844 1563 y Fp(:=)910 1495 y Fl(Z)937 1607 y Fg(R)972 1563 y Fo(d\025)14 b(j)1062 1570 y Fk(^)-21 b Fi(\045)1080 1563 y Fp(\()p Fo(\025)p Fp(\))p Fo(;)601 b Fz(\(4.4\))89 1699 y Fc(is)17 b(\034nite)i(and)f (satis\034es)g(the)g(estimate)702 1833 y Fm(j)p Fo(j)739 1840 y Fk(^)-21 b Fi(\045)756 1833 y Fm(j)13 b(\024)856 1799 y Fp(1)p 841 1822 54 2 v 841 1867 a(2)p Fo(\031)908 1765 y Fl(Z)936 1878 y Fg(R)970 1833 y Fo(d\025)i Fz(tr)1076 1841 y Fg(C)1097 1832 y Ff(2)1119 1833 y Fp(\()p Fo(\045)p Fp(\()p Fo(\025)p Fp(\)\))513 b Fz(\(4.5\))89 1996 y Fq(Pro)r(of.)21 b Fz(F)l(rom)15 b(the)h(de\034nition)g(\(4.2\))g(one)h (gets)f(that)128 2109 y Fo(j)151 2116 y Fk(^)-21 b Fi(\045)168 2109 y Fp(\()p Fo(x;)8 b(\025)p Fp(\))14 b(=)1425 b Fz(\(4.6\))223 2226 y Fm(=)p Fz(m)308 2141 y Fl( *)387 2156 y(\022)445 2196 y Fo(\026)474 2203 y Fi(b)491 2196 y Fp(\()p Fo(\025)p Fp(\))88 b(0)489 2256 y(0)e Fo(\026)628 2263 y Fi(a)649 2256 y Fp(\()p Fo(\025)p Fp(\))736 2156 y Fl(\023)781 2141 y( )879 2169 y Fk(1)p 848 2177 79 2 v 848 2206 a Fi(m)p Fk(\()p Fi(x)p Fk(\))932 2189 y Fo( )r Fp(\()p Fo(x;)8 b(\025;)g(e)1108 2196 y Fi(b)1124 2189 y Fp(\()p Fo(\025)p Fp(\)\))1209 2170 y Fj(0)877 2237 y Fk(1)p 846 2245 V 846 2273 a Fi(m)p Fk(\()p Fi(x)p Fk(\))930 2256 y Fo( )r Fp(\()p Fo(x;)g(\025;)g(e)1106 2263 y Fi(a)1126 2256 y Fp(\()p Fo(\025)p Fp(\)\))1211 2238 y Fj(0)1243 2141 y Fl(!)1291 2226 y Fo(;)1313 2156 y Fl(\022)1372 2196 y Fo( )r Fp(\()p Fo(x;)g(\025;)g(e)1548 2203 y Fi(b)1564 2196 y Fp(\()p Fo(\025)p Fp(\)\))1370 2256 y Fo( )r Fp(\()p Fo(x;)g(\025;)g(e)1546 2263 y Fi(a)1566 2256 y Fp(\()p Fo(\025)p Fp(\)\))1672 2156 y Fl(\023)1709 2141 y(+!)1797 2226 y Fo(:)89 2383 y Fz(T)l(aking)17 b(in)o(to)f(accoun)o(t)g (\(3.57\))h(and)f(\(3.61\))h(w)o(e)f(get)225 2539 y Fo(j)248 2546 y Fk(^)-21 b Fi(\045)266 2539 y Fp(\()p Fo(x;)8 b(\025)p Fp(\))14 b(=)f Fm(=)p Fz(m)532 2454 y Fl( )q(*)612 2539 y Fo(\045)637 2519 y Fi(t)652 2539 y Fp(\()p Fo(\025)p Fp(\))726 2454 y Fl( )824 2482 y Fk(1)p 794 2490 V 794 2519 a Fi(m)p Fk(\()p Fi(x)p Fk(\))878 2502 y Fo( )912 2483 y Fj(\000)941 2502 y Fp(\()p Fo(x;)8 b(\025;)g(b)p Fp(\))1100 2483 y Fj(0)822 2550 y Fk(1)p 792 2558 V 792 2586 a Fi(m)p Fk(\()p Fi(x)p Fk(\))875 2569 y Fo( )909 2551 y Fj(\000)938 2569 y Fp(\()p Fo(x;)g(\025;)g(a)p Fp(\))1102 2551 y Fj(0)1134 2454 y Fl(!)1182 2539 y Fo(;)1204 2469 y Fl(\022)1263 2509 y Fo( )1297 2491 y Fj(\000)1326 2509 y Fp(\()p Fo(x;)g(\025;)g(b)p Fp(\))1261 2569 y Fo( )1295 2551 y Fj(\000)1324 2569 y Fp(\()p Fo(x;)g(\025;)g(a)p Fp(\))1508 2469 y Fl(\023)1545 2454 y(+!)1761 2539 y Fz(\(4.7\))89 2693 y(whic)o(h)16 b(can)g(b)q(e)g(expressed)g(b)o(y)681 2806 y Fo(j)704 2813 y Fk(^)-21 b Fi(\045)722 2806 y Fp(\()p Fo(x;)8 b(\025)p Fp(\))14 b(=)f Fz(tr)941 2814 y Fg(C)962 2805 y Ff(2)985 2806 y Fp(\()p Fo(\045)p Fp(\()p Fo(\025)p Fp(\))p Fo(C)t Fp(\()p Fo(x;)8 b(\025)p Fp(\)\))492 b Fz(\(4.8\))p eop %%Page: 21 23 21 22 bop 89 23 a Fs(Density)18 b(and)g(current)h(of)f(a)g(dissip)l(a)l (tive)g(Schr)o(\366dinger)h(opera)l(tor)187 b Fz(21)89 147 y(where)147 249 y Fo(C)t Fp(\()p Fo(x;)8 b(\025)p Fp(\))13 b(:=)1394 b Fz(\(4.9\))289 365 y Fm(=)p Fz(m)374 280 y Fl( )q( )512 307 y Fk(1)p 481 315 79 2 v 481 344 a Fi(m)p Fk(\()p Fi(x)p Fk(\))565 327 y Fo( )599 309 y Fj(\000)628 327 y Fp(\()p Fo(x;)8 b(\025;)g(b)p Fp(\))787 309 y Fj(0)p 812 284 222 2 v 812 327 a Fo( )846 312 y Fj(\000)875 327 y Fp(\()p Fo(x;)g(\025;)g(b)p Fp(\))1115 307 y Fk(1)p 1085 315 79 2 v 1085 344 a Fi(m)p Fk(\()p Fi(x)p Fk(\))1168 327 y Fo( )1202 309 y Fj(\000)1231 327 y Fp(\()p Fo(x;)g(\025;)g(a)p Fp(\))1395 309 y Fj(0)p 1420 284 222 2 v 1420 327 a Fo( )1454 312 y Fj(\000)1483 327 y Fp(\()p Fo(x;)g(\025;)g(b)p Fp(\))509 380 y Fk(1)p 479 388 79 2 v 479 417 a Fi(m)p Fk(\()p Fi(x)p Fk(\))562 400 y Fo( )596 382 y Fj(\000)625 400 y Fp(\()p Fo(x;)g(\025;)g(b)p Fp(\))784 382 y Fj(0)p 809 356 227 2 v 809 400 a Fo( )843 385 y Fj(\000)872 400 y Fp(\()p Fo(x;)g(\025;)g(a)p Fp(\))1113 380 y Fk(1)p 1082 388 79 2 v 1082 417 a Fi(m)p Fk(\()p Fi(x)p Fk(\))1166 400 y Fo( )1200 382 y Fj(\000)1229 400 y Fp(\()p Fo(x;)g(\025;)g(a)p Fp(\))1393 382 y Fj(0)p 1418 356 227 2 v 1418 400 a Fo( )1452 385 y Fj(\000)1481 400 y Fp(\()p Fo(x;)g(\025;)g(a)p Fp(\))1665 280 y Fl(!!)89 507 y Fz(W)l(e)16 b(note)h(that)140 629 y Fo(C)t Fp(\()p Fo(x;)8 b(\025)p Fp(\))13 b(=)365 596 y(1)p 365 618 25 2 v 369 664 a Fo(i)403 559 y Fl(\022)463 599 y Fo(W)7 b Fp(\()p 535 556 222 2 v Fo( )569 585 y Fj(\000)597 599 y Fp(\()p Fo(x;)h(\025;)g(b)p Fp(\))p Fo(;)g( )812 581 y Fj(\000)841 599 y Fp(\()p Fo(x;)g(\025;)g(b)p Fp(\)\))46 b Fo(W)7 b Fp(\()p 1137 556 V Fo( )1171 585 y Fj(\000)1200 599 y Fp(\()p Fo(x;)h(\025;)g(b)p Fp(\))o Fo(;)g( )1414 581 y Fj(\000)1443 599 y Fp(\()p Fo(x;)g(\025;)g(a)p Fp(\)\))460 664 y Fo(W)f Fp(\()p 532 621 227 2 v Fo( )566 650 y Fj(\000)595 664 y Fp(\()p Fo(x;)h(\025;)g(a)p Fp(\))o Fo(;)g( )814 646 y Fj(\000)843 664 y Fp(\()p Fo(x;)g(\025;)g(b)p Fp(\)\))41 b Fo(W)7 b Fp(\()p 1134 621 V Fo( )1168 650 y Fj(\000)1197 664 y Fp(\()p Fo(x;)h(\025;)g(a)p Fp(\))p Fo(;)g( )1417 646 y Fj(\000)1445 664 y Fp(\()p Fo(x;)g(\025;)g(a)p Fp(\)\))1649 559 y Fl(\023)1736 629 y Fz(\(4.10\))89 765 y(where)16 b Fo(W)7 b Fp(\()p Fm(\001)p Fo(;)h Fm(\001)p Fp(\))16 b Fz(is)g(the)g(W)l(ronskian)g(de\034ned)g(b)o(y)g(\(2.5\).)21 b(Since)263 896 y Fo(W)7 b Fp(\()p 335 852 222 2 v Fo( )369 881 y Fj(\000)398 896 y Fp(\()p Fo(x;)h(\025;)g(b)p Fp(\))o Fo(;)g( )612 875 y Fj(\000)641 896 y Fp(\()p Fo(x;)g(\025;)g(b)p Fp(\)\))41 b(=)h Fm(\000)1002 862 y Fo(i)p 984 884 54 2 v 984 930 a Fp(2)p Fo(\031)1079 862 y(\013)1110 844 y Fk(2)1110 875 y Fi(b)1130 862 y Fo(\013)1161 844 y Fk(2)1161 874 y Fi(a)p 1047 884 167 2 v 1047 930 a Fm(j)p Fo(W)7 b Fp(\()p Fo(\025)p Fp(\))p Fm(j)1194 915 y Fk(2)1736 896 y Fz(\(4.11\))258 1025 y Fo(W)g Fp(\()p 330 981 227 2 v Fo( )364 1010 y Fj(\000)393 1025 y Fp(\()p Fo(x;)h(\025;)g(a)p Fp(\))o Fo(;)g( )612 1004 y Fj(\000)641 1025 y Fp(\()p Fo(x;)g(\025;)g(b)p Fp(\)\))41 b(=)959 991 y(1)p 945 1013 54 2 v 945 1059 a(2)p Fo(\031)1042 991 y(\013)1073 998 y Fi(b)1090 991 y Fo(\013)1121 998 y Fi(a)p 1008 1013 167 2 v 1008 1059 a Fm(j)p Fo(W)7 b Fp(\()p Fo(\025)p Fp(\))p Fm(j)1155 1045 y Fk(2)1188 984 y Fl(\010)1217 1025 y Fo(W)g Fp(\()p Fo(\025)p Fp(\))12 b Fm(\000)f Fo(i\013)1446 1004 y Fk(2)1446 1037 y Fi(b)1465 1025 y Fo(v)1489 1032 y Fi(a)1510 1025 y Fp(\()p Fo(b;)d(\025)p Fp(\))1619 984 y Fl(\011)1736 1025 y Fz(\(4.12\))258 1154 y Fo(W)f Fp(\()p 330 1110 222 2 v Fo( )364 1139 y Fj(\000)393 1154 y Fp(\()p Fo(x;)h(\025;)g(b)p Fp(\))p Fo(;)g( )608 1133 y Fj(\000)636 1154 y Fp(\()p Fo(x;)g(\025;)g(a)p Fp(\)\))41 b(=)h Fm(\000)998 1120 y Fp(1)p 984 1142 54 2 v 984 1188 a(2)p Fo(\031)1081 1120 y(\013)1112 1127 y Fi(b)1129 1120 y Fo(\013)1160 1127 y Fi(a)p 1047 1142 167 2 v 1047 1188 a Fm(j)p Fo(W)7 b Fp(\()p Fo(\025)p Fp(\))p Fm(j)1194 1174 y Fk(2)1227 1113 y Fl(\010)1256 1154 y Fo(W)g Fp(\()p Fo(\025)p Fp(\))k Fm(\000)g Fo(i\013)1484 1133 y Fk(2)1484 1166 y Fi(a)1505 1154 y Fo(v)1529 1161 y Fi(b)1546 1154 y Fp(\()p Fo(a;)d(\025)p Fp(\))1660 1113 y Fl(\011)1736 1154 y Fz(\(4.13\))253 1290 y Fo(W)f Fp(\()p 325 1247 227 2 v Fo( )359 1276 y Fj(\000)388 1290 y Fp(\()p Fo(x;)h(\025;)g(a)p Fp(\))p Fo(;)g( )608 1270 y Fj(\000)636 1290 y Fp(\()p Fo(x;)g(\025;)g(a)p Fp(\)\))41 b(=)963 1256 y Fo(i)p 945 1279 54 2 v 945 1324 a Fp(2)p Fo(\031)1040 1256 y(\013)1071 1238 y Fk(2)1071 1269 y Fi(a)1092 1256 y Fo(\013)1123 1238 y Fk(2)1123 1269 y Fi(b)p 1008 1279 167 2 v 1008 1324 a Fm(j)p Fo(W)7 b Fp(\()p Fo(\025)p Fp(\))p Fm(j)1155 1310 y Fk(2)1736 1290 y Fz(\(4.14\))89 1416 y(w)o(e)16 b(\034nd)140 1534 y Fo(C)t Fp(\()p Fo(x;)8 b(\025)p Fp(\))14 b(=)388 1500 y(1)p 365 1523 71 2 v 365 1568 a(2)p Fo(\031)r(i)479 1500 y(\013)510 1507 y Fi(b)527 1500 y Fo(\013)558 1507 y Fi(a)p 446 1523 167 2 v 446 1568 a Fm(j)p Fo(W)7 b Fp(\()p Fo(\025)p Fp(\))p Fm(j)593 1554 y Fk(2)626 1464 y Fl(\022)806 1504 y Fm(\000)p Fo(i\013)893 1511 y Fi(b)910 1504 y Fo(\013)941 1511 y Fi(a)1126 1504 y Fm(\000)p Fp(\()p Fo(W)g Fp(\()p Fo(\025)p Fp(\))k Fm(\000)g Fo(i\013)1412 1485 y Fk(2)1412 1516 y Fi(a)1432 1504 y Fo(v)1456 1511 y Fi(b)1473 1504 y Fp(\()p Fo(a;)d(\025)p Fp(\)\))683 1564 y Fo(W)f Fp(\()p Fo(\025)p Fp(\))k Fm(\000)g Fo(i\013)911 1546 y Fk(2)911 1577 y Fi(b)931 1564 y Fo(v)955 1571 y Fi(a)975 1564 y Fp(\()p Fo(b;)d(\025)p Fp(\))223 b Fo(i\013)1355 1571 y Fi(a)1376 1564 y Fo(\013)1407 1571 y Fi(b)1626 1464 y Fl(\023)1671 1534 y Fo(:)51 b Fz(\(4.15\))89 1661 y(This)17 b(yields)140 1762 y Fo(C)t Fp(\()p Fo(x;)8 b(\025)p Fp(\))14 b(=)1389 b Fz(\(4.16\))247 1864 y Fm(\000)314 1830 y Fp(1)p 291 1853 71 2 v 291 1898 a(2)p Fo(\031)r(i)405 1830 y(\013)436 1837 y Fi(b)453 1830 y Fo(\013)484 1837 y Fi(a)p 371 1853 167 2 v 371 1898 a Fm(j)p Fo(W)7 b Fp(\()p Fo(\025)p Fp(\))p Fm(j)518 1884 y Fk(2)551 1794 y Fl(\022)628 1834 y Fp(0)61 b(1)609 1894 y Fm(\000)p Fp(1)41 b(0)758 1794 y Fl(\023)8 b(\022)861 1834 y Fo(W)f Fp(\()p Fo(\025)p Fp(\))k Fm(\000)g Fo(i\013)1089 1816 y Fk(2)1089 1847 y Fi(b)1109 1834 y Fo(v)1133 1841 y Fi(a)1153 1834 y Fp(\()p Fo(b;)d(\025)p Fp(\))185 b Fo(i\013)1495 1841 y Fi(b)1512 1834 y Fo(\013)1543 1841 y Fi(a)1003 1894 y Fo(i\013)1051 1901 y Fi(b)1068 1894 y Fo(\013)1099 1901 y Fi(a)1303 1894 y Fo(W)7 b Fp(\()p Fo(\025)p Fp(\))12 b Fm(\000)f Fo(i\013)1532 1876 y Fk(2)1532 1906 y Fi(a)1552 1894 y Fo(v)1576 1901 y Fi(b)1593 1894 y Fp(\()p Fo(a;)d(\025)p Fp(\))1727 1794 y Fl(\023)1772 1864 y Fo(:)89 1994 y Fz(F)l(rom)15 b(\(2.21\))i(w)o(e)f(obtain)241 2125 y Fp(\002)279 2132 y Fi(H)313 2125 y Fp(\()p Fo(\025)p Fp(\))379 2104 y Fj(\003)413 2125 y Fp(=)517 2091 y(1)p 469 2113 120 2 v 469 2159 a Fo(W)7 b Fp(\()p Fo(\025)p Fp(\))602 2054 y Fl(\022)659 2094 y Fo(W)g Fp(\()p Fo(\025)p Fp(\))k Fm(\000)g Fo(i\013)887 2076 y Fk(2)887 2107 y Fi(b)907 2094 y Fo(v)931 2101 y Fi(a)951 2094 y Fp(\()p Fo(b;)d(\025)p Fp(\))185 b Fo(i\013)1293 2101 y Fi(b)1310 2094 y Fo(\013)1341 2101 y Fi(a)801 2154 y Fo(i\013)849 2161 y Fi(b)866 2154 y Fo(\013)897 2161 y Fi(a)1102 2154 y Fo(W)7 b Fp(\()p Fo(\025)p Fp(\))k Fm(\000)g Fo(i\013)1330 2136 y Fk(2)1330 2167 y Fi(a)1350 2154 y Fo(v)1374 2161 y Fi(b)1391 2154 y Fp(\()p Fo(a;)d(\025)p Fp(\))1526 2054 y Fl(\023)1571 2125 y Fo(:)151 b Fz(\(4.17\))89 2252 y(Hence,)15 b(one)h(has)642 2331 y Fo(C)t Fp(\()p Fo(x;)8 b(\025)p Fp(\))14 b(=)f Fm(\000)929 2298 y Fp(1)p 906 2320 71 2 v 906 2366 a(2)p Fo(\031)r(i)996 2298 y(\013)1027 2305 y Fi(b)1044 2298 y Fo(\013)1075 2305 y Fi(a)p 987 2320 120 2 v 987 2330 V 987 2373 a Fo(W)7 b Fp(\()p Fo(\025)p Fp(\))1111 2331 y Fo(E)s Fp(\002)1188 2338 y Fi(H)1221 2331 y Fp(\()p Fo(\025)p Fp(\))1287 2311 y Fj(\003)1736 2331 y Fz(\(4.18\))89 2447 y(where)783 2524 y Fo(E)17 b Fp(:=)901 2454 y Fl(\022)978 2494 y Fp(0)61 b(1)959 2554 y Fm(\000)p Fp(1)41 b(0)1108 2454 y Fl(\023)1153 2524 y Fo(:)569 b Fz(\(4.19\))89 2637 y(This)15 b(sho)o(ws)h(that)f Fo(C)t Fp(\()p Fo(x;)8 b(\025)p Fp(\))14 b Fz(is)h(actually)f(indep)q (enden)o(t)g(of)h Fo(x)e Fm(2)h Fp(\012)p Fz(.)21 b(By)15 b(\(4.8\))g(this)f(leads)h(to)g(the)89 2697 y(represen)o(tation)532 2776 y Fo(j)555 2783 y Fk(^)-21 b Fi(\045)572 2776 y Fp(\()p Fo(x;)8 b(\025)p Fp(\))14 b(=)g Fm(\000)821 2743 y Fp(1)p 798 2765 71 2 v 798 2811 a(2)p Fo(\031)r(i)888 2743 y(\013)919 2750 y Fi(b)936 2743 y Fo(\013)967 2750 y Fi(a)p 878 2765 120 2 v 878 2775 V 878 2818 a Fo(W)7 b Fp(\()p Fo(\025)p Fp(\))1002 2776 y Fz(tr)1040 2785 y Fg(C)1061 2775 y Ff(2)1092 2776 y Fp(\()p Fo(\045)p Fp(\()p Fo(\025)p Fp(\))p Fo(E)s Fp(\002)1279 2783 y Fi(H)1313 2776 y Fp(\()p Fo(\025)p Fp(\))1379 2756 y Fj(\003)1399 2776 y Fp(\))318 b Fz(\(4.20\))p eop %%Page: 22 24 22 23 bop 89 23 a Fz(22)692 b Fs(H.-Ch.)25 b(Kaiser,)18 b(H.)h(Neidhardt,)g(J.)g(Rehber)o(g)89 147 y Fz(whic)o(h)e(sho)o(ws)h (that)g(the)g(curren)o(t)e(densit)o(y)h(is)g(also)h(indep)q(enden)o(t)f (of)h Fo(x)e Fm(2)g Fp(\012)p Fz(.)26 b(Th)o(us)17 b(it)h(mak)o(es)89 208 y(sense)13 b(to)h(denote)f Fo(C)t Fp(\()p Fo(x;)8 b(\025)p Fp(\))13 b Fz(and)h Fo(j)705 215 y Fk(^)-21 b Fi(\045)722 208 y Fp(\()p Fo(x;)8 b(\025)p Fp(\))13 b Fz(b)o(y)g Fo(C)t Fp(\()p Fo(\025)p Fp(\))g Fz(and)h Fo(j)1149 215 y Fk(^)-21 b Fi(\045)1166 208 y Fp(\()p Fo(\025)p Fp(\))p Fz(,)14 b(resp)q(ectiv)o(ely)l(.)k(Hence,)12 b(w)o(e)h(ha)o(v)o(e)89 268 y(pro)o(v)o(ed)j(one)g(part)h(of)f(the)g (assertion)h(\(4.2\))f(and)h(\(4.3\).)22 b(T)l(aking)16 b(in)o(to)g(accoun)o(t)g(the)g(iden)o(tit)o(y)659 376 y Fp(1)p 612 398 120 2 v 612 408 V 612 452 a Fo(W)7 b Fp(\()p Fo(\025)p Fp(\))736 410 y Fo(E)s Fp(\002)813 417 y Fi(H)847 410 y Fp(\()p Fo(\025)p Fp(\))913 389 y Fj(\003)947 410 y Fp(=)13 b Fm(\000)1089 376 y Fp(1)p 1042 398 V 1042 444 a Fo(W)7 b Fp(\()p Fo(\025)p Fp(\))1166 410 y(\002)1204 417 y Fi(H)1238 410 y Fp(\()p Fo(\025)p Fp(\))p Fo(E)396 b Fz(\(4.21\))89 562 y(w)o(e)16 b(v)o(erify)e(the)i (other)h(part.)148 646 y(It)d(remains)g(to)h(sho)o(w)g(\(4.4\))g(and)g (\(4.5\).)21 b(T)l(o)15 b(this)g(end)g(w)o(e)f(note)h(that)g(from)e (\(4.17\))j(one)f(gets)411 795 y Fo(J)5 b Fp(\002)481 802 y Fi(H)514 795 y Fp(\()p Fo(\025)p Fp(\))580 775 y Fj(\003)614 795 y Fp(=)14 b Fo(J)h Fp(+)813 761 y Fo(i)p 762 784 V 762 829 a(W)7 b Fp(\()p Fo(\025)p Fp(\))895 725 y Fl(\022)1024 765 y Fo(\013)1055 772 y Fi(b)1072 765 y Fo(\013)1103 772 y Fi(a)1237 765 y Fm(\000)p Fo(\013)1307 747 y Fk(2)1307 777 y Fi(a)1327 765 y Fo(v)1351 772 y Fi(b)1368 765 y Fp(\()p Fo(a;)h(\025)p Fp(\))952 825 y Fm(\000)p Fo(\013)1022 807 y Fk(2)1022 838 y Fi(b)1042 825 y Fo(v)1066 832 y Fi(a)1086 825 y Fp(\()p Fo(b;)g(\025)p Fp(\))114 b Fo(\013)1340 832 y Fi(b)1358 825 y Fo(\013)1389 832 y Fi(a)1503 725 y Fl(\023)1736 795 y Fz(\(4.22\))89 941 y(where)806 1028 y Fo(J)19 b Fp(:=)917 958 y Fl(\022)974 997 y Fp(0)42 b(1)974 1057 y(1)g(0)1085 958 y Fl(\023)1130 1028 y Fo(:)592 b Fz(\(4.23\))89 1153 y(Hence)15 b(w)o(e)h(\034nd)703 1223 y(tr)741 1231 y Fg(C)762 1222 y Ff(2)784 1223 y Fp(\()p Fo(J)5 b Fp(\002)873 1230 y Fi(H)906 1223 y Fp(\()p Fo(\025)p Fp(\))972 1202 y Fj(\003)992 1223 y Fp(\))14 b(=)g(2)p Fo(i)1133 1189 y(\013)1164 1196 y Fi(b)1181 1189 y Fo(\013)1212 1196 y Fi(a)p 1123 1211 V 1123 1257 a Fo(W)7 b Fp(\()p Fo(\025)p Fp(\))1736 1223 y Fz(\(4.24\))89 1346 y(whic)o(h)16 b(yields)653 1402 y Fp(1)p 630 1424 71 2 v 630 1470 a(2)p Fo(\031)r(i)720 1402 y(\013)751 1409 y Fi(b)768 1402 y Fo(\013)799 1409 y Fi(a)p 711 1424 120 2 v 711 1470 a Fo(W)7 b Fp(\()p Fo(\025)p Fp(\))849 1436 y(=)13 b Fm(\000)959 1402 y Fp(1)p 944 1424 54 2 v 944 1470 a(4)p Fo(\031)1003 1436 y Fz(tr)1041 1444 y Fg(C)1062 1435 y Ff(2)1084 1436 y Fp(\()p Fo(J)5 b Fp(\002)1173 1443 y Fi(H)1206 1436 y Fp(\()p Fo(\025)p Fp(\))1272 1415 y Fj(\003)1292 1436 y Fp(\))p Fo(:)411 b Fz(\(4.25\))89 1563 y(Using)16 b(the)g(de\034nition)g(\(4.3\))g(w)o (e)g(\034nally)g(obtain)476 1705 y Fo(j)499 1712 y Fk(^)-21 b Fi(\045)516 1705 y Fp(\()p Fo(\025)p Fp(\))14 b(=)g Fm(\000)707 1671 y Fp(1)p 692 1693 V 692 1739 a(4)p Fo(\031)751 1705 y Fz(tr)789 1713 y Fg(C)810 1704 y Ff(2)832 1705 y Fp(\()p Fo(J)5 b Fp(\002)921 1712 y Fi(H)954 1705 y Fp(\()p Fo(\025)p Fp(\))1020 1684 y Fj(\003)1040 1705 y Fp(\))14 b Fz(tr)1111 1713 y Fg(C)1132 1704 y Ff(2)1154 1705 y Fp(\()p Fo(\045)p Fp(\()p Fo(\025)p Fp(\)\002)1302 1712 y Fi(H)1336 1705 y Fp(\()p Fo(\025)p Fp(\))p Fo(E)s Fp(\))p Fo(:)262 b Fz(\(4.26\))89 1837 y(This)17 b(leads)f(to)g(the)g (estimate)549 1967 y Fm(j)p Fo(j)586 1974 y Fk(^)-21 b Fi(\045)603 1967 y Fp(\()p Fo(\025)p Fp(\))p Fm(j)14 b(\024)770 1933 y Fp(1)p 755 1955 V 755 2001 a(4)p Fo(\031)814 1967 y Fm(k)p Fp(\002)877 1974 y Fi(H)910 1967 y Fp(\()p Fo(\025)p Fp(\))976 1946 y Fj(\003)996 1967 y Fm(k)1021 1975 y Fj(L)1045 1980 y Ff(1)1063 1975 y Fk(\()p Fg(C)1097 1966 y Ff(2)1118 1975 y Fk(\))1134 1967 y Fm(k)p Fo(\045)p Fp(\()p Fo(\025)p Fp(\))p Fm(k)1275 1975 y Fj(L)1299 1980 y Ff(1)1317 1975 y Fk(\()p Fg(C)1351 1966 y Ff(2)1372 1975 y Fk(\))1387 1967 y Fo(:)335 b Fz(\(4.27\))89 2102 y(Since)16 b Fm(k)p Fp(\002)280 2109 y Fi(H)313 2102 y Fp(\()p Fo(\025)p Fp(\))379 2084 y Fj(\003)399 2102 y Fm(k)424 2110 y Fj(L)448 2115 y Ff(1)466 2110 y Fk(\()p Fg(C)500 2101 y Ff(2)521 2110 y Fk(\))550 2102 y Fm(\024)e Fp(2)i Fz(and)h Fm(k)p Fo(\045)p Fp(\()p Fo(\025)p Fp(\))p Fm(k)879 2110 y Fj(L)903 2115 y Ff(1)921 2110 y Fk(\()p Fg(C)956 2101 y Ff(2)976 2110 y Fk(\))1006 2102 y Fp(=)d Fz(tr)1095 2110 y Fg(C)1116 2101 y Ff(2)1139 2102 y Fp(\()p Fo(\045)p Fp(\()p Fo(\025)p Fp(\)\))j Fz(w)o(e)e(\034nd)737 2248 y Fm(j)p Fo(j)774 2255 y Fk(^)-21 b Fi(\045)791 2248 y Fp(\()p Fo(\025)p Fp(\))p Fm(j)15 b(\024)958 2215 y Fp(1)p 943 2237 V 943 2283 a(2)p Fo(\031)1002 2248 y Fz(tr)1040 2257 y Fg(C)1061 2247 y Ff(2)1083 2248 y Fp(\()p Fo(\045)p Fp(\()p Fo(\025)p Fp(\)\))524 b Fz(\(4.28\))89 2384 y(whic)o(h)16 b(imme)o(diatel)o(y)d(pro)o(v)o(es)j(\(4.5\).)1051 b Fa(\003)148 2468 y Fz(In)20 b(corresp)q(ondence)g(to)g(the)g(carrier) f(densit)o(y)g(it)g(seems)g(to)h(b)q(e)g(useful)f(to)i(in)o(tro)q(duce) e(the)89 2528 y(follo)o(wing)d(de\034nition.)89 2685 y Fq(De\034nition)k(4.2)k Fz(Assume)15 b Fo(m)c Fp(+)743 2666 y Fk(1)p 736 2674 32 2 v 736 2703 a Fi(m)787 2685 y Fm(2)k Fo(L)868 2667 y Fj(1)905 2685 y Fp(\([)p Fo(a;)8 b(b)p Fp(]\))p Fz(,)15 b Fo(V)26 b Fm(2)15 b Fo(L)1204 2667 y Fk(2)1224 2685 y Fp(\([)p Fo(a;)8 b(b)p Fp(]\))15 b Fz(and)i Fo(\024)1497 2692 y Fi(a)1518 2685 y Fo(;)8 b(\024)1568 2692 y Fi(b)1600 2685 y Fm(2)15 b Fn(C)1681 2692 y Fk(+)1713 2685 y Fz(.)23 b(Then)89 2746 y(the)17 b(matrix)f Fo(C)t Fp(\()p Fo(\025)p Fp(\))h Fz(and)h(the)g(v)m(alue)f Fo(j)s(\045)p Fp(\()p Fo(\025)p Fp(\))h Fz(are)f(called)g(the)g(curren) o(t)f(densit)o(y)h(observ)m(able)g(and)89 2806 y(the)f(curren)o(t)f (densit)o(y)h(at)g(energy)g Fo(\025)e Fm(2)g Fn(R)g Fz(of)i(the)g (system)f(describ)q(ed)h(b)o(y)f Fo(H)t Fz(,)h(resp)q(ectiv)o(ely)l(.)p eop %%Page: 23 25 23 24 bop 89 23 a Fs(Density)18 b(and)g(current)h(of)f(a)g(dissip)l(a)l (tive)g(Schr)o(\366dinger)h(opera)l(tor)187 b Fz(23)89 147 y(The)15 b(de\034nition)e(is)i(again)g(justi\034ed)f(b)o(y)g(the)g (fact)g(that)h(the)f(curren)o(t)f(densit)o(y)h(at)h(energy)f Fo(\025)g Fm(2)g Fn(R)89 208 y Fz(is)23 b(the)f(exp)q(ectation)h(v)m (alue)f(of)h(the)g(curren)o(t)f(densit)o(y)f(observ)m(able)i(at)h (energy)e Fo(\025)j Fm(2)h Fn(R)p Fz(,)21 b(i.e.)89 268 y Fo(j)112 275 y Fk(^)-21 b Fi(\045)129 268 y Fp(\()p Fo(\025)p Fp(\))15 b(=)e Fn(E)292 275 y Fi(\045)p Fk(\()p Fi(\025)p Fk(\))363 268 y Fp(\()p Fo(C)t Fp(\()p Fo(\025)p Fp(\)\))j Fz(for)g Fo(\025)f Fm(2)f Fn(R)p Fz(.)k(Using)e(this)g (notation)h(form)o(ula)e(\(4.4\))h(tak)o(es)g(the)g(form)728 415 y Fo(j)751 422 y Fk(^)-21 b Fi(\045)782 415 y Fp(=)834 347 y Fl(Z)862 460 y Fg(R)896 415 y Fo(d\025)15 b Fn(E)994 423 y Fi(\045)p Fk(\()p Fi(\025)p Fk(\))1065 415 y Fp(\()p Fo(C)t Fp(\()p Fo(\025)p Fp(\)\))p Fo(:)514 b Fz(\(4.29\))89 557 y(In)16 b(the)g(follo)o(wing)g(corollary)g(w)o(e)g(consider)g(the)g (case)h(that)f(the)h(steady)f(state)h Fo(\045)f Fz(is)g(a)h(function)89 618 y(of)g Fo(K)t Fz(,)f(i.e.)867 687 y Fo(\045)e Fp(=)f Fo(f)5 b Fp(\()p Fo(K)t Fp(\))p Fo(;)653 b Fz(\(4.30\))89 785 y(where,)18 b(of)g(course,)g Fo(f)5 b Fp(\()p Fm(\001)p Fp(\))17 b Fm(2)g Fo(L)649 767 y Fj(1)686 785 y Fp(\()p Fn(R)p Fp(\))e Fz(and)k Fo(f)5 b Fp(\()p Fo(\025)p Fp(\))17 b Fm(\025)f Fp(0)j Fz(for)f(a.e.)26 b Fo(\025)17 b Fm(2)g Fn(R)e Fz(and.)27 b(In)17 b(this)h(case)g(the)89 845 y(densit)o(y)d(matrix)g Fo(\045)h Fz(b)q(elongs)h(to)g(the)f(bicomm)n (utan)o(t)e(of)i Fo(K)t Fz(.)89 983 y Fq(Corollary)k(4.3)k Fc(Assume)18 b Fo(m)12 b Fp(+)732 963 y Fk(1)p 725 971 32 2 v 725 1000 a Fi(m)776 983 y Fm(2)j Fo(L)857 965 y Fj(1)895 983 y Fp(\([)p Fo(a;)8 b(b)p Fp(]\))p Fc(,)16 b Fo(V)26 b Fm(2)15 b Fo(L)1196 965 y Fk(2)1216 983 y Fp(\([)p Fo(a;)8 b(b)p Fp(]\))16 b Fc(and)i Fo(\024)1490 990 y Fi(a)1511 983 y Fo(;)8 b(\024)1561 990 y Fi(b)1593 983 y Fm(2)15 b Fn(C)1674 990 y Fk(+)1707 983 y Fc(.)24 b(If)17 b(the)89 1043 y(ste)n(ady)j(state)g Fo(\045)g Fc(is)g(given)i(by)e(\(4.30\))f(with)i(a)f(non-ne)n(gative)i(function)f Fo(f)5 b Fp(\()p Fm(\001)p Fp(\))19 b Fm(2)g Fo(L)1623 1025 y Fj(1)1660 1043 y Fp(\()p Fn(R)p Fp(\))p Fc(,)f(then)89 1103 y Fo(j)112 1110 y Fk(^)-21 b Fi(\045)129 1103 y Fp(\()p Fo(\025)p Fp(\))15 b(=)e(0)18 b Fc(for)f(a.e.)23 b Fo(\025)14 b Fm(2)g Fn(R)p Fc(.)89 1241 y Fq(Pro)r(of.)21 b Fz(In)16 b(this)g(case)h(one)f(has)711 1360 y Fo(\045)p Fp(\()p Fo(\025)p Fp(\))f(=)f Fo(f)5 b Fp(\()p Fo(\025)p Fp(\))p Fo(I)986 1368 y Fg(C)1007 1359 y Ff(2)1029 1360 y Fo(;)57 b(\025)14 b Fm(2)g Fn(R)p Fo(:)494 b Fz(\(4.31\))89 1479 y(whic)o(h)16 b(giv)o(es)381 1618 y Fo(j)404 1625 y Fk(^)-21 b Fi(\045)421 1618 y Fp(\()p Fo(\025)p Fp(\))15 b(=)e Fo(f)5 b Fp(\()p Fo(\025)p Fp(\))p Fz(tr)q Fp(\()p Fo(C)t Fp(\()p Fo(\025)p Fp(\)\))14 b(=)f Fm(\000)962 1584 y Fp(1)p 939 1607 71 2 v 939 1652 a(2)p Fo(\031)r(i)1015 1618 y(f)5 b Fp(\()p Fo(\025)p Fp(\))1125 1584 y Fo(\013)1156 1591 y Fi(b)1173 1584 y Fo(\013)1204 1591 y Fi(a)p 1115 1607 120 2 v 1115 1617 V 1115 1660 a Fo(W)i Fp(\()p Fo(\025)p Fp(\))1240 1618 y Fz(tr)1277 1626 y Fg(C)1298 1617 y Ff(2)1321 1618 y Fp(\()p Fo(E)s Fp(\002)1417 1625 y Fi(H)1450 1618 y Fp(\()p Fo(\025)p Fp(\))1516 1598 y Fj(\003)1536 1618 y Fp(\))p Fo(:)167 b Fz(\(4.32\))89 1773 y(By)16 b(\(4.17\))h(w)o(e)e(immedi)o(ately)e(get)j(that)h(tr)881 1781 y Fg(C)902 1772 y Ff(2)924 1773 y Fp(\()p Fo(E)s Fp(\002)1020 1780 y Fi(H)1054 1773 y Fp(\()p Fo(\025)p Fp(\))1120 1755 y Fj(\003)1140 1773 y Fp(\))d(=)f(0)k Fz(for)g Fo(\025)d Fm(2)g Fn(R)p Fz(.)340 b Fa(\003)148 1857 y Fz(If)16 b(the)g(steady)g(state)h Fo(\045)f Fz(has)h(the)f(form) 680 2002 y Fo(\045)p Fp(\()p Fo(\025)p Fp(\))e(=)837 1932 y Fl(\022)894 1972 y Fo(\045)919 1954 y Fi(bb)952 1972 y Fp(\()p Fo(\025)p Fp(\))95 b(0)944 2032 y(0)d Fo(\045)1085 2014 y Fi(aa)1124 2032 y Fp(\()p Fo(\025)p Fp(\))1212 1932 y Fl(\023)1257 2002 y Fo(;)465 b Fz(\(4.33\))89 2148 y(then)16 b(the)g(curren)o(t)g(densit)o(y)f(is)h(giv)o(en)f(b)o(y) 566 2294 y Fo(j)589 2301 y Fk(^)-21 b Fi(\045)606 2294 y Fp(\()p Fo(\025)p Fp(\))14 b(=)g Fm(\000)796 2260 y Fp(1)p 782 2282 54 2 v 782 2328 a(2)p Fo(\031)877 2260 y(\013)908 2242 y Fk(2)908 2272 y Fi(a)929 2260 y Fo(\013)960 2242 y Fk(2)960 2273 y Fi(b)p 845 2282 167 2 v 845 2328 a Fm(j)p Fo(W)7 b Fp(\()p Fo(\025)p Fp(\))p Fm(j)992 2313 y Fk(2)1017 2294 y Fp(\()p Fo(\045)1061 2273 y Fi(bb)1093 2294 y Fp(\()p Fo(\025)p Fp(\))12 b Fm(\000)f Fo(\045)1246 2273 y Fi(aa)1285 2294 y Fp(\()p Fo(\025)p Fp(\)\))p Fo(:)352 b Fz(\(4.34\))89 2445 y(This)21 b(curren)o(t)f(densit)o(y)f (is)i(di\033eren)o(t)f(from)f(zero)i(if)f Fo(\045)1103 2427 y Fi(bb)1135 2445 y Fp(\()p Fo(\025)p Fp(\))i Fm(6)p Fp(=)g Fo(\045)1308 2427 y Fi(aa)1347 2445 y Fp(\()p Fo(\025)p Fp(\))f Fz(at)g(least)g(for)g(a)g(set)g(of)89 2505 y(p)q(ositiv)o(e)14 b(Leb)q(esgue)g(measure.)20 b(So)14 b(a)h(curren)o(t)e(densit)o(y)g(di\033eren)o(t)h(from)f(zero)g (arises)i(only)f(if)f(w)o(e)89 2565 y(ha)o(v)o(e)h(a)h(o)q(ccupation)g (disparit)o(y)f(b)q(et)o(w)o(een)f(the)i(t)o(w)o(o)f(eigenstates)h Fo( )1326 2547 y Fj(\000)1355 2565 y Fp(\()p Fo(x;)8 b(\025;)g(b)p Fp(\))14 b Fz(and)h Fo( )1655 2547 y Fj(\000)1684 2565 y Fp(\()p Fo(x;)8 b(\025;)g(a)p Fp(\))p Fz(.)89 2625 y(This)21 b(is)f(the)g(case)g(if)g(the)g(steady)g(state)h Fo(\045)f Fz(b)q(elongs)h(to)g(the)f(comm)o(utan)o(t)d(of)k Fo(K)j Fz(but)d(not)f(to)89 2685 y(the)i(bicomm)o(utan)o(t.)37 b(In)22 b(other)g(w)o(ords,)i(the)e(densit)o(y)g(matrix)e(\(1.8\))j (used)g(for)f(self-adjoin)o(t)89 2746 y(b)q(oundary)15 b(conditions)f(and)g(generalized)f(b)o(y)g(\(4.30\))i(to)f(the)g (dissipativ)o(e)e(case)i(leads)g(to)g(a)g(zero)89 2806 y(curren)o(t)h(densit)o(y)l(.)p eop %%Page: 24 26 24 25 bop 89 23 a Fz(24)692 b Fs(H.-Ch.)25 b(Kaiser,)18 b(H.)h(Neidhardt,)g(J.)g(Rehber)o(g)89 155 y Fx(5)77 b(Remarks)89 288 y Fz(1.)39 b(The)22 b(carrier)f(densit)o(y)g Fo(u)637 295 y Fk(^)-21 b Fi(\045)654 288 y Fp(\()p Fm(\001)p Fp(\))21 b Fz(p)q(erformed)g(b)o(y)g(\(3.41\))i(and)g(\(3.44\))f(is)g (a)g(straigh)o(tforw)o(ard)89 349 y(generalization)15 b(of)g(the)g(corresp)q(onding)h(de\034nition)e(\(1.5\))h(of)h(the)f (carrier)f(densit)o(y)g(in)g(the)h(self-)89 409 y(adjoin)o(t)h(case.)22 b(Indeed,)15 b(this)h(corresp)q(ondence)g(relies)f(on)i(the)f (replacemen)o(ts)743 508 y Fo(l)42 b Fm( )-8 b(!)41 b(f)p Fo(a;)8 b(b;)g(\025)p Fm(g)p Fo(;)606 598 y Fm(f)p Fo( )663 605 y Fi(l)676 598 y Fm(g)701 580 y Fj(1)701 611 y Fi(l)p Fk(=1)800 598 y Fm( )-8 b(!)41 b(f)964 585 y Fo(~)958 598 y( )q Fp(\()p Fm(\001)p Fo(;)8 b(\025;)g(\034)e Fp(\))p Fm(g)1167 605 y Fi(\025)p Fj(2)p Fg(R)-7 b Fi(;\034)s Fk(=)p Fi(a;b)1337 598 y Fo(;)648 645 y Fl(P)700 658 y Fj(1)700 696 y Fi(l)p Fk(=1)800 682 y Fm( )f(!)933 642 y Fl(R)956 699 y Fi(\025)p Fj(2)p Fg(R)1035 682 y Fo(d\025)1111 645 y Fl(P)1163 696 y Fi(\034)t Fk(=)p Fi(a;b)720 766 y Fo(\045)745 773 y Fi(l)800 766 y Fm( )g(!)41 b Fo(\045)p Fp(\()p Fo(\025)p Fp(\))1761 637 y Fz(\(5.1\))89 868 y(The)18 b(same)e(holds)i(for)g(the)f(curren)o(t)g(densit)o(y)g (de\034ned)g(b)o(y)g(\(4.1\))h(whic)o(h)f(is)g(a)h(straigh)o(tforw)o (ard)89 928 y(generalization)i(of)h(\(1.11\).)35 b Fc(However,)24 b(in)e(c)n(ontr)n(ast)f(to)g(the)h(self-adjoint)h(c)n(ase)f(the)g(curr) n(ent)89 989 y(density)c(now)g(is)f(not)h(ne)n(c)n(essarily)g(zer)n(o!) 148 1072 y Fz(2.)23 b(There)16 b(is)h(consensus)g(in)g(the)f(con)o (viction)f(that)j(scattering)e(states)h(are)g(resp)q(onsible)g(for)89 1133 y(the)12 b(curren)o(t.)19 b(This)12 b(usually)g(leads)g(to)h(a)f (a)h(relation)f(b)q(et)o(w)o(een)f(curren)o(t)g(densit)o(y)g(and)i (scattering)89 1193 y(matrix.)31 b(A)o(ctually)l(,)19 b(the)h(same)f(tak)o(es)h(place)g(here.)32 b(F)l(orm)o(ulae)18 b(\(4.2\))j(and)f(\(4.3\))h(relate)e(the)89 1253 y(curren)o(t)g(densit) o(y)g Fo(j)457 1260 y Fk(^)-21 b Fi(\045)474 1253 y Fp(\()p Fo(\025)p Fp(\))20 b Fz(with)f(the)h(c)o(haracteristic)e(function)h Fp(\002)1297 1260 y Fi(H)1329 1251 y Fd(\003)1349 1253 y Fp(\()p Fo(\025)p Fp(\))h Fz(of)g Fo(H)1538 1235 y Fj(\003)1558 1253 y Fz(.)31 b(It)19 b(turns)h(out)89 1313 y(that)j(the)f(c)o(haracteristic)f(function)h Fp(\002)832 1320 y Fi(H)864 1311 y Fd(\003)884 1313 y Fp(\()p Fo(\025)p Fp(\))h Fz(of)f Fo(H)1078 1295 y Fj(\003)1121 1313 y Fz(can)g(b)q(e)h(regarded)g(as)g(the)f(scattering)89 1373 y(matrix)15 b(of)j(an)f(asso)q(ciated)h(scattering)f(system.)22 b(Indeed,)16 b(with)h(the)g(self-adjoin)o(t)f(dilation)h Fo(K)89 1434 y Fz(one)g(can)f(asso)q(ciate)h(a)g(so-called)f (Lax-Phillips)f(scattering)i(theory)f([2)o(,)g(16)q(].)21 b(T)l(o)c(this)f(end)g(one)89 1494 y(in)o(tro)q(duces)g(the)g(Hilb)q (ert)f(space)h Fh(K)740 1501 y Fk(0)760 1494 y Fz(,)619 1597 y Fh(K)652 1604 y Fk(0)686 1597 y Fp(:=)d Fo(L)784 1576 y Fk(2)804 1597 y Fp(\()p Fn(R)p Fo(;)7 b Fn(C)914 1576 y Fk(2)936 1597 y Fp(\))14 b(=)g Fm(D)1059 1604 y Fj(\000)1100 1597 y Fm(\010)d(D)1188 1604 y Fk(+)1232 1597 y Fm(\022)i Fh(K)p Fo(:)430 b Fz(\(5.2\))89 1700 y(and)17 b(the)f(iden)o(ti\034cation)f(op)q(erators)j Fo(J)805 1707 y Fj(\006)848 1700 y Fp(:)13 b Fh(K)908 1707 y Fk(0)942 1700 y Fm(\000)-8 b(!)13 b Fh(K)p Fz(,)567 1794 y Fo(~)560 1807 y(f)19 b Fp(=)14 b Fo(J)682 1814 y Fj(\000)712 1807 y Fo(f)47 b Fp(:=)41 b Fo(P)914 1785 y Fe(K)937 1790 y Ff(0)907 1820 y Fj(D)934 1824 y Fd(\000)962 1807 y Fo(f)17 b Fm(\010)11 b Fp(0)g Fm(\010)g Fp(0)p Fo(;)57 b(f)19 b Fm(2)14 b Fh(K)1356 1814 y Fk(0)1376 1807 y Fo(;)567 1884 y(~)560 1897 y(f)19 b Fp(=)14 b Fo(J)682 1904 y Fk(+)712 1897 y Fo(f)47 b Fp(:=)41 b(0)11 b Fm(\010)g Fp(0)g Fm(\010)g Fo(P)1084 1875 y Fe(K)1107 1880 y Ff(0)1077 1911 y Fj(D)1104 1915 y Ff(+)1132 1897 y Fo(f)s(;)57 b(f)19 b Fm(2)14 b Fh(K)1353 1904 y Fk(0)1373 1897 y Fo(:)1761 1851 y Fz(\(5.3\))89 2000 y(The)f(subspaces)h Fm(D)444 2007 y Fj(\000)487 2000 y Fz(and)g Fm(D)617 2007 y Fk(+)660 2000 y Fz(are)f(called)f(incoming)g(and)h(outgoing)i (subspaces,)f(resp)q(ectiv)o(ely)l(.)89 2060 y(On)i(the)g(Hilb)q(ert)f (space)i Fh(K)586 2067 y Fk(0)622 2060 y Fz(one)f(de\034nes)g(the)g (self-adjoin)o(t)g(op)q(erator)i Fo(K)1444 2067 y Fk(0)1464 2060 y Fz(,)738 2185 y Fp(\()p Fo(K)798 2192 y Fk(0)819 2185 y Fo(f)5 b Fp(\)\()p Fo(x)p Fp(\))13 b(=)h Fm(\000)p Fo(i)1072 2152 y(d)p 1059 2174 54 2 v 1059 2220 a(dx)1117 2185 y(f)5 b Fp(\()p Fo(x)p Fp(\))549 b Fz(\(5.4\))89 2306 y(with)16 b(the)g(domain)g(dom)o Fp(\()p Fo(K)610 2313 y Fk(0)630 2306 y Fp(\))e(:=)f Fo(W)781 2288 y Fk(1)p Fi(;)p Fk(2)828 2306 y Fp(\()p Fn(R)p Fo(;)8 b Fn(C)938 2288 y Fk(2)960 2306 y Fp(\))p Fz(.)22 b(The)16 b(Lax-Phillips)g(w)o(a) o(v)o(e)f(op)q(erators)672 2409 y Fo(W)718 2416 y Fj(\006)761 2409 y Fp(=)f Fo(s)d Fm(\000)33 b Fp(lim)897 2439 y Fi(t)p Fj(!\0061)1016 2409 y Fo(e)1039 2389 y Fi(itK)1098 2409 y Fo(J)1125 2416 y Fj(\006)1154 2409 y Fo(e)1177 2389 y Fj(\000)p Fi(itK)1259 2394 y Ff(0)1761 2409 y Fz(\(5.5\))89 2532 y(exist)11 b(and)g(are)h(complete,)d(i.e.)18 b(ran)q Fp(\()p Fo(W)799 2539 y Fj(\006)828 2532 y Fp(\))c(=)g Fh(K)p Fz(.)20 b(The)11 b(corresp)q(onding)h(Lax-Phillips)f(scattering) 89 2592 y(op)q(erator)22 b Fo(S)j Fp(=)d Fo(W)458 2574 y Fj(\003)451 2605 y Fk(+)480 2592 y Fo(W)526 2599 y Fj(\000)577 2592 y Fp(:)g Fh(K)646 2599 y Fk(0)687 2592 y Fm(\000)-8 b(!)794 2579 y Fp(^)789 2592 y Fh(K)q Fz(,)21 b(is)g(unitary)g(and)g(comm)o(utes)d(with)j(the)f(self-adjoin)o(t)89 2660 y(op)q(erator)e Fo(K)327 2667 y Fk(0)347 2660 y Fz(.)j(By)16 b Fm(F)i Fp(:)c Fh(K)574 2667 y Fk(0)607 2660 y Fm(\000)-8 b(!)706 2647 y Fp(^)701 2660 y Fh(K)17 b Fz(w)o(e)f(denote)g(the)g(F)l(ourier)g(transform)335 2784 y Fp(\()p Fm(F)5 b Fo(f)g Fp(\)\()p Fo(\025)p Fp(\))14 b(=)585 2771 y(^)575 2784 y Fo(f)5 b Fp(\()p Fo(\025)p Fp(\))14 b(:=)790 2751 y(1)p 755 2773 96 2 v 755 2783 a Fm(p)p 796 2783 54 2 v 796 2824 a Fp(2)p Fo(\031)863 2717 y Fl(Z)891 2829 y Fg(R)925 2784 y Fo(dx)g(e)1015 2764 y Fj(\000)p Fi(i\025x)1097 2784 y Fo(f)5 b Fp(\()p Fo(x)p Fp(\))p Fo(;)56 b(f)20 b Fm(2)14 b Fh(K)1386 2791 y Fk(0)1406 2784 y Fo(;)56 b(\025)15 b Fm(2)f Fn(R)p Fo(:)142 b Fz(\(5.6\))p eop %%Page: 25 27 25 26 bop 89 23 a Fs(Density)18 b(and)g(current)h(of)f(a)g(dissip)l(a)l (tive)g(Schr)o(\366dinger)h(opera)l(tor)187 b Fz(25)89 147 y(W)l(e)22 b(note)h(that)f Fm(F)5 b Fo(K)487 154 y Fk(0)507 147 y Fm(F)548 129 y Fj(\000)p Fk(1)620 147 y Fp(=)24 b Fo(M)j Fz(where)22 b Fo(M)28 b Fz(is)22 b(de\034ned)g(b)o (y)g(\(2.43\).)40 b(Since)22 b Fo(S)j Fz(comm)o(utes)89 208 y(with)c Fo(K)246 215 y Fk(0)288 208 y Fz(the)g(op)q(erator)586 195 y Fp(^)578 208 y Fo(S)k Fp(=)d Fm(F)5 b Fo(S)s Fm(F)808 189 y Fj(\000)p Fk(1)877 208 y Fp(:)918 195 y(^)913 208 y Fh(K)23 b Fm(\000)-9 b(!)1076 195 y Fp(^)1071 208 y Fh(K)22 b Fz(comm)o(ute)o(s)d(with)i Fo(M)5 b Fz(.)36 b(Hence)1737 195 y Fp(^)1728 208 y Fo(S)24 b Fz(is)d(a)89 268 y(m)o(ultiplic)o(ation)14 b(op)q(erator)j(giv)o(en)f(b)o(y)560 368 y Fp(\()587 355 y(^)579 368 y Fo(S)622 354 y Fp(^)611 368 y Fo(f)6 b Fp(\)\()p Fo(\025)p Fp(\))14 b(=)g Fo(S)s Fp(\()p Fo(\025)p Fp(\))902 354 y(^)891 368 y Fo(f)5 b Fp(\()p Fo(\025)p Fp(\))p Fo(;)1068 354 y Fp(^)1057 368 y Fo(f)20 b Fm(2)1152 355 y Fp(^)1148 368 y Fh(K)p Fo(;)57 b(\025)14 b Fm(2)g Fn(R)p Fo(;)367 b Fz(\(5.7\))89 467 y(where)21 b Fm(f)p Fo(S)s Fp(\()p Fo(\025)p Fp(\))p Fm(g)384 474 y Fi(\025)p Fj(2)p Fg(R)476 467 y Fz(is)h(a)g(measurable)e (family)f(of)j(unitary)f(op)q(erators)i(whic)o(h)e(is)h(called)e(the)89 528 y(Lax-Phillips)c(scattering)g(matrix.)k(A)15 b(rather)i(in)o(v)o (olv)o(ed)d(computation)h(sho)o(ws)i(that)815 627 y Fo(S)s Fp(\()p Fo(\025)p Fp(\))d(=)f(\002)1017 634 y Fi(H)1049 625 y Fd(\003)1069 627 y Fp(\()p Fo(\025)p Fp(\))626 b Fz(\(5.8\))89 727 y(for)17 b(a.e.)j Fo(\025)15 b Fm(2)f Fn(R)p Fz(,)f(see)i(for)i(example)d([1].)21 b(By)15 b(the)h(w)o(a)o(y)g (one)g(has)640 827 y Fp(\010)675 834 y Fj(\000)719 827 y Fp(=)e Fm(F)5 b Fo(W)865 807 y Fj(\003)858 839 y Fk(+)936 827 y Fz(and)49 b Fp(\010)1098 834 y Fk(+)1142 827 y Fp(=)13 b Fm(F)5 b Fo(W)1287 807 y Fj(\003)1280 839 y(\000)1761 827 y Fz(\(5.9\))89 927 y(where)18 b(is)f Fp(\010)317 934 y Fj(\000)365 927 y Fz(is)g(the)h(incoming)e(F)l(ourier)h (transformation,)g(cf.)26 b(\(2.41\))18 b(and)g(\(2.42\),)h(and)f Fp(\010)1831 934 y Fk(+)89 987 y Fz(is)e(the)g(so-called)g(outgoing)i (F)l(ourier)d(transform)h(whic)o(h)g(w)o(as)g(in)o(tro)q(duced)g(in)g ([12].)148 1071 y(3.)22 b(As)16 b(men)o(tioned)e(ab)q(o)o(v)o(e)i(the)g (quan)o(tit)o(y)f Fo(N)973 1078 y Fi(\045)993 1071 y Fp(\()p Fo(!)r Fp(\))p Fz(,)615 1197 y Fo(N)654 1204 y Fi(\045)674 1197 y Fp(\()p Fo(!)r Fp(\))f(:=)f Fm(k)p Fo(u)879 1204 y Fk(^)-21 b Fi(\045)896 1197 y Fm(k)921 1205 y Fi(L)945 1195 y Ff(1)962 1205 y Fk(\()p Fi(!)q Fk(\))1028 1197 y Fp(=)1080 1129 y Fl(Z)1108 1242 y Fi(!)1141 1197 y Fo(dx)14 b(u)1239 1204 y Fk(^)-21 b Fi(\045)1256 1197 y Fp(\()p Fo(x)p Fp(\))p Fo(;)400 b Fz(\(5.10\))89 1323 y(has)19 b(the)g(meaning)e(of)i(the)f(n)o(um)o(b)q(er)f(of)h (carriers)g(on)h(the)g(Borel)e(set)h(subset)h Fo(!)h Fm(\022)e Fp(\012)f(=)h([)p Fo(a;)8 b(b)p Fp(])p Fz(.)89 1383 y(By)16 b(\(3.44\))h(one)f(has)h(the)f(represen)o(tation)766 1483 y Fo(N)805 1490 y Fi(\045)825 1483 y Fp(\()p Fo(!)r Fp(\))e(=)g Fz(tr)p Fp(\()p Fo(\045U)5 b Fp(\()p Fo(!)r Fp(\)\))p Fo(:)552 b Fz(\(5.11\))89 1583 y(Ob)o(viously)14 b Fo(N)19 b Fp(:=)13 b Fo(N)479 1590 y Fi(\045)499 1583 y Fp(\(\012\))i Fz(is)g(the)f(total)h(n)o(um)o(b)q(er)e(of)h(carriers)h (on)g(the)f(in)o(terv)m(al)f Fp([)p Fo(a;)8 b(b)p Fp(])p Fz(.)19 b(W)l(e)c(note)89 1643 y(that)i(under)f(the)f (condition\(3.23\))i(b)o(y)e(Prop)q(osition)i(3.7)g(the)f(total)g(n)o (um)o(b)q(er)e(of)i(carriers)g(in)g Fp(\012)89 1703 y Fz(is)g(alw)o(a)o(ys)g(\034nite.)148 1787 y(In)g(accordance)g(with)g ([15)q(])g(the)g(time)e(ev)o(olution)h(of)h(the)g(densit)o(y)g(matrix)e Fo(\045)j Fz(is)f(giv)o(en)f(b)o(y)695 1887 y Fo(\045)p Fp(\()p Fo(t)p Fp(\))e(=)h Fo(e)864 1866 y Fj(\000)p Fi(itK)950 1887 y Fo(\045e)998 1866 y Fi(itK)1057 1887 y Fo(;)56 b(t)14 b Fm(2)g Fn(R)p Fo(:)477 b Fz(\(5.12\))89 1986 y(Hence,)15 b(the)h(n)o(um)o(b)q(er)e(of)j(particles)e(at)i(time)d Fo(t)f Fm(2)h Fn(R)f Fz(on)k Fo(!)h Fz(is)e(giv)o(en)g(b)o(y)619 2086 y Fo(N)658 2094 y Fi(\045)p Fk(\()p Fi(t)p Fk(\))718 2086 y Fp(\()p Fo(!)r Fp(\))e(:=)g 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Fo(dx)14 b(\025)g Fz(tr)668 401 y Fg(C)689 391 y Ff(2)712 392 y Fp(\()p Fo(\045)p Fp(\()p Fo(\025)p Fp(\))p Fo(D)q Fp(\()p Fo(x;)8 b(\025)p Fp(\)\))k(+)f Fo(i)1084 324 y Fl(Z)1112 437 y Fg(R)1146 392 y Fo(d\025)1222 324 y Fl(Z)1250 437 y Fi(!)1283 392 y Fo(dx)j(\025)g Fz(tr)1430 401 y Fg(C)1451 391 y Ff(2)1474 392 y Fp(\()p Fo(\045)p Fp(\()p Fo(\025)p Fp(\))p Fo(D)q Fp(\()p Fo(x;)8 b(\025)p Fp(\)\))p Fo(:)89 532 y Fz(By)16 b(form)o(ula)e(\(3.42\))j(w)o(e)f(\034nd)271 603 y Fl(Z)299 716 y Fi(!)332 671 y Fo(dx)e(\025)p Fp(\()p Fz(tr)484 679 y Fg(C)505 669 y Ff(2)528 671 y Fp(\()p Fo(\045)p Fp(\()p Fo(\025)p Fp(\))p Fo(D)q Fp(\()p Fo(x;)8 b(\025)p Fp(\)\))15 b(=)869 b Fz(\(5.18\))472 737 y Fl(Z)500 850 y Fi(!)533 805 y Fo(dx)14 b(\025)636 735 y Fl(\034)674 805 y Fo(\045)699 784 y Fi(t)714 805 y Fp(\()p Fo(\025)p Fp(\))788 735 y Fl(\022)849 774 y Fo( )883 756 y Fj(\000)912 774 y Fp(\()p Fo(x;)8 b(\025;)g(b)p Fp(\))846 834 y Fo( )880 816 y Fj(\000)909 834 y Fp(\()p Fo(x;)g(\025;)g(a)p 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b(b)o(y)f(parts)h(and)h(taking)e(in)o(to)h(accoun)o(t)f(form)o(ula)g (\(4.7\))89 1951 y(w)o(e)e(get)558 2003 y Fo(d)p 549 2025 V 549 2071 a(dt)597 2037 y(N)636 2045 y Fi(\045)p Fk(\()p Fi(t)p Fk(\))696 2037 y Fp(\()p Fo(!)r Fp(\))e(=)832 1969 y Fl(Z)860 2082 y Fg(R)894 2037 y Fo(d\025)23 b Fm(f)p Fo(j)1018 2044 y Fk(^)-21 b Fi(\045)1035 2037 y Fp(\()p Fo(c;)8 b(\025)p Fp(\))k Fm(\000)e Fo(j)1228 2044 y Fk(^)-21 b Fi(\045)1246 2037 y Fp(\()p Fo(d;)8 b(\025)p Fp(\))p Fm(g)g Fo(:)330 b Fz(\(5.21\))89 2155 y(The)16 b(total)h(curren)o(t)e Fo(j)498 2162 y Fk(^)-21 b Fi(\045)516 2155 y Fp(\()p Fo(x)p Fp(\))15 b Fz(at)i(the)f(p)q(oin)o (t)g Fo(x)e Fm(2)g Fp([)p Fo(a;)8 b(b)p Fp(])15 b Fz(is)h(de\034ned)f (b)o(y)733 2295 y Fo(j)756 2302 y Fk(^)-21 b Fi(\045)773 2295 y Fp(\()p Fo(x)p Fp(\))13 b(:=)918 2227 y Fl(Z)945 2340 y Fg(R)980 2295 y Fo(d\025)h(j)1070 2302 y Fk(^)-21 b Fi(\045)1088 2295 y Fp(\()p Fo(x;)8 b(\025)p Fp(\))p Fo(:)518 b Fz(\(5.22\))89 2432 y(This)17 b(yields)e(This)h(yields)713 2484 y Fo(d)p 704 2507 V 704 2552 a(dt)752 2518 y(N)791 2526 y Fi(\045)p Fk(\()p Fi(t)p Fk(\))851 2518 y Fp(\()p Fo(!)r Fp(\))e(=)g Fo(j)1010 2525 y Fk(^)-21 b Fi(\045)1027 2518 y Fp(\()p Fo(c)p Fp(\))11 b Fm(\000)g Fo(j)1170 2525 y Fk(^)-21 b Fi(\045)1188 2518 y Fp(\()p Fo(d)p Fp(\))485 b Fz(\(5.23\))89 2625 y(whic)o(h)13 b(sho)o(ws)i(that)g(the)e (c)o(hange)h(of)h(the)e(n)o(um)o(b)q(er)f(of)j(carriers)e(in)h(the)f (set)h Fp([)p Fo(c;)8 b(d)p Fp(])14 b Fz(is)f(equal)h(to)g(the)89 2685 y(di\033erence)k(b)q(et)o(w)o(een)g(the)g(incoming)g(curren)o(t)g Fo(j)994 2692 y Fk(^)-21 b Fi(\045)1011 2685 y Fp(\()p Fo(c)p Fp(\))19 b Fz(at)g(p)q(oin)o(t)g Fo(c)g Fz(and)g(the)g(outgoing) h(curren)o(t)89 2746 y Fo(j)112 2753 y Fk(^)-21 b Fi(\045)129 2746 y Fp(\()p Fo(d)p Fp(\))23 b Fz(at)f(p)q(oin)o(t)g Fo(d)g Fz(whic)o(h)f(v)o(ery)f(w)o(ell)h(corresp)q(onds)i(to)f(the)f (ph)o(ysical)g(in)o(tuition.)37 b(Since)20 b(b)o(y)89 2806 y(Prop)q(osition)f(4.1)g(the)f(curren)o(t)f(densit)o(y)g Fo(j)884 2813 y Fk(^)-21 b Fi(\045)901 2806 y Fp(\()p Fo(x;)8 b(\025)p Fp(\))19 b Fz(do)q(es)f(not)h(dep)q(end)f(on)h Fo(x)e Fm(2)g Fp([)p Fo(a;)8 b(b)p Fp(])16 b Fz(one)j(gets)p eop %%Page: 27 29 27 28 bop 89 23 a Fs(Density)18 b(and)g(current)h(of)f(a)g(dissip)l(a)l (tive)g(Schr)o(\366dinger)h(opera)l(tor)187 b Fz(27)89 147 y Fo(j)112 154 y Fk(^)-21 b Fi(\045)129 147 y Fp(\()p Fo(d)p Fp(\))16 b(=)e Fo(j)283 154 y Fk(^)-21 b Fi(\045)301 147 y Fp(\()p Fo(c)p Fp(\))16 b Fz(whic)o(h)g(again)i(v)o(eri\034es)d (\(5.14\).)24 b(The)17 b(relation)f(\(5.23\))i(is)e(the)h(in)o(tegral)f (form)g(of)89 208 y(the)g(so-called)g(con)o(tin)o(uit)o(y)e(equation)j (whic)o(h)e(has)i(the)f(di\033eren)o(tial)f(form)506 303 y Fo(@)p 497 325 47 2 v 497 371 a(@)s(t)548 337 y(u)579 344 y Fk(^)-21 b Fi(\045)p Fk(\()p Fi(t)p Fk(\))637 337 y Fp(\()p Fo(x)p Fp(\))10 b(+)781 303 y Fo(@)p 767 325 57 2 v 767 371 a(@)s(x)828 337 y(j)851 344 y Fk(^)-21 b Fi(\045)q Fk(\()p Fi(t)p Fk(\))909 337 y Fp(\()p Fo(x)p Fp(\))13 b(=)h(0)p Fo(;)57 b(t)14 b Fm(2)g Fn(R)p Fo(;)53 b(x)14 b Fm(2)g Fp(\012)p Fo(;)278 b Fz(\(5.24\))89 459 y(where)21 b Fo(u)266 467 y Fk(^)-21 b Fi(\045)p Fk(\()p Fi(t)p Fk(\))323 459 y Fp(\()p Fo(x)p Fp(\))20 b Fz(is)h(the)g(carrier) f(densit)o(y)g(at)h(time)e Fo(t)i Fm(2)h Fn(R)c Fz(and)j Fo(j)1334 467 y Fi(\045)p Fk(\()p Fi(t)p Fk(\))1395 459 y Fp(\()p Fo(x)p Fp(\))f Fz(is)h(curren)o(t)f(densit)o(y)89 519 y(at)h(time)e Fo(t)h Fz(giv)o(en)f(b)o(y)h(\(5.22\).)35 b(Since)20 b Fo(u)846 527 y Fk(^)-21 b Fi(\045)p Fk(\()p Fi(t)p Fk(\))903 519 y Fp(\()p Fo(x)p Fp(\))21 b(=)g Fo(u)1080 526 y Fk(^)-21 b Fi(\045)1097 519 y Fp(\()p Fo(x)p Fp(\))20 b Fz(is)h(indep)q(enden)o(t)e(of)i Fo(t)g Fz(and)g Fo(j)1735 527 y Fi(\045)p Fk(\()p Fi(t)p Fk(\))1795 519 y Fp(\()p Fo(x)p Fp(\))89 579 y Fz(is)g(indep)q(enden)o(t)f(of)h Fo(x)f Fz(the)h(con)o(tin)o(uit)o(y)e(equation)i(\(5.24\))g(ob)o (viously)f(holds)h(in)g(the)f(presen)o(t)89 640 y(situation.)148 723 y(4.)i(On)15 b(the)g(basis)h(of)f(the)g(considerations)h(ab)q(o)o (v)o(e)f(w)o(e)g(are)g(able)g(to)h(giv)o(e)e(a)i(mathematicall)o(y)89 784 y(rigourous)23 b(foundation)f(of)g(dissipativ)o(e)f(Sc)o(hr\366)q (dinger-P)o(oisson)h(systems.)36 b(T)l(o)22 b(this)g(end)f(w)o(e)89 844 y(consider)c(di\033eren)o(t)e(sp)q(ecies)h Fm(\006)h Fz(of)g(particles)f(\(holes)g(and)h(electrons\))f(and)h(assume)f(that)h (that)89 904 y(for)k(these)g(sp)q(ecies)f(the)h(e\033ectiv)o(e)e (masses)h Fo(m)954 911 y Fj(\006)983 904 y Fz(,)i(external)d(p)q(oten)o (tials)i Fo(V)1480 883 y Fj(\006)1469 916 y Fk(0)1531 904 y Fz(and)g(co)q(e\036cien)o(ts)89 964 y Fo(\024)117 946 y Fj(\006)117 977 y Fi(a)147 964 y Fo(;)8 b(\024)197 944 y Fj(\006)197 978 y Fi(b)240 964 y Fm(2)14 b Fn(C)320 971 y Fk(+)364 964 y Fz(are)d(giv)o(en.)18 b(F)l(or)11 b(eac)o(h)g(sp)q(ecies)f(this)h(leads)g(to)g(di\033eren)o(t)f (dissipativ)o(e)g(Sc)o(hr\366)q(dinger)89 1024 y(op)q(erators)17 b Fo(H)348 1006 y Fj(\006)378 1024 y Fp(\()p Fo(V)425 1031 y Fj(\006)455 1024 y Fp(\))f Fz(de\034ned)f(in)g(accordance)h (with)f(\(1.12\),)h(\(2.1\))g(and)g(\(2.2\).)22 b(The)15 b(p)q(oten)o(tial)89 1085 y Fo(V)29 b Fz(en)o(tering)16 b(in)o(to)h(the)g(de\034nition)g(of)h(the)f(Sc)o(hr\366)q(dinger)g(op)q (erators)i(is)e(di\033eren)o(t)f(for)i(di\033eren)o(t)89 1145 y(sp)q(ecies.)j(It)16 b(tak)o(es)g(the)g(form)786 1205 y Fo(V)814 1212 y Fj(\006)858 1205 y Fp(:=)d Fo(V)962 1184 y Fj(\006)951 1217 y Fk(0)1003 1205 y Fm(\006)e Fo(')p Fp(\()p Fo(u)p Fp(\))p Fo(:)571 b Fz(\(5.25\))89 1291 y(where)17 b(the)h(electrostatic)e(p)q(oten)o(tial)i Fo(')f Fz(is)h(a)g(solution)g(of)g(the)f(P)o(oisson)i(equation)e (\(1.1\))h(with)89 1351 y(b)q(oundary)f(condition)f(\(1.3\).)21 b(The)16 b(carrier)g(densities)f Fo(u)1138 1333 y Fj(\006)1183 1351 y Fz(en)o(tering)g(in)o(to)h(P)o(oisson's)g(equation)89 1411 y(are)f(obtained)f(from)f(the)h(dissipativ)o(e)g(Sc)o(hr\366)q (dinger)g(op)q(erators)i Fo(H)1324 1393 y Fj(\006)1353 1411 y Fp(\()p Fo(V)1400 1418 y Fj(\006)1430 1411 y Fp(\))f Fz(in)f(accordance)g(with)89 1471 y(Section)k(3.)27 b(T)l(o)19 b(this)f(end)g(w)o(e)g(assume)f(that)i(the)f(families)e(of)i(matrices)e Fm(f)p Fo(\045)1514 1478 y Fj(\006)1544 1471 y Fp(\()p Fo(\025)p Fp(\))p Fm(g)1635 1478 y Fi(\025)p Fj(2)p Fg(R)1705 1471 y Fz(,)j(whic)o(h)89 1532 y(ob)q(ey)570 1592 y Fo(C)608 1599 y Fk(^)-21 b Fi(\045)623 1603 y Fd(\006)665 1592 y Fp(:=)13 b(sup)733 1633 y Fi(\025)p Fj(2)p Fg(R)812 1548 y Fm(p)p 853 1548 133 2 v 853 1592 a Fo(\025)881 1577 y Fk(2)912 1592 y Fp(+)e(1)q Fm(k)p Fo(\045)1036 1599 y Fj(\006)1065 1592 y Fp(\()p Fo(\025)p Fp(\))p Fm(k)1156 1600 y Fj(B)q Fk(\()p Fg(C)1215 1591 y Ff(2)1236 1600 y Fk(\))1265 1592 y Fo(<)j Fm(1)p Fo(;)355 b Fz(\(5.26\))89 1710 y(are)22 b(giv)o(en)f(and)h(w)o(e)g(de\034ne)f(the)h(carrier)f (densities)g Fo(u)1115 1692 y Fj(\006)1144 1710 y Fp(\()p Fo(x)p Fp(\))i(:=)g Fo(u)1336 1690 y Fj(\006)1339 1724 y Fk(^)-21 b Fi(\045)1354 1728 y Fd(\006)1382 1710 y Fp(\()p Fo(V)1429 1717 y Fj(\006)1459 1710 y Fp(\)\()p Fo(x)p Fp(\))21 b Fz(in)h(accordance)89 1771 y(with)c(Section)f(3,)h (i.e.,)e(if)i Fo(D)612 1753 y Fj(\006)642 1771 y Fp(\()p Fo(V)689 1778 y Fj(\006)719 1771 y Fp(\)\()p Fo(x)p Fp(\))f Fz(are)h(the)g(carrier)f(densit)o(y)g(observ)m(ables)h(at)g Fo(x)e Fm(2)h Fp(\012)h Fz(and)89 1831 y(at)f(energy)f Fo(\025)e Fm(2)g Fn(R)p Fz(,)f(then)j(the)g(carrier)f(densities)g(are)i (computed)e(b)o(y)513 1959 y Fo(u)541 1939 y Fj(\006)544 1973 y Fk(^)-21 b Fi(\045)559 1977 y Fd(\006)587 1959 y Fp(\()p Fo(V)634 1966 y Fj(\006)664 1959 y Fp(\)\()p Fo(x)p Fp(\))13 b(=)814 1892 y Fl(Z)841 2004 y Fg(R)876 1959 y Fo(d\025)h(u)971 1939 y Fj(\006)974 1973 y Fk(^)-21 b Fi(\045)989 1977 y Fd(\006)1017 1959 y Fp(\()p Fo(V)1064 1966 y Fj(\006)1094 1959 y Fp(\)\()p Fo(x;)8 b(\025)p Fp(\))p Fo(;)57 b(x)13 b Fm(2)i Fp(\012)p Fo(;)298 b Fz(\(5.27\))89 2090 y(where)273 2197 y Fo(u)301 2176 y Fj(\006)304 2211 y Fk(^)-21 b Fi(\045)319 2215 y Fd(\006)347 2197 y Fp(\()p Fo(V)394 2204 y Fj(\006)424 2197 y Fp(\)\()p Fo(x;)8 b(\025)p Fp(\))14 b(=)g Fz(tr)663 2205 y Fg(C)684 2196 y Ff(2)706 2197 y Fp(\()p Fo(\045)750 2204 y Fj(\006)780 2197 y Fp(\()p Fo(\025)p Fp(\))p Fo(D)887 2176 y Fj(\006)917 2197 y Fp(\()p Fo(V)964 2204 y Fj(\006)994 2197 y Fp(\)\()p Fo(x;)8 b(\025)p Fp(\)\))p Fo(;)57 b(x)13 b Fm(2)h Fp(\012)p Fo(;)57 b(\025)15 b Fm(2)f Fn(R)p Fo(:)180 b Fz(\(5.28\))89 2310 y(Moreo)o(v)o(er,)15 b(if)g(tr)399 2318 y Fg(C)420 2309 y Ff(2)443 2310 y Fp(\()p Fo(\045)487 2317 y Fj(\006)516 2310 y Fp(\()p Fm(\001)p Fp(\)\))f Fm(2)g Fo(L)681 2292 y Fk(1)701 2310 y Fp(\()p Fn(R)p Fp(\))f Fz(is)j(v)m(alid,)f(then)i (the)f(curren)o(t)f(densities)g Fo(j)1557 2290 y Fj(\006)1557 2324 y Fk(^)-21 b Fi(\045)1572 2328 y Fd(\006)1601 2310 y Fz(,)730 2451 y Fo(j)753 2430 y Fj(\006)753 2465 y Fk(^)g Fi(\045)768 2469 y Fd(\006)810 2451 y Fp(=)862 2383 y Fl(Z)890 2496 y Fg(R)924 2451 y Fo(d\025)15 b(j)1015 2430 y Fj(\006)1015 2465 y Fk(^)-21 b Fi(\045)1030 2469 y Fd(\006)1058 2451 y Fp(\()p Fo(V)1105 2458 y Fj(\006)1135 2451 y Fp(\)\()p Fo(\025)p Fp(\))516 b Fz(\(5.29\))89 2581 y(are)19 b(also)g(w)o(ell-de\034ned)e(and)i(\034nite,)f(cf.)28 b(Section)18 b(4.)28 b(The)19 b(so)g(describ)q(ed)f(system)f(is)h (called)f(a)89 2641 y(dissipativ)o(e)g(Sc)o(hr\366)q(dinger-P)o(oisson) h(system.)25 b(W)l(e)17 b(note)h(that)h(the)e(total)h(n)o(um)o(b)q(er)e (of)i(carriers)89 2701 y Fo(N)133 2683 y Fj(\006)163 2701 y Fp(\()p Fo(V)210 2708 y Fj(\006)240 2701 y Fp(\))e Fz(is)g(giv)o(en)f(b)o(y)674 2786 y Fo(N)718 2765 y Fj(\006)748 2786 y Fp(\()p Fo(V)795 2793 y Fj(\006)825 2786 y Fp(\))f(=)910 2718 y Fl(Z)937 2831 y Fk(\012)973 2786 y Fo(dx)g(u)1068 2765 y Fj(\006)1071 2800 y Fk(^)-21 b Fi(\045)1086 2804 y Fd(\006)1114 2786 y Fp(\()p Fo(V)1161 2793 y Fj(\006)1191 2786 y Fp(\)\()p Fo(x)p Fp(\))460 b Fz(\(5.30\))p eop %%Page: 28 30 28 29 bop 89 23 a Fz(28)692 b Fs(H.-Ch.)25 b(Kaiser,)18 b(H.)h(Neidhardt,)g(J.)g(Rehber)o(g)89 147 y Fz(and)e(is)f(not)h (\034xed.)148 231 y(In)e(a)h(forthcoming)e(pap)q(er)h([13)q(])f(w)o(e)h (sho)o(w)h(that)f(under)g(suitable)g(conditions)g(on)h Fo(\017)p Fp(\()p Fm(\001)p Fp(\))p Fz(,)e Fo(C)t Fp(\()p Fm(\001)p Fp(\))p Fz(,)89 291 y Fo(m)132 298 y Fj(\006)161 291 y Fp(\()p Fm(\001)p Fp(\))p Fz(,)23 b Fo(V)289 271 y Fj(\006)278 304 y Fk(0)319 291 y Fp(\()p Fm(\001)p Fp(\))p Fz(,)f Fo(\024)435 271 y Fj(\006)435 305 y Fi(b)465 291 y Fz(,)h Fo(\024)530 273 y Fj(\006)530 304 y Fi(a)581 291 y Fz(and)g Fo(\045)707 298 y Fj(\006)758 291 y Fz(this)f (dissipativ)o(e)f(Sc)o(hr\366)q(dinger-P)o(oisson)i(system)d(alw)o(a)o (ys)89 352 y(admits)15 b(a)i(self-consisten)o(t)e(solution.)89 556 y Fx(A)n(c)n(kno)n(wledgemen)n(t)89 692 y Fz(The)23 b(authors)g(thank)g(M.)f(Baro)h(for)f(carefully)f(reading)i(the)f(man)o (uscript.)39 b(The)22 b(\034nancial)89 752 y(supp)q(ort)c(of)e(DF)o(G)g (is)g(grateful)h(ac)o(kno)o(wledged.)89 957 y Fx(References)114 1097 y Fz([1])23 b(A)o(dam)o(y)o(an,)f(V.M.;)h(Neidhardt,)g(H.:)33 b(On)22 b(the)g(absolutely)g(con)o(tin)o(uous)h(subspace)g(for)189 1157 y(non-selfadjoin)o(t)17 b(op)q(erators.)g Fc(Math.)g(Nachr.)f Fq(210)p Fz(,)g(5-42)i(\(2000\).)114 1273 y([2])23 b(Baumg\344rtel,)g (H.;)h(W)l(ollen)o(b)q(erg,)f(M.:)34 b Fc(Mathematic)n(al)23 b(sc)n(attering)h(the)n(ory)p Fz(,)f(Ak)m(ademie)189 1333 y(V)l(erlag,)16 b(Berlin)e(1983.)114 1449 y([3])23 b(Birman,)i(M.S.;)j(Solom)o(y)o(ak,)c(M.Z.:)38 b Fc(Sp)n(e)n(ctr)n(al) 25 b(the)n(ory)f(of)h(self-adjoint)i(op)n(er)n(ators)c(in)189 1509 y(Hilb)n(ert)18 b(sp)n(ac)n(e)p Fz(,)e(D.)g(Reidel)f(Publishing)h (Co.,)g(Dordrec)o(h)o(t,)f(1987.)114 1625 y([4])23 b(Ben)29 b(Ab)q(dallah,)j(N.;)j(Degond,)e(P)l(.,)e(Mark)o(o)o(wic)o(h,)g(P)l (.A.:)46 b(On)30 b(a)f(one-dimensional)189 1685 y(Sc)o(hr\366)q (dinger-P)o(oisson)16 b(scattering)e(mo)q(del.)f Fc(Z.)i(A)o(ngew.)j (Math.)d(Phys.)f Fq(48)h Fz(\(1997\),)g(No.1,)189 1745 y(135-155.)114 1861 y([5])23 b(Ben)g(Ab)q(dallah,)h(N.:)34 b(A)23 b(h)o(ybrid)g(kinetic-quan)o(tum)d(mo)q(del)i(for)h(stationary)h (electron)189 1921 y(transp)q(ort.)18 b Fc(J.)f(Stat.)h(Phys.)d Fq(90)i Fz(\(1998\),)g(No.3-4,)f(627-662.)114 2037 y([6])23 b(Ben)15 b(Ab)q(dallah,)f(N.;)g(Ke\034,)h(J.:)20 b(Limite)12 b(semi-classique)h(du)i(probleme)e(de)h(Sc)o(hro)q(edinger)189 2097 y(a)o(v)o(ec)h(masse)h(v)m(ariable.)f Fc(C.)i(R.)g(A)n(c)n(ad.)f (Sci.,)i(Paris,)f(Ser.)g(I,)g(Math.)f Fq(331)p Fz(,)g(No.2,)f(165-170) 189 2158 y(\(2000\).)114 2273 y([7])23 b(Buslaev,)15 b(V.;)f(F)l(omin,)f(V.:)20 b(An)15 b(in)o(v)o(erse)e(scattering)i (problem)f(for)h(the)g(one-dimensional)189 2334 y(Sc)o(hr\366)q(dinger) 20 b(equation)g(on)g(the)g(en)o(tire)e(axis.)i Fc(V)l(estnik)i(L)n (eningr)n(ad.)e(Univ.)h Fq(17)p Fz(,)g(No.)e(1,)189 2394 y(56\02564)g(\(1962\).)114 2509 y([8])k(F)l(oias,)13 b(C.;)g(Sz.-Nagy)l(,)f(B.:)18 b Fc(Harmonic)13 b(analysis)h(of)f(op)n (er)n(ators)f(on)i(Hilb)n(ert)g(sp)n(ac)n(e)p Fz(,)e(North-)189 2570 y(Holland)k(Publishing)g(Compan)o(y)l(,)f(Amsterdam-London,)g (1970.)114 2685 y([9])23 b(Kaiser,)16 b(H.-Ch.;)g(Reh)o(b)q(erg,)g(J.:) 21 b(Ab)q(out)c(a)g(one-dimensional)e(stationary)j(Sc)o(hr\366)q (dinger-)189 2746 y(P)o(oisson)i(system)e(with)h(K)o(ohn-Sham)g(p)q (oten)o(tial.)f Fc(Z.)i(A)o(ngew.)i(Math.)d(Phys.)g Fq(50)p Fz(,)g(No.3,)189 2806 y(423-458)g(\(1999\).)p eop %%Page: 29 31 29 30 bop 89 23 a Fs(Density)18 b(and)g(current)h(of)f(a)g(dissip)l(a)l (tive)g(Schr)o(\366dinger)h(opera)l(tor)187 b Fz(29)89 147 y([10])24 b(Kaiser,)d(H.-Ch.;)g(Reh)o(b)q(erg,)f(J.:)29 b(Ab)q(out)21 b(a)f(stationary)h(Sc)o(hr\366)q(dinger-P)o(oisson)g (system)189 208 y(with)i(K)o(ohn-Sham)f(p)q(oten)o(tial)h(in)f(a)i(b)q (ounded)f(t)o(w)o(o-)g(or)g(three-dimensional)e(domain.)189 268 y Fc(Nonline)n(ar)d(A)o(nal.,)g(The)n(ory)f(Metho)n(ds)g(Appl.)f Fq(41A)p Fz(,)g(No.)g(1-2,)h(33-72)h(\(2000\).)89 369 y([11])24 b(Kaiser,)43 b(H-Ch.;)49 b(Neidhardt,)43 b(H.;)48 b(Reh)o(b)q(erg,)43 b(J.:)65 b 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b(systems)d(are)h(of)h(great)g(in)o(terest)e(in)h(semiconductor)f(ph)o (ysics.)24 b(In)17 b(the)89 347 y(follo)o(wing)c(w)o(e)f(are)h(in)o (terested)e(in)h(a)h(Sc)o(hr\366)q(dinger-P)o(oisson)h(system)d(on)j (the)e(in)o(terv)m(al)g Fp(\012)i(=)g([)p Fo(a;)8 b(b)p Fp(])89 408 y Fz(on)16 b(the)g(real)f(axis)g Fn(R)p Fz(.)j(Systems)c (of)i(this)g(t)o(yp)q(e)f(w)o(ere)f(considered)h(in)h([9].)k(By)15 b Fo(')h Fz(w)o(e)f(denote)g(the)89 468 y(electrostatic)g(p)q(oten)o (tial)h(on)h Fp(\012)f Fz(whic)o(h)g(is)g(determined)d(b)o(y)j(P)o (oisson's)h(equation)484 602 y Fm(\000)542 568 y Fo(d)p 528 591 54 2 v 528 636 a(dx)586 602 y(\017)p Fp(\()p Fo(x)p Fp(\))690 568 y Fo(d)p 677 591 V 677 636 a(dx)735 602 y(')c Fp(=)h Fo(q)864 562 y Fl(\000)887 602 y Fo(C)t Fp(\()p Fo(x)p Fp(\))c(+)h Fo(u)1079 581 y Fk(+)1119 602 y Fm(\000)g Fo(u)1197 581 y Fj(\000)1226 562 y Fl(\001)1258 602 y Fo(;)56 b(x)14 b Fm(2)g Fp(\012)p Fo(;)295 b Fz(\(1.1\))89 731 y(where)16 b Fo(u)258 712 y Fk(+)303 731 y Fz(and)h 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1350 y(The)12 b(second)g(imp)q(ortan)o(t)f (ingredien)o(t)g(of)h(the)g(Sc)o(hr\366)q(dinger-P)o(oisson)h(system)d (are)i(Sc)o(hr\366dinger-)89 1410 y(t)o(yp)q(e)20 b(op)q(erators)h Fo(H)k Fz(whic)o(h)19 b(are)h(de\034ned)g(on)g(the)g(Hilb)q(ert)f (space)h Fh(H)g Fp(=)h Fo(L)1500 1392 y Fk(2)1519 1410 y Fp(\([)p Fo(a;)8 b(b)p Fp(]\))19 b Fz(and)h(lo)q(ok)89 1471 y(lik)o(e)702 1545 y Fo(H)e Fp(:=)c Fm(\000)870 1512 y Fp(1)p 870 1534 25 2 v 870 1580 a(2)918 1512 y Fo(d)p 904 1534 54 2 v 904 1580 a(dx)1009 1512 y Fp(1)p 967 1534 109 2 v 967 1580 a Fo(m)p Fp(\()p Fo(x)p Fp(\))1099 1512 y Fo(d)p 1085 1534 54 2 v 1085 1580 a(dx)1154 1545 y Fp(+)d Fo(V)s(;)513 b Fz(\(1.4\))89 1660 y(where)19 b(the)g(real)f(p)q(oten)o(tial)h Fo(V)30 b Fz(is)19 b(sligh)o(tly)f (di\033eren)o(t)g(for)h(holes)g(and)g(electrons,)g(see)f(Section)89 1720 y(5,)25 b(and)e Fo(m)g Fz(equals)f(either)g(to)i(the)e Fo(x)p Fz(-dep)q(enden)o(t)h(e\033ectiv)o(e)d(mass)j(of)g(the)g(holes)g (or)g(of)g(the)89 1781 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Fp(\))g(=)h Fz(tr)p Fp(\()p Fo(f)5 b Fp(\()p Fo(H)t Fp(\)\))p Fz(.)28 b(Usually)l(,)17 b(non-)89 598 y(negativ)o(e)11 b(self-adjoin)o(t)h(n)o(uclear)f(op)q (erators)i(are)f(called)f(densit)o(y)g(matrices.)18 b(Since)11 b Fo(\045)h Fz(comm)o(utes)89 658 y(with)18 b Fo(H)j Fz(the)d(densit)o(y)e(matrix)g Fo(\045)i Fz(remains)e(unc)o(hanged)i (in)f(time.)23 b(In)18 b(this)f(case)h Fo(\045)g Fz(is)f(called)f(a)89 718 y(steady)j(state.)28 b(If)18 b(the)g(densit)o(y)f(matrix)g(is)h (kno)o(wn,)h(then)f(one)h(restores)f(the)g(carrier)g(densit)o(y)89 778 y(as)f(follo)o(ws:)k(the)16 b(n)o(um)o(b)q(er)f(of)h(carriers)g Fo(N)858 785 y Fi(\045)878 778 y Fp(\()p Fo(!)r Fp(\))g Fz(in)g(the)g(set)g Fo(!)g Fm(\022)e Fp([)p Fo(a;)8 b(b)p Fp(])14 b Fz(is)i(computed)f(b)o(y)792 883 y Fo(N)831 890 y Fi(\045)852 883 y Fp(\()p Fo(!)r Fp(\))f(=)f Fz(tr)p Fp(\()p Fo(\045\037)1100 890 y Fi(!)1125 883 y Fp(\))p Fo(;)603 b Fz(\(1.9\))89 988 y(where)11 b Fo(\037)256 995 y Fi(!)281 988 y Fp(\()p Fm(\001)p Fp(\))f Fz(is)h(the)g(indicator) g(function)f(of)i(the)e(subset)i Fo(!)r Fz(.)19 b(It)11 b(turns)g(out)h(that)f(\(1.9\))g(de\034nes)g(an)89 1048 y(absolutely)j(con)o(tin)o(uous)f(measure)g(\(with)g(resp)q(ect)h(to)g (the)f(Leb)q(esgue)i(measure\).)k(One)13 b(v)o(eri\034es)89 1108 y(that)k(its)f(Radon-Nik)o(o)q(dym)f(deriv)m(ativ)o(e)g Fo(u)880 1115 y Fi(\045)916 1108 y Fz(coincides)h(with)g(the)g(carrier) g(densit)o(y)f(de\034ned)h(b)o(y)89 1169 y(\(1.5\),)g(i.e.)k Fo(u)334 1176 y Fk(^)-21 b Fi(\045)365 1169 y Fp(=)14 b Fo(u)445 1176 y Fi(\045)465 1169 y Fz(.)148 1252 y(The)22 b(system)e(describ)q(ed)i(b)o(y)f Fo(H)26 b Fz(is)c(closed.)38 b(Hence)20 b(there)i(is)f(no)i(in)o(teraction)d(with)i(the)89 1313 y(en)o(vironmen)o(t,)15 b(in)i(particular,)g(no)h(carrier)f(exc)o (hange.)24 b(Consequen)o(tly)l(,)17 b(the)g(curren)o(t)g(densit)o(y)89 1373 y Fo(j)112 1380 y Fk(^)-21 b Fi(\045)129 1373 y Fp(\()p Fo(x)p Fp(\))16 b Fz(de\034ned)g(b)o(y)640 1472 y Fo(j)663 1479 y Fk(^)-21 b Fi(\045)680 1472 y Fp(\()p Fo(x)p Fp(\))14 b(:=)843 1410 y Fj(1)825 1425 y Fl(X)833 1531 y Fi(l)p Fk(=1)905 1472 y Fo(\045)930 1479 y Fi(l)943 1472 y Fo(j)963 1479 y Fi(l)977 1472 y Fp(\()p Fo(x)p Fp(\))p Fo(;)56 b(x)13 b Fm(2)h Fp([)p Fo(a;)8 b(b)p Fp(])p Fo(;)424 b Fz(\(1.10\))89 1603 y(is)24 b(iden)o(tical)e(zero,)j (where)e(in)g(accordance)h(with)g([15])f(the)h(curren)o(t)f(densit)o(y) f Fo(j)1627 1610 y Fi(l)1641 1603 y Fp(\()p Fo(x)p Fp(\))h Fz(of)h(the)89 1663 y(eigenstate)16 b Fo( )349 1670 y Fi(l)378 1663 y Fz(is)g(giv)o(en)f(b)o(y)379 1793 y Fo(j)399 1800 y Fi(l)412 1793 y Fp(\()p Fo(x)p Fp(\))f(:=)f Fm(=)p Fz(m)642 1723 y Fl(\022)725 1760 y Fp(1)p 684 1782 109 2 v 684 1828 a Fo(m)p Fp(\()p Fo(x)p Fp(\))797 1793 y Fo( )831 1773 y Fj(0)829 1806 y Fi(l)842 1793 y Fp(\()p Fo(x)p Fp(\))p 908 1750 111 2 v Fo( )940 1800 y Fi(l)952 1793 y Fp(\()p Fo(x)p Fp(\))1018 1723 y Fl(\023)1063 1793 y Fo(;)22 b(x)13 b Fm(2)h Fp([)p Fo(a;)8 b(b)p Fp(])p Fo(;)20 b(l)15 b Fp(=)e(1)p Fo(;)8 b Fp(2)p Fo(;)g(:)g(:)g(:)g(:)165 b Fz(\(1.11\))89 1924 y(Ho)o(w)o(ev)o(er,)12 b(from)f(the)i(p)q(oin)o (t)g(of)h(view)e(of)h(semiconductor)e(ph)o(ysics)i(this)g(consequence)f (is)g(totally)89 1984 y(unacceptable)19 b(since)f(a)i(net)f(curren)o(t) f(\035o)o(w)h(through)i(the)d(b)q(oundary)j(is)e(natural.)30 b(Th)o(us)20 b(one)89 2044 y(has)d(to)g(devise)e(b)q(oundary)i (conditions)g(whic)o(h)e(allo)o(w)h(those)h(\035o)o(ws.)148 2128 y(A)23 b(simple)f(prop)q(osal)j(w)o(as)f(made)e(in)h([9])g(whic)o (h)g(adds)i(up)e(to)h(replace)f(the)g(self-adjoin)o(t)89 2188 y(b)q(oundary)18 b(conditions)e(b)o(y)g(non-selfadjoin)o(t)g (ones,)g(i.e.)k(to)d(consider)f Fo(H)k Fz(on)d(the)f(domain)360 2365 y(dom)p Fp(\()p Fo(H)t Fp(\))e(=)600 2248 y Fl(8)600 2293 y(>)600 2308 y(<)600 2397 y(>)600 2412 y(:)645 2365 y Fo(g)i Fm(2)e Fo(W)784 2344 y Fk(1)p Fi(;)p Fk(2)831 2365 y Fp(\([)p Fo(a;)8 b(b)p Fp(]\))k(:)1061 2269 y Fk(1)p 1031 2277 79 2 v 1031 2306 a Fi(m)p Fk(\()p Fi(x)p Fk(\))1114 2289 y Fo(g)1139 2271 y Fj(0)1151 2289 y Fp(\()p Fo(x)p Fp(\))i Fm(2)g Fo(W)1331 2271 y Fk(1)p Fi(;)p Fk(2)1378 2289 y Fp(\([)p Fo(a;)8 b(b)p Fp(]\))p Fo(;)1070 2341 y Fk(1)p 1031 2349 96 2 v 1031 2378 a(2)p Fi(m)p Fk(\()p Fi(a)p Fk(\))1131 2361 y Fo(g)1156 2343 y Fj(0)1168 2361 y Fp(\()p Fo(a)p Fp(\))13 b(=)h Fm(\000)p Fo(\024)1364 2368 y Fi(a)1384 2361 y Fo(g)r Fp(\()p Fo(a)p Fp(\))p Fo(;)1068 2413 y Fk(1)p 1031 2421 92 2 v 1031 2450 a(2)p Fi(m)p Fk(\()p Fi(b)p Fk(\))1127 2433 y Fo(g)1152 2415 y Fj(0)1164 2433 y Fp(\()p Fo(b)p Fp(\))g(=)f Fo(\024)1316 2440 y Fi(b)1334 2433 y Fo(g)r Fp(\()p Fo(b)p Fp(\))1545 2248 y Fl(9)1545 2293 y(>)1545 2308 y(=)1545 2397 y(>)1545 2412 y(;)1736 2365 y Fz(\(1.12\))89 2541 y(where)k Fo(\024)259 2548 y Fi(a)279 2541 y Fo(;)8 b(\024)329 2548 y Fi(b)361 2541 y Fm(2)15 b Fn(C)9 b Fz(.)26 b(If)16 b(at)h(least)g(one)g(of)g (the)f(imaginary)g(parts)h(is)f(di\033eren)o(t)g(from)g(zero,)g(then)89 2602 y(the)g(op)q(erator)i Fo(H)j Fz(is)16 b(non-selfadjoin)o(t.)22 b(Ho)o(w)o(ev)o(er,)14 b(this)i(non-selfadjoin)o(tness)h(implies)d(sev) o(eral)89 2662 y(complications.)20 b(In)15 b(particular,)h(the)g (notion)h(of)f(carrier)g(densities)f(b)q(ecomes)g(unclear.)148 2746 y(This)e(situation)g(can)g(b)q(e)h(handled)e(if)h(w)o(e)f (restrict)g(ourselv)o(es)g(to)i(dissipativ)o(e)d(op)q(erators.)22 b(Let)89 2806 y(us)15 b(recall)f(that)h(an)h(op)q(erator)g(is)e(called) g(dissipativ)o(e)g(if)g(the)g(imaginary)g(part)h(of)g(its)g(asso)q (ciated)p eop %%Page: 4 6 4 5 bop 89 23 a Fz(4)716 b Fs(H.-Ch.)25 b(Kaiser,)18 b(H.)h(Neidhardt,)g(J.)g(Rehber)o(g)89 147 y Fz(quadratic)f(form)e(is)h (non-p)q(ositiv)o(e.)26 b(In)17 b(the)g(presen)o(t)g(case)h(the)f(op)q (erator)i Fo(H)j Fz(is)17 b(dissipativ)o(e)f(if)89 208 y Fo(\024)117 215 y Fi(a)138 208 y Fo(;)8 b(\024)188 215 y Fi(b)225 208 y Fm(2)19 b Fn(C)310 215 y Fk(+)363 208 y Fp(:=)g Fm(f)p Fo(z)i Fm(2)f Fn(C)31 b Fp(:)19 b Fm(=)p Fz(m)o Fp(\()p Fo(z)r Fp(\))g Fo(>)h Fp(0)p Fm(g)p Fz(,)g(see)f([11)q(].)30 b(Moreo)o(v)o(er,)19 b(under)h(this)f(assumption)89 268 y(the)h(op)q(erator)h Fo(H)j Fz(b)q(ecomes)18 b(maximal)f(dissipativ)o(e,)h(i.e.)31 b(it)19 b(admits)g(no)h(prop)q(er)g(dissipativ)o(e)89 328 y(extension,)12 b(see)f([11].)19 b(The)12 b(main)e(tec)o(hnical)f (to)q(ol)j(to)g(o)o(v)o(ercome)d(the)i(di\036culties)f(is)h(the)g (dilation)89 388 y(theory)k(for)h(maximal)c(dissipativ)o(e)i(op)q (erator.)22 b(In)15 b([12])g(the)g(minim)o(al)d(self-adjoin)o(t)j (dilation)g Fo(K)89 448 y Fz(of)j Fo(H)j Fz(w)o(as)d(explicitly)d (constructed)i(and)h(analysed)f(in)g(detail.)24 b(F)l(rom)16 b(the)h(ph)o(ysical)f(p)q(oin)o(t)i(of)89 509 y(view)f(the)g(minim)o (al)e(self-adjoin)o(t)h(dilation)h(pla)o(ys)g(the)g(role)g(of)h(the)f (Hamiltonian)f(of)h(a)h(larger)89 569 y(closed)h(system)e(whic)o(h)i (con)o(tains)g(the)f(original)h(system)f(describ)q(ed)g(b)o(y)h Fo(H)t Fz(.)29 b(Using)19 b(this)g(fact)89 629 y(one)f(de\034nes)g (steady)g(states,)h(carrier)e(densities)g(and)i(curren)o(t)e (densities.)26 b(It)17 b(turns)i(out)f(that)89 689 y(the)f(curren)o(t)f (densit)o(y)g(is)h(indep)q(enden)o(t)g(of)g Fo(x)e Fm(2)h Fp([)p Fo(a;)8 b(b)p Fp(])15 b Fz(and,)j(in)f(general,)f(di\033eren)o (t)h(from)f(zero.)89 749 y(So)22 b(w)o(e)e(ha)o(v)o(e)g(a)i(constan)o (t)f(curren)o(t)f(through)i Fp([)p Fo(a;)8 b(b)p Fp(])20 b Fz(whic)o(h)g(is)h(quite)f(satisfactory)h(from)f(the)89 809 y(ph)o(ysical)15 b(p)q(oin)o(t)i(of)f(view.)148 893 y(It)i(turns)h(out)g(that)f(this)h(formal)e(approac)o(h)i(to)g(the)f (net)g(curren)o(t)f(\035o)o(w)i(problem)e(\034ts)h(in)o(to)89 953 y(mo)q(dels)e(discussed)i(in)f(the)g(literature.)23 b(Indeed,)17 b(a)g(w)o(ell)f(adopted)i(mo)q(del)e(in)h(semiconductor)89 1014 y(ph)o(ysics)d(is)g(the)g(so-called)g(Kirkner-Len)o(t)f(mo)q(del)g ([14])h(whic)o(h)g(w)o(as)g(mathematically)d(analysed)89 1074 y(in)k([4],)f([5])g(and)h([6].)20 b(In)15 b(this)f(mo)q(del)g(one) h(replaces)f(the)g(maximal)e(dissipativ)o(e)h(op)q(erator)j Fo(H)j Fz(b)o(y)89 1134 y(a)d(family)d(of)i(maximal)d(dissipativ)o(e)i (op)q(erators)i Fm(f)p Fo(H)t Fp(\()p Fo(z)r Fp(\))p Fm(g)1138 1147 y Fi(z)q Fj(2)p 1179 1120 50 2 v Fg(C)1201 1151 y Ff(+)1246 1134 y Fz(de\034ned)f(on)h Fh(H)d Fp(=)h Fo(L)1614 1116 y Fk(2)1634 1134 y Fp(\([)p Fo(a;)8 b(b)p Fp(]\))13 b Fz(and)89 1194 y(giv)o(en)j(b)o(y)269 1335 y(dom)o Fp(\()p Fo(H)t Fp(\()p Fo(z)r Fp(\)\))41 b(:=)640 1264 y Fl(\032)678 1335 y Fo(g)16 b Fm(2)e Fo(W)817 1317 y Fk(2)p Fi(;)p Fk(2)864 1335 y Fp(\([0)p Fo(;)8 b Fp(1]\))13 b(:)1061 1304 y Fo(g)1086 1286 y Fj(0)1098 1304 y Fp(\(1\))h(=)g Fo(\024)1254 1311 y Fk(1)1274 1304 y Fp(\()p Fo(z)r Fp(\))p Fo(g)r Fp(\(1\))1061 1364 y Fo(g)1086 1346 y Fj(0)1098 1364 y Fp(\(0\))g(=)g Fm(\000)p Fo(\024)1293 1371 y Fk(0)1313 1364 y Fp(\()p Fo(z)r Fp(\))p Fo(g)r Fp(\(0\))p Fo(;)1497 1264 y Fl(\033)1543 1335 y Fo(;)269 1471 y Fp(\()p Fo(H)t Fp(\()p Fo(z)r Fp(\))p Fo(g)r Fp(\)\()p Fo(x)p Fp(\))42 b(:=)f Fm(\000)694 1451 y Fi(d)712 1440 y Ff(2)p 684 1459 56 2 v 684 1488 a Fi(dx)722 1479 y Ff(2)744 1471 y Fo(g)r Fp(\()p Fo(x)p Fp(\))11 b(+)g Fo(V)h Fp(\()p Fo(x)p Fp(\))p Fo(g)r Fp(\()p Fo(x)p Fp(\))p Fo(;)56 b(g)16 b Fm(2)e Fz(dom)o Fp(\()p Fo(H)t Fp(\()p Fo(z)r Fp(\)\))p Fo(;)1736 1388 y Fz(\(1.13\))89 1584 y Fo(m)p Fp(\()p Fo(x)p Fp(\))f Fm(\021)h Fp(1)p Fo(=)p Fp(2)p Fz(,)j(with)433 1703 y Fo(\024)461 1710 y Fk(1)481 1703 y Fp(\()p Fo(z)r Fp(\))c(:=)h Fo(i)640 1665 y Fm(p)p 681 1665 25 2 v 681 1703 a Fo(z)50 b Fz(and)g Fo(\024)910 1710 y Fk(0)930 1703 y Fp(\()p Fo(z)r Fp(\))13 b(:=)h Fo(i)1089 1659 y Fl(p)p 1138 1659 144 2 v 1138 1703 a Fo(z)f Fm(\000)e Fo(V)1252 1710 y Fj(\000)1282 1703 y Fo(;)56 b(z)16 b Fm(2)p 1438 1663 66 2 v 14 w Fn(C)1471 1710 y Fk(+)1504 1703 y Fo(;)218 b Fz(\(1.14\))89 1817 y(where)17 b(the)g(cut)h(of)f(the)g(square)h(ro)q(ot)g(is)g(along)g Fp([0)p Fo(;)8 b Fm(1)p Fp(\))17 b Fz(and)h Fm(=)p Fz(m)n Fp(\()1322 1781 y Fm(p)p 1364 1781 25 2 v 36 x Fo(z)r Fp(\))e Fm(\025)f Fp(0)j Fz(for)g Fo(z)f Fm(2)f Fn(C)1718 1824 y Fk(+)1751 1817 y Fz(.)25 b(Let)89 1877 y(us)17 b(consider)f(the)g(self-adjoin)o(t)g(Sc)o(hr\366)q(dinger)g(op)q (erator)h Fo(K)t Fz(,)391 1987 y(dom)o Fp(\()p Fo(K)t Fp(\))44 b(:=)d Fo(W)756 1969 y Fk(2)p Fi(;)p Fk(2)803 1987 y Fp(\()p Fn(R)p Fp(\))p Fo(;)391 2076 y Fp(\()p Fo(K)t(f)5 b Fp(\)\()p Fo(x)p Fp(\))41 b(:=)g Fm(\000)757 2057 y Fi(d)775 2045 y Ff(2)p 747 2065 56 2 v 747 2094 a Fi(dx)785 2084 y Ff(2)807 2076 y Fo(f)5 b Fp(\()p Fo(x)p Fp(\))11 b(+)g Fo(V)990 2083 y Fi(K)r(L)1049 2076 y Fp(\()p Fo(x)p Fp(\))p Fo(f)5 b Fp(\()p Fo(x)p Fp(\))p Fo(;)56 b(f)20 b Fm(2)14 b Fz(dom)o Fp(\()p Fo(K)t Fp(\))p Fo(;)1736 2032 y Fz(\(1.15\))89 2192 y(on)j(the)f(Hilb)q(ert)f(space)h Fh(K)e Fp(=)g Fo(L)671 2174 y Fk(2)691 2192 y Fp(\()p Fn(R)p Fp(\))f Fz(whose)k(p)q(oten)o(tial)f Fo(V)1159 2199 y Fi(K)r(L)1231 2192 y Fm(2)e Fo(C)t Fp(\()p Fn(R)p Fp(\))f Fz(lo)q(oks)k(lik)o(e)635 2365 y Fo(V)663 2372 y Fi(K)r(L)735 2365 y Fp(=)787 2263 y Fl(8)787 2308 y(<)787 2398 y(:)852 2305 y Fo(V)880 2312 y Fj(\000)959 2305 y Fp(:)48 b Fo(x)13 b Fm(2)h Fn(R)1148 2312 y Fj(\000)852 2365 y Fo(V)79 b Fp(:)48 b Fo(x)13 b Fm(2)h Fp([0)p Fo(;)8 b Fp(1])852 2425 y(0)83 b(:)48 b Fo(x)13 b Fm(2)h Fp(\(1)p Fo(;)8 b Fp(+)p Fm(1)p Fp(\))1736 2365 y Fz(\(1.16\))89 2535 y(where)16 b Fo(V)258 2542 y Fj(\000)302 2535 y Fo(>)e Fp(0)p Fz(,)672 2600 y Fo(V)d Fp(\(0\))j(=)g Fo(V)867 2607 y Fj(\000)946 2600 y Fz(and)49 b Fo(V)11 b Fp(\(1\))j(=)g(0)p Fo(:)458 b Fz(\(1.17\))89 2692 y(Op)q(erators)17 b(of)g(this)f(t)o(yp)q (e)g(w)o(ere)f(in)o(v)o(estigate)g(in)h([7].)k(It)c(turns)h(out)g(that) 521 2806 y Fo(P)559 2785 y Fe(K)552 2818 y(H)584 2806 y Fp(\()p Fo(K)e Fm(\000)c Fo(z)r Fp(\))753 2785 y Fj(\000)p Fk(1)800 2806 y Fm(j)p Fh(H)j Fp(=)g(\()p Fo(H)t Fp(\()p Fo(z)r Fp(\))d Fm(\000)g Fo(z)r Fp(\))1147 2785 y Fj(\000)p Fk(1)1194 2806 y Fo(;)56 b(z)16 b Fm(2)e Fn(C)1383 2813 y Fk(+)1415 2806 y Fo(:)307 b Fz(\(1.18\))p eop %%Page: 5 7 5 6 bop 89 23 a Fs(Density)18 b(and)g(current)h(of)f(a)g(dissip)l(a)l (tive)g(Schr)o(\366dinger)h(opera)l(tor)211 b Fz(5)89 147 y(whic)o(h)18 b(means)f(that)h(the)g(op)q(erator)h Fo(K)j Fz(can)d(b)q(e)f(regarded)g(as)h(a)f(self-adjoin)o(t)g(dilation) g(of)g(the)89 208 y(family)c Fm(f)p Fo(H)t Fp(\()p Fo(z)r Fp(\))p Fm(g)395 221 y Fi(z)q Fj(2)p 436 193 50 2 v Fg(C)457 225 y Ff(+)488 208 y Fz(.)21 b(Moreo)o(v)o(er,)15 b(it)g(holds)772 292 y Fl(_)746 400 y Fi(z)q Fj(2)p Fg(C)s Fj(n)p Fg(R)853 340 y Fp(\()p Fo(K)g Fm(\000)c Fo(z)r Fp(\))1022 319 y Fj(\000)p Fk(1)1069 340 y Fh(H)j Fp(=)g Fh(K)532 b Fz(\(1.19\))89 507 y(whic)o(h)15 b(means)g(that)h Fo(K)j Fz(is)d(a)g(minim)o(al)d(self-adjoin)o(t)i(dilation)g(and,)h(hence,)f (that)h Fo(K)j Fz(is)d(unique)89 567 y(up)22 b(to)f(a)h(certain)e (isomorphism.)34 b(Th)o(us,)22 b(if)e(w)o(e)h(w)o(an)o(t)g(to)h (compute)d(quan)o(tities)h(whic)o(h)h(are)89 627 y(related)13 b(to)h(the)g(subspace)g Fh(H)p Fz(,)f(for)h(instance,)g(carrier)e (densities)h(on)h(the)g(in)o(terv)m(al)e Fp([0)p Fo(;)c Fp(1])p Fz(,)13 b(w)o(e)h(can)89 687 y(do)h(this)f(using)g(either)f (the)h(self-adjoin)o(t)g(op)q(erator)h Fo(K)j Fz(or)d(the)f(family)e Fm(f)p Fo(H)t Fp(\()p Fo(\025)p Fp(\))p Fm(g)1540 694 y Fi(\025)p Fj(2)p Fg(R)1625 687 y Fz(of)i(maximal)89 747 y(dissipativ)o(e)h(op)q(erators.)22 b(Moreo)o(v)o(er,)15 b(if)g(w)o(e)h(are)g(only)g(in)o(terested)f(in)g(quan)o(tities)g(with)h (resp)q(ect)89 808 y(to)k(a)g(small)e(energy)h(in)o(terv)m(al)f(around) j(the)e(energy)g Fo(\025)1108 815 y Fk(0)1148 808 y Fm(2)g Fn(R)e Fz(\(in)i(the)g(limit)e(only)i(quan)o(tities)89 868 y(for)g(the)f(energy)h Fo(\025)439 875 y Fk(0)459 868 y Fz(\),)g(then)f(it)g(is)h(enough)g(to)g(consider)f(the)h(maximal) c(dissipativ)o(e)j(op)q(erator)89 928 y Fo(H)t Fp(\()p Fo(\025)180 935 y Fk(0)201 928 y Fp(\))e Fz(whic)o(h)f(leads)h(to)h(a)g (mo)q(del)e(describ)q(ed)g(at)i(the)f(b)q(eginning.)148 1012 y(The)f(pap)q(er)h(is)f(organized)h(as)g(follo)o(ws.)k(In)15 b(Section)g(2)h(w)o(e)e(rigorously)i(de\034ne)f(Sc)o(hr\366)q(dinger-) 89 1072 y(t)o(yp)q(e)j(op)q(erators)h(and)g(brie\035y)e(recall)g(their) h(prop)q(erties,)g(cf.)26 b([11].)h(Moreo)o(v)o(er,)17 b(w)o(e)g(in)o(tro)q(duce)89 1132 y(the)h(imp)q(ortan)o(t)g(notion)h (of)f(the)g(c)o(haracteristic)f(function)h Fp(\002)1238 1139 y Fi(H)1272 1132 y Fp(\()p Fo(z)r Fp(\))g Fz(and)h(brie\035y)e (describ)q(e)h(the)89 1192 y(minimal)8 b(self-adjoin)o(t)j(dilation)f Fo(K)16 b Fz(of)11 b Fo(H)t Fz(.)20 b(F)l(or)11 b(further)g(purp)q (oses)h(w)o(e)f(indicate)f(its)h(generalized)89 1253 y(eigenfunction)k(expansion.)21 b(Essen)o(tially)l(,)14 b(w)o(e)i(follo)o(w)f(here)g(the)g(pap)q(er)i([12].)k(In)15 b(Section)g(3)h(w)o(e)89 1313 y(in)o(tro)q(duce)22 b(generalized)f (steady)h(states)g(and)h(de\034ne)f(carrier)f(densities.)38 b(In)21 b(Section)h(4)g(w)o(e)89 1373 y(de\034ne)e(the)g(curren)o(t)f (densit)o(y)h(and)g(compute)f(it)h(in)g(terms)e(of)j(the)f(c)o (haracteristic)f(function.)89 1433 y(In)g(particular,)f(it)g(turns)h (out)g(that)h(if)e(the)g(steady)h(state)g(is)f(giv)o(en)g(b)o(y)g Fo(\045)g Fp(=)g Fo(f)5 b Fp(\()p Fo(K)t Fp(\))p Fz(,)20 b(then)e(the)89 1493 y(curren)o(t)k(densit)o(y)g(is)h(zero.)42 b(In)22 b(Section)h(5)g(w)o(e)g(commen)o(t)d(the)j(results,)h(in)e (particular,)i(w)o(e)89 1554 y(clarify)18 b(the)i(relation)f(to)h(the)f (Lax-Phillips)g(scattering)g(theory)h(and)g(v)o(erify)e(the)h(con)o (tin)o(uit)o(y)89 1614 y(equation.)35 b(Finally)l(,)21 b(w)o(e)f(giv)o(e)g(a)h(rigorous)h(setting)f(of)g(so-called)g (dissipativ)o(e)f(Sc)o(hr\366)q(dinger-)89 1674 y(P)o(oisson)d(systems) e(whic)o(h)h(ha)o(v)o(e)g(the)g(adv)m(an)o(tage)h(that)g(their)e (curren)o(t)h(densities)f(are)i(di\033eren)o(t)89 1734 y(from)e(zero)h(for)h(suitable)f(c)o(hosen)g(steady)g(states.)89 1932 y Fx(2)77 b(Sc)n(hr\366)r(dinger-t)n(yp)r(e)26 b(op)r(erators)89 2067 y Fz(F)l(ollo)o(wing)15 b(the)f(prop)q(osal)i(of)g([9])e(w)o(e)g (consider)h(the)g(non-selfadjoin)o(t)g(Sc)o(hr\366)q(dinger-t)o(yp)q(e) f(op)q(er-)89 2127 y(ator)j Fo(H)k Fz(on)16 b(the)g(Hilb)q(ert)f(space) i Fh(H)c Fp(=)h Fo(L)837 2109 y Fk(2)857 2127 y Fp(\([)p Fo(a;)8 b(b)p Fp(]\))14 b Fz(giv)o(en)i(b)o(y)g(the)g(domain)f (\(1.12\))i(and)592 2242 y Fp(\()p Fo(H)t(g)r Fp(\)\()p Fo(x)p Fp(\))d(=)g(\()p Fo(l)q Fp(\()p Fo(g)r Fp(\)\)\()p Fo(x)p Fp(\))p Fo(;)56 b(g)16 b Fm(2)e Fz(dom)o Fp(\()p Fo(H)t Fp(\))p Fo(;)403 b Fz(\(2.1\))89 2357 y(where)505 2434 y Fp(\()p Fo(l)q Fp(\()p Fo(g)r Fp(\)\)\()p Fo(x)p Fp(\))14 b(:=)f Fm(\000)811 2400 y Fp(1)p 811 2423 25 2 v 811 2468 a(2)859 2400 y Fo(d)p 845 2423 54 2 v 845 2468 a(dx)950 2400 y Fp(1)p 908 2423 109 2 v 908 2468 a Fo(m)p Fp(\()p Fo(x)p Fp(\))1040 2400 y Fo(d)p 1026 2423 54 2 v 1026 2468 a(dx)1084 2434 y(g)r Fp(\()p Fo(x)p Fp(\))e(+)g Fo(V)h Fp(\()p Fo(x)p Fp(\))p Fo(g)r Fp(\()p Fo(x)p Fp(\))p Fo(;)315 b Fz(\(2.2\))89 2558 y(and)12 b Fo(V)25 b Fm(2)14 b Fo(L)312 2540 y Fk(2)332 2558 y Fp(\([)p Fo(a;)8 b(b)p Fp(]\))h Fz(is)i(a)h(real)f(p)q(oten)o(tial,)g (the)g(e\033ectiv)o(e)e(mass)i Fo(m)p Fp(\()p Fo(x)p Fp(\))i Fo(>)h Fp(0)d Fz(ob)q(eys)h Fo(m)p Fp(\()p Fo(x)p Fp(\))q(+)1761 2539 y Fk(1)p 1730 2547 79 2 v 1730 2575 a Fi(m)p Fk(\()p Fi(x)p Fk(\))1828 2558 y Fm(2)89 2625 y Fo(L)122 2607 y Fj(1)160 2625 y Fp(\([)p Fo(a;)c(b)p Fp(]\))15 b Fz(and)i Fo(\024)433 2632 y Fi(a)454 2625 y Fo(;)8 b(\024)504 2632 y Fi(b)537 2625 y Fm(2)15 b Fn(C)618 2632 y Fk(+)650 2625 y Fz(.)24 b(The)17 b(op)q(erator)h Fo(H)j Fz(is)c(maximal)d(dissipativ)o(e)h(and)j(completely)89 2685 y(non-selfadjoin)o(t,)g(see)f([11].)25 b(The)17 b(sp)q(ectrum)g(of)g Fo(H)22 b Fz(consists)c(of)g(isolated)f(eigen)o(v) m(alues)g(in)g(the)89 2746 y(lo)o(w)o(er)j(half-plane)g(with)h(the)f (only)g(accum)o(ulation)f(p)q(oin)o(t)h(at)h(in\034nit)o(y)l(.)33 b(Since)19 b(the)i(op)q(erator)89 2806 y Fo(H)h Fz(is)17 b(completely)d(non-selfadjoin)o(t)j(there)g(do)g(not)h(exist)e(real)h (eigen)o(v)m(alues.)23 b(T)l(o)17 b(analyze)g(the)p eop %%Page: 6 8 6 7 bop 89 23 a Fz(6)716 b Fs(H.-Ch.)25 b(Kaiser,)18 b(H.)h(Neidhardt,)g(J.)g(Rehber)o(g)89 147 y Fz(op)q(erator)f Fo(H)i Fz(it)c(is)g(useful)g(to)g(in)o(tro)q(duce)g(the)g(elemen)o (tary)d(solutions)k Fo(v)1421 154 y Fi(a)1441 147 y Fp(\()p Fo(x;)8 b(z)r Fp(\))16 b Fz(and)h Fo(v)1689 154 y Fi(b)1706 147 y Fp(\()p Fo(x;)8 b(z)r Fp(\))p Fz(,)258 274 y Fo(l)q Fp(\()p Fo(v)317 281 y Fi(a)337 274 y Fp(\()p Fo(x;)g(z)r Fp(\)\))i Fm(\000)h Fo(z)r(v)578 281 y Fi(a)598 274 y Fp(\()p Fo(x;)d(z)r Fp(\))14 b(=)f(0)p Fo(;)57 b(v)895 281 y Fi(a)916 274 y Fp(\()p Fo(a;)8 b(z)r Fp(\))13 b(=)h(1)p Fo(;)1245 240 y Fp(1)p 1192 263 131 2 v 1192 308 a(2)p Fo(m)p Fp(\()p Fo(a)p Fp(\))1327 274 y Fo(v)1353 253 y Fj(0)1351 286 y Fi(a)1372 274 y Fp(\()p Fo(a;)8 b(z)r Fp(\))13 b(=)h Fm(\000)p Fo(\024)1615 281 y Fi(a)1635 274 y Fo(;)112 b Fz(\(2.3\))258 403 y Fo(l)q Fp(\()p Fo(v)317 410 y Fi(b)333 403 y Fp(\()p Fo(x;)8 b(z)r Fp(\)\))j Fm(\000)f Fo(z)r(v)574 410 y Fi(b)591 403 y Fp(\()p Fo(x;)e(z)r Fp(\))13 b(=)h(0)p Fo(;)57 b(v)888 410 y Fi(b)905 403 y Fp(\()p Fo(b;)8 b(z)r Fp(\))13 b(=)h(1)p Fo(;)1227 369 y Fp(1)p 1176 392 126 2 v 1176 437 a(2)p Fo(m)p Fp(\()p Fo(b)p Fp(\))1307 403 y Fo(v)1333 383 y Fj(0)1331 415 y Fi(b)1347 403 y Fp(\()p Fo(b;)8 b(z)r Fp(\))14 b(=)f Fo(\024)1546 410 y Fi(b)1564 403 y Fo(;)183 b Fz(\(2.4\))89 535 y Fo(x)16 b Fm(2)g Fp([)p Fo(a;)8 b(b)p Fp(])p Fz(,)16 b Fo(z)h Fm(2)g Fn(C)9 b Fz(,)20 b(whic)o(h)d(alw)o(a)o(ys)g(exist.)24 b(The)18 b(W)l(ronskian)f Fo(W)7 b Fp(\()p Fo(z)r Fp(\))18 b Fz(of)f Fo(v)1461 542 y Fi(a)1482 535 y Fp(\()p Fo(x;)8 b(z)r Fp(\))17 b Fz(and)h Fo(v)1732 542 y Fi(b)1748 535 y Fp(\()p Fo(x;)8 b(z)r Fp(\))89 596 y Fz(is)16 b(de\034ned)g(b)o(y)395 720 y Fo(W)7 b Fp(\()p Fo(z)r Fp(\))13 b(=)h Fo(v)600 727 y Fi(a)620 720 y Fp(\()p Fo(x;)8 b(z)r Fp(\))792 686 y(1)p 738 708 133 2 v 738 754 a(2)p Fo(m)p Fp(\()p Fo(x)p Fp(\))875 720 y Fo(v)901 699 y Fj(0)899 732 y Fi(b)916 720 y Fp(\()p Fo(x;)g(z)r Fp(\))i Fm(\000)h Fo(v)1113 727 y Fi(b)1130 720 y Fp(\()p Fo(x;)d(z)r Fp(\))1302 686 y(1)p 1248 708 V 1248 754 a(2)p Fo(m)p Fp(\()p Fo(x)p Fp(\))1385 720 y Fo(v)1411 699 y Fj(0)1409 732 y Fi(a)1429 720 y Fp(\()p Fo(x;)g(z)r Fp(\))p Fo(:)205 b Fz(\(2.5\))89 852 y(W)l(e)15 b(note)h(that)g(the)g(W)l(roskian)f(do)q(es)i(not)f(dep) q(end)g(on)g Fo(x)p Fz(.)k(Similarly)l(,)13 b(the)i(functions)g Fo(v)1710 859 y Fj(\003)p Fi(a)1748 852 y Fp(\()p Fo(x;)8 b(z)r Fp(\))89 912 y Fz(and)17 b Fo(v)208 919 y Fj(\003)p Fi(b)242 912 y Fp(\()p Fo(x;)8 b(z)r Fp(\))p Fz(,)377 1017 y Fo(v)401 1024 y Fj(\003)p Fi(a)439 1017 y Fp(\()p Fo(x;)g(z)r Fp(\))14 b(:=)p 631 973 157 2 v 13 w Fo(v)655 1024 y Fi(a)675 1017 y Fp(\()p Fo(x;)p 744 989 25 2 v 8 w(z)r Fp(\))48 b Fz(and)i Fo(v)988 1024 y Fj(\003)p Fi(b)1022 1017 y Fp(\()p Fo(x;)8 b(z)r Fp(\))13 b(:=)p 1214 973 154 2 v 14 w Fo(v)1238 1024 y Fi(b)1254 1017 y Fp(\()p Fo(x;)p 1323 989 25 2 v 8 w(z)r Fp(\))p Fo(;)56 b(z)16 b Fm(2)e Fn(C)9 b Fo(;)191 b Fz(\(2.6\))89 1121 y Fo(x)14 b Fm(2)g Fp([)p Fo(a;)8 b(b)p Fp(])p Fz(,)14 b Fo(z)i Fm(2)e Fn(C)9 b Fz(,)18 b(are)f(elemen)o(tary)c(solutions)k (of)223 1248 y Fo(l)q Fp(\()p Fo(v)282 1255 y Fj(\003)p Fi(a)319 1248 y Fp(\()p Fo(x;)8 b(z)r Fp(\)\))j Fm(\000)f Fo(z)r(v)560 1255 y Fj(\003)p Fi(a)598 1248 y Fp(\()p Fo(x;)e(z)r Fp(\))14 b(=)f(0)p Fo(;)57 b(v)895 1255 y Fj(\003)p Fi(a)933 1248 y Fp(\()p Fo(a;)8 b(z)r Fp(\))14 b(=)f(1)p Fo(;)1262 1214 y Fp(1)p 1209 1236 131 2 v 1209 1282 a(2)p Fo(m)p Fp(\()p Fo(a)p Fp(\))1345 1248 y Fo(v)1371 1227 y Fj(0)1369 1260 y(\003)p Fi(a)1407 1248 y Fp(\()p Fo(a;)8 b(z)r Fp(\))13 b(=)h Fm(\000)p 1622 1220 49 2 v Fo(\024)1650 1255 y Fi(a)1670 1248 y Fo(;)77 b Fz(\(2.7\))223 1400 y Fo(l)q Fp(\()p Fo(v)282 1407 y Fj(\003)p Fi(b)316 1400 y Fp(\()p Fo(x;)8 b(z)r Fp(\)\))i Fm(\000)h Fo(z)r(v)557 1407 y Fj(\003)p Fi(b)591 1400 y Fp(\()p Fo(x;)d(z)r Fp(\))13 b(=)h(0)p Fo(;)57 b(v)888 1407 y Fj(\003)p Fi(b)923 1400 y Fp(\()p Fo(b;)8 b(z)r Fp(\))13 b(=)h(1)p Fo(;)1244 1367 y Fp(1)p 1193 1389 126 2 v 1193 1435 a(2)p Fo(m)p Fp(\()p Fo(b)p Fp(\))1324 1400 y Fo(v)1350 1380 y Fj(0)1348 1413 y(\003)p Fi(b)1383 1400 y Fp(\()p Fo(b;)8 b(z)r Fp(\))13 b(=)p 1554 1373 46 2 v 14 w Fo(\024)1582 1407 y Fi(b)1599 1400 y Fo(;)148 b Fz(\(2.8\))89 1533 y Fo(x)22 b Fm(2)h Fp([)p Fo(a;)8 b(b)p Fp(])p Fz(.)35 b(The)21 b(W)l(ronskian)h(of)g Fp(\()p Fo(v)798 1540 y Fj(\003)p Fi(a)836 1533 y Fp(\()p Fo(x;)8 b(z)r Fp(\))21 b Fz(and)h Fo(v)1094 1540 y Fj(\003)p Fi(b)1128 1533 y Fp(\()p Fo(x;)8 b(z)r Fp(\))21 b Fz(is)g(denoted)g(b)o(y)g Fo(W)1624 1540 y Fj(\003)1644 1533 y Fp(\()p Fo(z)r Fp(\))g Fz(and)h(is)89 1593 y(also)16 b(indep)q(enden)o(t)e(of)h Fo(x)p Fz(.)21 b(By)14 b(these)g(elemen)o(tary)e(solutions)k(one)f(gets)g(for)h(the)e (resolv)o(en)o(ts)g(the)89 1653 y(represen)o(tations)203 1758 y Fp(\(\()p Fo(H)h Fm(\000)c Fo(z)r Fp(\))390 1737 y Fj(\000)p Fk(1)437 1758 y Fo(f)5 b Fp(\)\()p Fo(x)p Fp(\))14 b(=)1158 b Fz(\(2.9\))404 1864 y Fm(\000)448 1831 y Fo(v)472 1838 y Fi(b)489 1831 y Fp(\()p Fo(x;)8 b(z)r Fp(\))p 448 1853 154 2 v 467 1899 a Fo(W)f Fp(\()p Fo(z)r Fp(\))614 1797 y Fl(Z)664 1810 y Fi(x)642 1909 y(a)694 1864 y Fo(dy)k(v)778 1871 y Fi(a)798 1864 y Fp(\()p Fo(y)r(;)d(z)r Fp(\))p Fo(f)d Fp(\()p Fo(y)r Fp(\))10 b Fm(\000)1067 1831 y Fo(v)1091 1838 y Fi(a)1111 1831 y Fp(\()p Fo(x;)e(z)r Fp(\))p 1067 1853 157 2 v 1087 1899 a Fo(W)f Fp(\()p Fo(z)r Fp(\))1237 1797 y Fl(Z)1287 1810 y Fi(b)1265 1909 y(x)1312 1864 y Fo(dy)j(v)1395 1871 y Fi(b)1412 1864 y Fp(\()p Fo(y)r(;)e(z)r Fp(\))p Fo(f)d Fp(\()p Fo(y)r Fp(\))p Fo(;)89 1999 y Fz(for)17 b Fo(z)e Fm(2)f Fo(\045)p Fp(\()p Fo(H)t Fp(\))p Fz(,)j Fo(f)i Fm(2)14 b Fo(L)510 1981 y Fk(2)530 1999 y Fp(\([)p Fo(a;)8 b(b)p Fp(]\))14 b Fz(and)168 2103 y Fp(\(\()p Fo(H)250 2083 y Fj(\003)281 2103 y Fm(\000)d Fo(z)r Fp(\))375 2083 y Fj(\000)p Fk(1)422 2103 y Fo(f)5 b Fp(\)\()p Fo(x)p Fp(\))13 b(=)1149 b Fz(\(2.10\))369 2210 y Fm(\000)413 2177 y Fo(v)437 2184 y Fj(\003)p Fi(b)471 2177 y Fp(\()p Fo(x;)8 b(z)r Fp(\))p 413 2199 171 2 v 434 2244 a Fo(W)480 2251 y Fj(\003)499 2244 y Fp(\()p Fo(z)r Fp(\))597 2142 y Fl(Z)646 2156 y Fi(x)624 2255 y(a)677 2210 y Fo(dy)i(v)760 2217 y Fj(\003)p Fi(a)798 2210 y Fp(\()p Fo(y)r(;)e(z)r Fp(\))p Fo(f)d Fp(\()p Fo(y)r Fp(\))10 b Fm(\000)1067 2177 y Fo(v)1091 2184 y Fj(\003)p Fi(a)1129 2177 y Fp(\()p Fo(x;)e(z)r Fp(\))p 1067 2199 175 2 v 1090 2244 a Fo(W)1136 2251 y Fj(\003)1156 2244 y Fp(\()p Fo(z)r Fp(\))1255 2142 y Fl(Z)1304 2156 y Fi(b)1282 2255 y(x)1330 2210 y Fo(dy)i(v)1413 2217 y Fj(\003)p Fi(b)1448 2210 y Fp(\()p Fo(y)r(;)e(z)r Fp(\))p Fo(f)d Fp(\()p Fo(y)r Fp(\))p Fo(;)89 2345 y Fz(for)17 b Fo(z)e Fm(2)f Fo(\045)p Fp(\()p Fo(H)337 2327 y Fj(\003)358 2345 y Fp(\))i Fz(and)h Fo(f)i Fm(2)14 b Fo(L)611 2327 y Fk(2)631 2345 y Fp(\([)p Fo(a;)8 b(b)p Fp(]\))p Fz(,)14 b(see)i([12].)148 2429 y(Since)e Fo(H)19 b Fz(is)14 b(completely)e(non-selfadjoin)o(t)j(the)f(maximal)e (dissipativ)o(e)h(op)q(erator)j Fo(H)j Fz(can)c(b)q(e)89 2489 y(completely)j(c)o(haracterized)g(b)o(y)i(its)g(c)o(haracteristic) f(function)h Fp(\002)1320 2496 y Fi(H)1354 2489 y Fp(\()p Fo(z)r Fp(\))p Fz(,)h Fo(z)i Fm(2)e Fo(\045)p Fp(\()p Fo(H)t Fp(\))14 b Fm(\\)g Fo(\045)p Fp(\()p Fo(H)1808 2471 y Fj(\003)1828 2489 y Fp(\))p Fz(,)89 2549 y(cf.)21 b([8].)f(The)15 b(de\034nition)g(of)g(the)g(c)o(haracteristic)f (function)g(relies)g(on)i(the)f(so-called)g(b)q(oundary)89 2609 y(op)q(erators)20 b Fo(T)7 b Fp(\()p Fo(z)r Fp(\))16 b(:)h Fh(H)g Fm(\000)-9 b(!)17 b Fn(C)636 2591 y Fk(2)659 2609 y Fz(,)h Fo(z)h Fm(2)e Fo(\045)p Fp(\()p Fo(H)t Fp(\))i Fz(and)g Fo(T)1035 2616 y Fj(\003)1054 2609 y Fp(\()p Fo(z)r Fp(\))e(:)f Fh(H)h Fm(\000)-8 b(!)17 b Fn(C)1348 2591 y Fk(2)1371 2609 y Fz(,)h Fo(z)h Fm(2)e Fo(\045)p Fp(\()p Fo(H)1583 2591 y Fj(\003)1603 2609 y Fp(\))p Fz(,)h(whic)o(h)g(are)89 2669 y(de\034ned)e(in)g([12].)21 b(W)l(riting)440 2797 y Fo(\024)468 2804 y Fi(a)502 2797 y Fp(=)14 b Fo(q)576 2804 y Fi(a)608 2797 y Fp(+)666 2763 y Fo(i)p 662 2785 25 2 v 662 2831 a Fp(2)691 2797 y Fo(\013)722 2776 y Fk(2)722 2809 y Fi(a)792 2797 y Fz(and)49 b Fo(\024)947 2804 y Fi(b)978 2797 y Fp(=)14 b Fo(q)1052 2804 y Fi(b)1080 2797 y Fp(+)1138 2763 y Fo(i)p 1134 2785 V 1134 2831 a Fp(2)1163 2797 y Fo(\013)1194 2776 y Fk(2)1194 2809 y Fi(b)1214 2797 y Fo(;)57 b(\013)1316 2804 y Fi(a)1337 2797 y Fo(;)8 b(\013)1390 2804 y Fi(b)1421 2797 y Fo(>)13 b Fp(0)p Fo(;)226 b Fz(\(2.11\))p eop %%Page: 7 9 7 8 bop 89 23 a Fs(Density)18 b(and)g(current)h(of)f(a)g(dissip)l(a)l (tive)g(Schr)o(\366dinger)h(opera)l(tor)211 b Fz(7)89 147 y(the)16 b(b)q(oundary)i(op)q(erators)f(are)g(de\034ned)e(b)o(y)596 285 y Fo(T)7 b Fp(\()p Fo(z)r Fp(\))p Fo(f)18 b Fp(:=)803 215 y Fl(\022)884 255 y Fo(\013)915 262 y Fi(b)932 255 y Fp(\(\()p Fo(H)d Fm(\000)c Fo(z)r Fp(\))1119 237 y Fj(\000)p Fk(1)1166 255 y Fo(f)5 b Fp(\)\()p Fo(b)p Fp(\))860 315 y Fm(\000)p Fo(\013)930 322 y Fi(a)951 315 y Fp(\(\()p Fo(H)15 b Fm(\000)c Fo(z)r Fp(\))1138 297 y Fj(\000)p Fk(1)1185 315 y Fo(f)5 b Fp(\)\()p Fo(a)p Fp(\))1317 215 y Fl(\023)1736 285 y Fz(\(2.12\))89 423 y(and)569 501 y Fo(T)598 508 y Fj(\003)617 501 y Fp(\()p Fo(z)r Fp(\))p Fo(f)19 b Fp(:=)788 431 y Fl(\022)869 471 y Fo(\013)900 478 y Fi(b)918 471 y Fp(\(\()p Fo(H)1000 453 y Fj(\003)1031 471 y Fm(\000)10 b Fo(z)r Fp(\))1124 453 y Fj(\000)p Fk(1)1171 471 y Fo(f)5 b Fp(\)\()p Fo(b)p Fp(\))846 531 y Fm(\000)p Fo(\013)916 538 y Fi(a)936 531 y Fp(\(\()p Fo(H)1018 513 y Fj(\003)1049 531 y Fm(\000)11 b Fo(z)r Fp(\))1143 513 y Fj(\000)p Fk(1)1190 531 y Fo(f)5 b Fp(\)\()p Fo(a)p Fp(\))1323 431 y Fl(\023)1368 501 y Fo(;)354 b Fz(\(2.13\))89 621 y Fo(f)19 b Fm(2)14 b Fo(L)212 603 y Fk(2)232 621 y Fp(\([)p Fo(a;)8 b(b)p Fp(]\))p Fz(.)20 b(Using)c(the)g(resolv)o(en)o(t)f(represen)o(tations)g(\(2.9\))i(and)g (\(2.10\))g(w)o(e)e(obtain)518 776 y Fo(T)7 b Fp(\()p Fo(z)r Fp(\))p Fo(f)19 b Fp(=)762 743 y(1)p 716 765 116 2 v 716 811 a Fo(W)7 b Fp(\()p Fo(z)r Fp(\))845 691 y Fl( )905 746 y Fm(\000)p Fo(\013)975 753 y Fi(b)1001 706 y Fl(R)1034 719 y Fi(b)1024 763 y(a)1059 746 y Fo(dy)16 b(v)1148 753 y Fi(a)1169 746 y Fp(\()p Fo(y)r(;)8 b(z)r Fp(\))p Fo(f)d Fp(\()p Fo(y)r Fp(\))925 814 y Fo(\013)956 821 y Fi(a)985 774 y Fl(R)1018 787 y Fi(b)1008 831 y(a)1044 814 y Fo(dy)15 b(v)1132 821 y Fi(b)1149 814 y Fp(\()p Fo(y)r(;)8 b(z)r Fp(\))p Fo(f)d Fp(\()p Fo(y)r Fp(\))1392 691 y Fl(!)1736 776 y Fz(\(2.14\))89 929 y(and)486 1021 y Fo(T)515 1028 y Fj(\003)534 1021 y Fp(\()p Fo(z)r Fp(\))p Fo(f)19 b Fp(=)749 988 y(1)p 697 1010 129 2 v 697 1056 a Fo(W)743 1063 y Fj(\003)762 1056 y Fp(\()p Fo(z)r Fp(\))838 936 y Fl( )899 991 y Fm(\000)p Fo(\013)969 998 y Fi(b)994 951 y Fl(R)1027 964 y Fi(b)1017 1008 y(a)1053 991 y Fo(dy)c(v)1141 998 y Fj(\003)p Fi(a)1179 991 y Fp(\()p Fo(y)r(;)8 b(z)r Fp(\))p Fo(f)d Fp(\()p Fo(y)r Fp(\))920 1059 y Fo(\013)951 1066 y Fi(b)976 1019 y Fl(R)1010 1032 y Fi(b)1000 1076 y(a)1035 1059 y Fo(dy)16 b(v)1124 1066 y Fj(\003)p Fi(b)1158 1059 y Fp(\()p Fo(y)r(;)8 b(z)r Fp(\))p Fo(f)d Fp(\()p Fo(y)r Fp(\))1403 936 y Fl(!)1451 1021 y Fo(;)271 b Fz(\(2.15\))89 1156 y Fo(f)19 b Fm(2)14 b Fo(L)212 1138 y Fk(2)232 1156 y Fp(\([)p Fo(a;)8 b(b)p Fp(]\))p Fz(.)20 b(The)c(adjoin)o(t)g(op)q (erators)i(are)e(giv)o(en)f(b)o(y)420 1290 y Fp(\()p Fo(T)7 b Fp(\()p Fo(z)r Fp(\))538 1269 y Fj(\003)557 1290 y Fo(\030)r Fp(\))i(\()p Fo(x)p Fp(\))41 b(=)845 1256 y(1)p 799 1278 116 2 v 799 1288 V 799 1332 a Fo(W)7 b Fp(\()p Fo(z)r Fp(\))928 1235 y Fl(\020)958 1290 y Fm(\000)p Fo(\013)1028 1297 y Fi(b)p 1045 1247 157 2 v 1045 1290 a Fo(v)1069 1297 y Fi(a)1089 1290 y Fp(\()p Fo(x;)h(z)r Fp(\))p Fo(;)g(\013)1255 1297 y Fi(a)p 1275 1247 154 2 v 1275 1290 a Fo(v)1299 1297 y Fi(b)1316 1290 y Fp(\()p Fo(x;)g(z)r Fp(\))1428 1235 y Fl(\021)1466 1290 y Fo(\030)249 b Fz(\(2.16\))715 1427 y Fp(=)851 1393 y(1)p 799 1415 129 2 v 799 1461 a Fo(W)845 1468 y Fj(\003)865 1461 y Fp(\()p 884 1433 25 2 v Fo(z)r Fp(\))941 1427 y(\()p Fm(\000)p Fo(\013)1030 1434 y Fi(b)1047 1427 y Fo(v)1071 1434 y Fj(\003)p Fi(a)1109 1427 y Fp(\()p Fo(x;)p 1178 1399 V 8 w(z)q Fp(\))p Fo(;)8 b(\013)1274 1434 y Fi(a)1295 1427 y Fo(v)1319 1434 y Fj(\003)p Fi(b)1353 1427 y Fp(\()p Fo(x;)p 1422 1399 V 8 w(z)r Fp(\)\))g Fo(\030)r(;)89 1563 y Fz(and)409 1684 y Fp(\()p Fo(T)457 1691 y Fj(\003)476 1684 y Fp(\()p Fo(z)r Fp(\))539 1664 y Fj(\003)559 1684 y Fo(\030)r Fp(\))h(\()p Fo(x)p Fp(\))41 b(=)854 1651 y(1)p 802 1673 129 2 v 802 1683 V 802 1726 a Fo(W)848 1733 y Fj(\003)867 1726 y Fp(\()p Fo(z)r Fp(\))943 1629 y Fl(\020)973 1684 y Fm(\000)p Fo(\013)1043 1691 y Fi(b)p 1060 1641 175 2 v 1060 1684 a Fo(v)1084 1691 y Fj(\003)p Fi(a)1122 1684 y Fp(\()p Fo(x;)8 b(z)r Fp(\))p Fo(;)g(\013)1288 1691 y Fi(a)p 1308 1641 171 2 v 1308 1684 a Fo(v)1332 1691 y Fj(\003)p Fi(b)1367 1684 y Fp(\()p Fo(x;)g(z)r Fp(\))1479 1629 y Fl(\021)1517 1684 y Fo(\030)198 b Fz(\(2.17\))717 1821 y Fp(=)847 1787 y(1)p 802 1810 116 2 v 802 1855 a Fo(W)7 b Fp(\()p 874 1828 25 2 v Fo(z)q Fp(\))931 1821 y(\()o Fm(\000)p Fo(\013)1019 1828 y Fi(b)1036 1821 y Fo(v)1060 1828 y Fi(a)1081 1821 y Fp(\()p Fo(x;)p 1150 1794 V 8 w(z)q Fp(\))p Fo(;)h(\013)1246 1828 y Fi(a)1267 1821 y Fo(v)1291 1828 y Fi(b)1308 1821 y Fp(\()p Fo(x;)p 1377 1794 V 8 w(z)q Fp(\)\))g Fo(\030)r(;)89 1958 y Fz(where)786 2036 y Fo(\030)16 b Fp(=)875 1966 y Fl(\022)934 2005 y Fo(\030)957 1987 y Fi(b)932 2066 y Fo(\030)955 2048 y Fi(a)998 1966 y Fl(\023)1048 2036 y Fm(2)e Fn(C)1128 2015 y Fk(2)1151 2036 y Fo(:)571 b Fz(\(2.18\))89 2151 y(The)17 b(c)o(haracteristic)f(function)g Fp(\002)720 2158 y Fi(H)771 2151 y Fz(of)h(the)g(maximal)d(dissipativ)o(e)h(op)q (erator)k Fo(H)i Fz(is)c(a)g(t)o(w)o(o-b)o(y-)89 2211 y(t)o(w)o(o)f(matrix-v)m(alued)f(function)h(whic)o(h)f(satis\034es)i (the)f(relation)513 2323 y Fp(\002)551 2330 y Fi(H)584 2323 y Fp(\()p Fo(z)r Fp(\))p Fo(T)7 b Fp(\()p Fo(z)r Fp(\))p Fo(f)18 b Fp(=)c Fo(T)869 2330 y Fj(\003)888 2323 y Fp(\()p Fo(z)r Fp(\))p Fo(f)s(;)56 b(z)16 b Fm(2)e Fo(\045)p Fp(\()p Fo(H)t Fp(\))e Fm(\\)f Fo(\045)p Fp(\()p Fo(H)1385 2302 y Fj(\003)1405 2323 y Fp(\))p Fo(;)298 b Fz(\(2.19\))89 2434 y Fo(f)20 b Fm(2)15 b Fo(L)214 2416 y Fk(2)234 2434 y Fp(\([)p Fo(a;)8 b(b)p Fp(]\))p Fz(.)21 b(It)16 b(dep)q(ends)h(meromorphically)c(on)k Fo(z)g Fm(2)e Fo(\045)p Fp(\()p Fo(H)t Fp(\))c Fm(\\)h Fo(\045)p Fp(\()p Fo(H)1426 2416 y Fj(\003)1446 2434 y Fp(\))17 b Fz(and)g(is)f(con)o(tractiv)o(e)89 2494 y(in)g Fn(C)179 2501 y Fj(\000)212 2494 y Fz(,)f(i.e.)677 2556 y Fm(k)p Fp(\002)740 2563 y Fi(H)774 2556 y Fp(\()p Fo(z)r Fp(\))p Fm(k)e(\024)h Fp(1)49 b Fz(for)g Fo(z)16 b Fm(2)e Fn(C)1227 2563 y Fj(\000)1259 2556 y Fo(:)463 b Fz(\(2.20\))89 2645 y(Using)16 b(the)g(elemen)o(tary)d(solutions)k (the)f(c)o(haracteristic)f(function)h Fp(\002)1376 2652 y Fi(H)1426 2645 y Fz(tak)o(es)f(the)h(form)422 2783 y Fp(\002)460 2790 y Fi(H)494 2783 y Fp(\()p Fo(z)r Fp(\))d(=)h Fo(I)644 2791 y Fg(C)665 2782 y Ff(2)698 2783 y Fp(+)d Fo(i)821 2749 y Fp(1)p 769 2772 129 2 v 769 2817 a Fo(W)815 2824 y Fj(\003)834 2817 y Fp(\()p Fo(z)r Fp(\))911 2713 y Fl(\022)968 2753 y Fo(\013)999 2734 y Fk(2)999 2765 y Fi(b)1019 2753 y Fo(v)1043 2760 y Fj(\003)p Fi(a)1081 2753 y Fp(\()p Fo(b;)d(z)r Fp(\))82 b Fm(\000)p Fo(\013)1339 2760 y Fi(b)1356 2753 y Fo(\013)1387 2760 y Fi(a)1008 2813 y Fm(\000)p Fo(\013)1078 2820 y Fi(b)1095 2813 y Fo(\013)1126 2820 y Fi(a)1228 2813 y Fo(\013)1259 2795 y Fk(2)1259 2825 y Fi(a)1280 2813 y Fo(v)1304 2820 y Fj(\003)p Fi(b)1338 2813 y Fp(\()p Fo(a;)8 b(z)r Fp(\))1469 2713 y Fl(\023)1514 2783 y Fo(:)208 b Fz(\(2.21\))p eop %%Page: 8 10 8 9 bop 89 23 a Fz(8)716 b Fs(H.-Ch.)25 b(Kaiser,)18 b(H.)h(Neidhardt,)g(J.)g(Rehber)o(g)89 147 y Fz(for)e Fo(z)e Fm(2)f Fo(\045)p Fp(\()p Fo(H)t Fp(\))e Fm(\\)f Fo(\045)p Fp(\()p Fo(H)500 129 y Fj(\003)520 147 y Fp(\))p Fz(,)16 b(cf.)21 b([12])148 231 y(Since)16 b Fo(H)22 b Fz(is)17 b(a)g(maximal)d(dissipativ)o(e)i(op)q(erator)i(there)f(is)g (a)g(larger)g(Hilb)q(ert)f(space)h Fh(K)f Fm(\023)f Fh(H)89 291 y Fz(and)i(a)g(self-adjoin)o(t)e(op)q(erator)j Fo(K)i Fz(on)d Fh(K)g Fz(suc)o(h)e(that)513 396 y Fo(P)551 375 y Fe(K)544 408 y(H)577 396 y Fp(\()p Fo(K)g Fm(\000)c Fo(z)r Fp(\))746 375 y Fj(\000)p Fk(1)793 396 y Fm(j)p Fh(H)i Fp(=)h(\()p Fo(H)h Fm(\000)c Fo(z)r Fp(\))1076 375 y Fj(\000)p Fk(1)1123 396 y Fo(;)57 b Fm(=)p Fz(m)n Fp(\()p Fo(z)r Fp(\))14 b Fo(>)g Fp(0)p Fo(;)299 b Fz(\(2.22\))89 500 y(see)18 b([8].)27 b(The)18 b(op)q(erator)h Fo(K)j Fz(is)c(called)f(a)i(self-adjoin)o(t)f(dilation)g(of)g(the)g(maximal)d (dissipativ)o(e)89 560 y(op)q(erator)j Fo(H)t Fz(.)j(Ob)o(viously)l(,) 15 b(from)g(the)h(condition)g(\(2.22\))h(one)f(gets)503 664 y Fo(P)541 644 y Fe(K)534 677 y(H)567 664 y Fp(\()p Fo(K)f Fm(\000)c Fo(z)r Fp(\))736 644 y Fj(\000)p Fk(1)783 664 y Fm(j)p Fh(H)i Fp(=)h(\()p Fo(H)961 644 y Fj(\003)992 664 y Fm(\000)d Fo(z)r Fp(\))1086 644 y Fj(\000)p Fk(1)1133 664 y Fo(;)57 b Fm(=)p Fz(m)n Fp(\()p Fo(z)r Fp(\))14 b Fo(<)g Fp(0)p Fo(:)289 b Fz(\(2.23\))89 769 y Fo(K)21 b Fz(is)16 b(called)f(a)h(minimal)d(self-adjoin)o(t)j(dilation)g(of)g Fo(H)21 b Fz(if)15 b(the)h(condition)772 831 y Fl(_)746 939 y Fi(z)q Fj(2)p Fg(C)s Fj(n)p Fg(R)853 879 y Fp(\()p Fo(K)f Fm(\000)c Fo(z)r Fp(\))1022 858 y Fj(\000)p Fk(1)1069 879 y Fh(H)j Fp(=)g Fh(K)532 b Fz(\(2.24\))89 1035 y(is)12 b(satis\034ed.)20 b(Minimal)9 b(self-adjoin)o(t)i(dilations)h(of)g (maximal)d(dissipativ)o(e)h(op)q(erators)j(are)f(deter-)89 1095 y(mined)j(up)i(to)h(a)f(certain)f(isomorphism,)f(in)h(particular,) h(all)f(minimal)d(self-adjoin)o(t)k(dilations)89 1156 y(are)f(unitarily)g(equiv)m(alen)o(t.)148 1239 y(In)j(the)g(presen)o(t) f(case)h(the)g(minim)o(al)d(self-adjoin)o(t)j(dilation)f(of)i(the)e (maximal)e(dissipativ)o(e)89 1300 y(op)q(erator)j Fo(H)j Fz(can)c(b)q(e)g(constructed)f(in)h(an)g(explicit)e(manner.)24 b(F)l(ollo)o(wing)17 b([12)q(])g(w)o(e)g(in)o(tro)q(duce)89 1360 y(the)f(larger)g(Hilb)q(ert)f(space)772 1420 y Fh(K)f Fp(=)g Fm(D)909 1427 y Fj(\000)950 1420 y Fm(\010)d Fh(H)g Fm(\010)g(D)1135 1427 y Fk(+)1164 1420 y Fo(;)558 b Fz(\(2.25\))89 1505 y(where)16 b Fm(D)268 1512 y Fj(\006)312 1505 y Fp(:=)d Fo(L)410 1487 y Fk(2)430 1505 y Fp(\()p Fn(R)488 1512 y Fj(\006)515 1505 y Fo(;)8 b Fn(C)569 1487 y Fk(2)592 1505 y Fp(\))p Fz(.)21 b(In)o(tro)q(ducing)16 b(the)g(graph)1138 1492 y Fp(^)1133 1505 y(\012)p Fz(,)p 539 1666 333 4 v 705 1644 a Fn(R)744 1651 y Fj(\000)p 871 1666 V 1037 1644 a Fn(R)1076 1651 y Fk(+)p 539 1832 V 705 1810 a Fn(R)744 1817 y Fj(\000)p 871 1832 V 1037 1810 a Fn(R)1076 1817 y Fk(+)p 870 1831 4 167 v 892 1756 a Fp([)p Fo(a;)8 b(b)p Fp(])89 1992 y Fz(one)17 b(can)f(write)g(the)g(Hilb)q(ert)f (space)h Fh(K)g Fz(as)h Fo(L)917 1974 y Fk(2)937 1992 y Fp(\()961 1980 y(^)956 1992 y(\012\))p Fz(.)22 b(F)l(urthermore,)13 b(w)o(e)j(de\034ne)788 2097 y Fo(~)-22 b(g)16 b Fp(:=)d Fo(g)917 2104 y Fj(\000)958 2097 y Fm(\010)e Fo(g)i Fm(\010)e Fo(g)1117 2104 y Fk(+)1147 2097 y Fo(;)575 b Fz(\(2.26\))89 2201 y(where)416 2278 y Fo(g)439 2285 y Fj(\000)469 2278 y Fp(\()p Fo(x)p Fp(\))14 b(:=)614 2207 y Fl(\022)672 2247 y Fo(g)697 2229 y Fi(b)695 2260 y Fj(\000)724 2247 y Fp(\()p Fo(x)p Fp(\))672 2307 y Fo(g)697 2289 y Fi(a)695 2320 y Fj(\000)724 2307 y Fp(\()p Fo(x)p Fp(\))811 2207 y Fl(\023)953 2278 y Fz(and)50 b Fo(g)1104 2285 y Fk(+)1134 2278 y Fp(\()p Fo(x)p Fp(\))13 b(:=)1278 2207 y Fl(\022)1336 2247 y Fo(g)1361 2229 y Fi(b)1359 2260 y Fk(+)1389 2247 y Fp(\()p Fo(x)p Fp(\))1336 2307 y Fo(g)1361 2289 y Fi(a)1359 2320 y Fk(+)1389 2307 y Fp(\()p Fo(x)p Fp(\))1475 2207 y Fl(\023)1520 2278 y Fo(;)202 b Fz(\(2.27\))89 2395 y(for)17 b Fo(x)c Fm(2)h Fn(R)291 2402 y Fj(\000)334 2395 y Fz(and)j Fo(x)c Fm(2)h Fn(R)556 2402 y Fk(+)583 2395 y Fz(,)i(resp)q(ectiv)o(ely)l(.)i(Let)f(the)f(matrices)e Fo(K)1305 2377 y Fi(a)1301 2408 y Fj(\006)1347 2395 y Fz(and)j Fo(K)1487 2377 y Fi(b)1483 2408 y Fj(\006)1529 2395 y Fz(giv)o(en)f(b)o(y)384 2532 y Fo(K)429 2511 y Fi(a)425 2544 y Fj(\000)469 2532 y Fp(:=)553 2498 y(1)p 539 2520 52 2 v 539 2566 a Fo(\013)570 2573 y Fi(a)604 2462 y Fl(\022)662 2501 y Fp(0)54 b(0)662 2562 y(1)42 b Fo(\024)756 2569 y Fi(a)797 2462 y Fl(\023)940 2532 y Fz(and)98 b Fo(K)1161 2511 y Fi(a)1157 2544 y Fk(+)1201 2532 y Fp(:=)1285 2498 y(1)p 1271 2520 V 1271 2566 a Fo(\013)1302 2573 y Fi(a)1336 2462 y Fl(\022)1394 2501 y Fp(0)54 b(0)1394 2562 y(1)p 1460 2534 49 2 v 42 w Fo(\024)1488 2569 y Fi(a)1529 2462 y Fl(\023)1736 2532 y Fz(\(2.28\))89 2662 y(as)17 b(w)o(ell)e(as)342 2783 y Fo(K)387 2762 y Fi(b)383 2795 y Fj(\000)427 2783 y Fp(:=)509 2749 y(1)p 497 2772 V 497 2817 a Fo(\013)528 2824 y Fi(b)558 2713 y Fl(\022)616 2753 y Fp(1)42 b Fm(\000)p Fo(\024)749 2760 y Fi(b)616 2813 y Fp(0)72 b(0)786 2713 y Fl(\023)929 2783 y Fz(and)98 b Fo(K)1150 2762 y Fi(b)1146 2795 y Fk(+)1190 2783 y Fp(:=)1272 2749 y(1)p 1260 2772 V 1260 2817 a Fo(\013)1291 2824 y Fi(b)1322 2713 y Fl(\022)1379 2753 y Fp(1)42 b Fm(\000)p 1484 2725 46 2 v Fo(\024)1512 2760 y Fi(b)1379 2813 y Fp(0)72 b(0)1550 2713 y Fl(\023)1595 2783 y Fo(:)127 b Fz(\(2.29\))p eop %%Page: 9 11 9 10 bop 89 23 a Fs(Density)18 b(and)g(current)h(of)f(a)g(dissip)l(a)l (tive)g(Schr)o(\366dinger)h(opera)l(tor)211 b Fz(9)89 147 y(Using)16 b(these)g(notations)h(the)f(self-adjoin)o(t)g(dilation)g Fo(K)k Fz(is)c(de\034ned)g(b)o(y)462 320 y(dom)o Fp(\()p Fo(K)t Fp(\))e(:=)716 188 y Fl(8)716 233 y(>)716 248 y(>)716 263 y(<)716 353 y(>)716 368 y(>)716 382 y(:)758 320 y Fo(~)-21 b(g)16 b Fm(2)e Fh(K)g Fp(:)942 230 y Fo(g)965 237 y Fj(\006)1009 230 y Fm(2)g Fo(W)1109 211 y Fk(1)p Fi(;)p Fk(2)1156 230 y Fp(\()p Fn(R)1214 237 y Fj(\006)1240 230 y Fo(;)8 b Fn(C)1295 211 y Fk(2)1318 230 y Fp(\))p Fo(;)942 290 y(g)r(;)1001 270 y Fk(1)p 994 278 32 2 v 994 307 a Fi(m)1030 290 y Fo(g)1055 272 y Fj(0)1081 290 y Fm(2)14 b Fo(W)1181 272 y Fk(1)p Fi(;)p Fk(2)1228 290 y Fp(\([)p Fo(a;)8 b(b)p Fp(]\))p Fo(;)942 350 y(K)987 332 y Fi(a)983 362 y Fj(\000)1013 350 y Fo(g)1036 357 y Fi(a)1068 350 y Fp(+)j Fo(K)1162 332 y Fi(b)1158 362 y Fj(\000)1188 350 y Fo(g)1211 357 y Fi(b)1242 350 y Fp(=)j Fo(g)1317 357 y Fj(\000)1347 350 y Fp(\(0\))p Fo(;)942 410 y(K)987 392 y Fi(a)983 422 y Fk(+)1013 410 y Fo(g)1036 417 y Fi(a)1068 410 y Fp(+)d Fo(K)1162 392 y Fi(b)1158 422 y Fk(+)1188 410 y Fo(g)1211 417 y Fi(b)1242 410 y Fp(=)j Fo(g)1317 417 y Fk(+)1347 410 y Fp(\(0\))1444 188 y Fl(9)1444 233 y(>)1444 248 y(>)1444 263 y(=)1444 353 y(>)1444 368 y(>)1444 382 y(;)1736 320 y Fz(\(2.30\))89 493 y(and)432 565 y Fo(K)q(~)-21 b(g)16 b Fp(:=)d Fm(\000)p Fo(i)655 532 y(d)p 642 554 54 2 v 642 599 a(dx)700 565 y(g)723 572 y Fj(\000)763 565 y Fm(\010)e Fo(l)q Fp(\()p Fo(g)r Fp(\))g Fm(\010)g(\000)p Fo(i)1027 532 y(d)p 1014 554 V 1014 599 a(dx)1071 565 y(g)1094 572 y Fk(+)1124 565 y Fo(;)103 b(~)-21 b(g)16 b Fm(2)e Fz(dom)o Fp(\()p Fo(K)t Fp(\))p Fo(;)217 b Fz(\(2.31\))89 655 y(where)448 735 y Fo(g)471 742 y Fi(a)506 735 y Fp(=)558 665 y Fl(\022)659 681 y Fk(1)p 620 690 96 2 v 620 718 a(2)p Fi(m)p Fk(\()p Fi(a)p Fk(\))720 701 y Fo(g)745 683 y Fj(0)757 701 y Fp(\()p Fo(a)p Fp(\))674 769 y Fo(g)r Fp(\()p Fo(a)p Fp(\))841 665 y Fl(\023)935 735 y Fz(and)49 b Fo(g)1085 742 y Fi(b)1117 735 y Fp(=)1168 665 y Fl(\022)1268 681 y Fk(1)p 1231 690 92 2 v 1231 718 a(2)p Fi(m)p Fk(\()p Fi(b)p Fk(\))1327 701 y Fo(g)1352 683 y Fj(0)1364 701 y Fp(\()p Fo(b)p Fp(\))1282 769 y Fo(g)r Fp(\()p Fo(b)p Fp(\))1443 665 y Fl(\023)1488 735 y Fo(:)234 b Fz(\(2.32\))89 842 y(With)16 b(resp)q(ect)g(to)h(a)f(graph)h(picture)f(the)g(op)q (erator)h Fo(K)j Fz(lo)q(oks)d(lik)o(e)p 282 1048 416 4 v 697 1054 a Fp(\))282 971 y Fo(\013)313 978 y Fi(b)331 971 y Fo(g)356 953 y Fi(b)354 984 y Fj(\000)383 971 y Fp(\(0\))e(=)553 952 y Fk(1)p 516 960 92 2 v 516 988 a(2)p Fi(m)p Fk(\()p Fi(b)p Fk(\))613 971 y Fo(g)638 953 y Fj(0)649 971 y Fp(\()p Fo(b)p Fp(\))c Fm(\000)g Fo(\024)797 978 y Fi(b)814 971 y Fo(g)r Fp(\()p Fo(b)p Fp(\))365 1108 y Fm(\000)p Fo(i)436 1089 y Fi(d)p 426 1097 39 2 v 426 1125 a(dx)469 1108 y Fo(g)494 1090 y Fi(b)492 1121 y Fj(\000)p 1154 1048 416 4 v 1146 1054 a Fz(\()1071 952 y Fk(1)p 1035 960 92 2 v 1035 988 a(2)p Fi(m)p Fk(\()p Fi(b)p Fk(\))1131 971 y Fo(g)1156 953 y Fj(0)1168 971 y Fp(\()p Fo(b)p Fp(\))f Fm(\000)p 1287 944 46 2 v 11 w Fo(\024)1315 978 y Fi(b)1332 971 y Fo(g)r Fp(\()p Fo(b)p Fp(\))k(=)g Fo(\013)1513 978 y Fi(b)1530 971 y Fo(g)1555 953 y Fi(b)1553 984 y Fk(+)1583 971 y Fp(\(0\))1362 1108 y Fm(\000)p Fo(i)1432 1089 y Fi(d)p 1423 1097 39 2 v 1423 1125 a(dx)1465 1108 y Fo(g)1490 1090 y Fi(b)1488 1121 y Fk(+)p 282 1297 416 4 v 697 1303 a Fp(\))262 1386 y Fo(\013)293 1393 y Fi(a)313 1386 y Fo(g)338 1368 y Fi(a)336 1399 y Fj(\000)366 1386 y Fp(\(0\))g(=)538 1367 y Fk(1)p 499 1375 96 2 v 499 1404 a(2)p Fi(m)p Fk(\()p Fi(a)p Fk(\))599 1386 y Fo(g)624 1368 y Fj(0)636 1386 y Fp(\()p Fo(a)p Fp(\))d(+)g Fo(\024)788 1393 y Fi(a)809 1386 y Fo(g)r Fp(\()p Fo(a)p Fp(\))365 1254 y Fm(\000)p Fo(i)436 1234 y Fi(d)p 426 1242 39 2 v 426 1271 a(dx)469 1254 y Fo(g)494 1235 y Fi(a)492 1266 y Fj(\000)p 1154 1297 416 4 v 1146 1303 a Fp(\()1073 1367 y Fk(1)p 1035 1375 96 2 v 1035 1404 a(2)p Fi(m)p Fk(\()p Fi(a)p Fk(\))1135 1386 y Fo(g)1160 1368 y Fj(0)1171 1386 y Fp(\()p Fo(a)p Fp(\))g(+)p 1295 1359 49 2 v 11 w Fo(\024)1323 1393 y Fi(a)1344 1386 y Fo(g)r Fp(\()p Fo(a)p Fp(\))i(=)h Fo(\013)1529 1393 y Fi(a)1550 1386 y Fo(g)1575 1368 y Fi(a)1573 1399 y Fk(+)1603 1386 y Fp(\(0\))1362 1254 y Fm(\000)p Fo(i)1432 1234 y Fi(d)p 1423 1242 39 2 v 1423 1271 a(dx)1465 1254 y Fo(g)1490 1235 y Fi(a)1488 1266 y Fk(+)p 945 1295 4 250 v 988 1171 a Fo(l)q Fp(\()p Fo(g)r Fp(\))89 1525 y Fz(The)f(self-adjoin)o(t)f(op)q(erator)i Fo(K)j Fz(is)12 b(absolutely)h(con)o(tin)o(uous)f(and)i(its)e(sp)q(ectrum)f(coincides)h (with)89 1585 y(the)k(real)g(axis,)g(i.e.)k Fo(\033)r Fp(\()p Fo(K)t Fp(\))13 b(=)h Fn(R)p Fz(.)k(The)e(m)o(ultiplic)o(it)o (y)d(of)j(its)g(sp)q(ectrum)f(is)h(t)o(w)o(o.)21 b(The)16 b(resolv)o(en)o(t)89 1645 y(of)h Fo(K)j Fz(admits)15 b(the)h(represen)o(tation)293 1731 y Fp(\()p Fo(K)f Fm(\000)c Fo(z)r Fp(\))462 1711 y Fj(\000)p Fk(1)517 1731 y Fp(\()p Fo(f)560 1738 y Fj(\000)601 1731 y Fm(\010)g Fo(f)16 b Fm(\010)11 b Fo(f)765 1738 y Fk(+)794 1731 y Fp(\))j(=)871 b Fz(\(2.33\))278 1837 y Fo(i)303 1769 y Fl(Z)353 1782 y Fi(x)331 1881 y Fj(\0001)404 1837 y Fo(dy)16 b(e)492 1816 y Fi(i)p Fk(\()p Fi(x)p Fj(\000)p Fi(y)q Fk(\))p Fi(z)617 1837 y Fo(f)641 1844 y Fj(\000)670 1837 y Fp(\()p Fo(y)r Fp(\))25 b Fm(\010)f Fp(\()p Fo(H)16 b Fm(\000)11 b Fo(z)r Fp(\))991 1816 y Fj(\000)p Fk(1)1038 1837 y Fo(f)16 b Fp(+)11 b Fo(iT)1173 1844 y Fj(\003)1192 1837 y Fp(\()p 1211 1809 25 2 v Fo(z)r Fp(\))1255 1816 y Fj(\003)1283 1769 y Fl(Z)1333 1782 y Fk(0)1310 1881 y Fj(\0001)1384 1837 y Fo(dy)k(e)1471 1816 y Fj(\000)p Fi(iy)q(z)1549 1837 y Fo(f)1573 1844 y Fj(\000)1602 1837 y Fp(\()p Fo(y)r Fp(\))25 b Fm(\010)278 1978 y Fo(i)303 1910 y Fl(Z)353 1923 y Fi(x)331 2023 y Fk(0)383 1978 y Fo(dy)16 b(e)471 1957 y Fi(i)p Fk(\()p Fi(x)p Fj(\000)p Fi(y)q Fk(\))p Fi(z)596 1978 y Fo(f)620 1985 y Fk(+)650 1978 y Fp(\()p Fo(y)r Fp(\))10 b(+)h Fo(ie)813 1957 y Fi(iz)q(x)864 1978 y Fo(T)c Fp(\()p Fo(z)r Fp(\))p Fo(f)16 b Fp(+)11 b Fo(i)p Fp(\002)1107 1985 y Fi(H)1140 1978 y Fp(\()p 1159 1950 V Fo(z)r Fp(\))1203 1957 y Fj(\003)1231 1910 y Fl(Z)1281 1923 y Fk(0)1259 2023 y Fj(\0001)1332 1978 y Fo(dy)k(e)1419 1957 y Fi(i)p Fk(\()p Fi(x)p Fj(\000)p Fi(y)q Fk(\))p Fi(z)1544 1978 y Fo(f)1568 1985 y Fj(\000)1598 1978 y Fp(\()p Fo(y)r Fp(\))89 2094 y Fz(for)i Fm(=)p Fz(m)n Fp(\()p Fo(z)r Fp(\))d Fo(>)g Fp(0)i Fz(and)296 2180 y Fp(\()p Fo(K)f Fm(\000)c Fo(z)r Fp(\))465 2160 y Fj(\000)p Fk(1)520 2180 y Fp(\()p Fo(f)563 2187 y Fj(\000)604 2180 y Fm(\010)g Fo(f)16 b Fm(\010)11 b Fo(f)768 2187 y Fk(+)798 2180 y Fp(\))i(=)868 b Fz(\(2.34\))282 2285 y Fm(\000)p Fo(i)346 2217 y Fl(Z)395 2231 y Fk(0)373 2330 y Fi(x)423 2285 y Fo(dy)16 b(e)511 2265 y Fi(i)p Fk(\()p Fi(x)p Fj(\000)p Fi(y)q Fk(\))p Fi(z)636 2285 y Fo(f)660 2292 y Fj(\000)689 2285 y Fp(\()p Fo(y)r Fp(\))11 b Fm(\000)g Fo(ie)854 2265 y Fi(iz)q(x)905 2285 y Fo(T)934 2292 y Fj(\003)953 2285 y Fp(\()p Fo(z)r Fp(\))p Fo(f)16 b Fm(\000)11 b Fo(i)p Fp(\002)1161 2292 y Fi(H)1194 2285 y Fp(\()p Fo(z)r Fp(\))1265 2217 y Fl(Z)1315 2231 y Fj(1)1293 2330 y Fk(0)1361 2285 y Fo(dy)16 b(e)1449 2265 y Fi(i)p Fk(\()p Fi(x)p Fj(\000)p Fi(y)q Fk(\))p Fi(z)1573 2285 y Fo(f)1597 2292 y Fk(+)1627 2285 y Fp(\()p Fo(y)r Fp(\))11 b Fm(\010)282 2416 y Fp(\()p Fo(H)345 2395 y Fj(\003)376 2416 y Fm(\000)f Fo(z)r Fp(\))469 2395 y Fj(\000)p Fk(1)516 2416 y Fo(f)17 b Fm(\000)11 b Fo(iT)c Fp(\()p 679 2388 V Fo(z)q Fp(\))722 2395 y Fj(\003)749 2348 y Fl(Z)799 2361 y Fj(1)777 2461 y Fk(0)845 2416 y Fo(dy)16 b(e)933 2395 y Fj(\000)p Fi(iy)q(z)1010 2416 y Fo(f)1034 2423 y Fk(+)1064 2416 y Fp(\()p Fo(y)r Fp(\))24 b Fm(\010)h(\000)p Fo(i)1280 2348 y Fl(Z)1329 2361 y Fj(1)1307 2461 y Fi(x)1375 2416 y Fo(dy)16 b(e)1463 2395 y Fi(i)p Fk(\()p Fi(x)p Fj(\000)p Fi(y)q Fk(\))p Fi(z)1588 2416 y Fo(f)1612 2423 y Fk(+)1641 2416 y Fp(\()p Fo(y)r Fp(\))89 2540 y Fz(for)g Fm(=)p Fz(m)n Fp(\()p Fo(z)r Fp(\))e Fo(<)g Fp(0)p Fz(.)21 b(The)15 b(generalized)f(eigenfunctions)1101 2527 y Fo(~)1095 2540 y( )1129 2522 y Fj(\000)1158 2540 y Fp(\()p Fm(\001)p Fo(;)8 b(\025;)g(\034)e Fp(\))p Fz(,)14 b Fo(\025)h Fm(2)f Fn(R)p Fz(,)e Fo(\034)19 b Fp(=)14 b Fo(a;)8 b(b)p Fz(,)14 b(of)i Fo(K)j Fz(are)89 2600 y(giv)o(en)d(b)o(y)261 2673 y Fo(~)255 2686 y( )289 2666 y Fj(\000)318 2686 y Fp(\()p Fo(x;)8 b(\025;)g(\034)e Fp(\))14 b(:=)f Fo( )596 2666 y Fj(\000)594 2699 y(\000)625 2686 y Fp(\()p Fo(x;)8 b(\025;)g(\034)e Fp(\))11 b Fm(\010)f Fo( )884 2666 y Fj(\000)913 2686 y Fp(\()p Fo(x;)e(\025;)g(\034)e Fp(\))11 b Fm(\010)g Fo( )1173 2666 y Fj(\000)1171 2699 y Fk(+)1202 2686 y Fp(\()p Fo(x;)d(\025;)g(\034)e Fp(\))13 b(=)318 b Fz(\(2.35\))497 2751 y Fp(1)p 461 2773 96 2 v 461 2783 a Fm(p)p 503 2783 54 2 v 41 x Fp(2)p Fo(\031)562 2784 y(e)585 2764 y Fi(ix\025)639 2784 y Fo(e)662 2791 y Fi(\034)694 2784 y Fm(\010)784 2751 y Fp(1)p 749 2773 96 2 v 749 2783 a Fm(p)p 790 2783 54 2 v 790 2824 a Fp(2)p Fo(\031)849 2784 y Fp(\()p Fo(T)897 2791 y Fj(\003)917 2784 y Fp(\()p Fo(\025)p Fp(\))983 2764 y Fj(\003)1003 2784 y Fo(e)1026 2791 y Fi(\034)1047 2784 y Fp(\)\()p Fo(x)p Fp(\))10 b Fm(\010)1233 2751 y Fp(1)p 1197 2773 96 2 v 1197 2783 a Fm(p)p 1239 2783 54 2 v 41 x Fp(2)p Fo(\031)1298 2784 y(e)1321 2764 y Fi(ix\025)1375 2784 y Fp(\002)1413 2791 y Fi(H)1446 2784 y Fp(\()p Fo(\025)p Fp(\))1512 2764 y Fj(\003)1532 2784 y Fo(e)1555 2791 y Fi(\034)p eop %%Page: 10 12 10 11 bop 89 23 a Fz(10)692 b Fs(H.-Ch.)25 b(Kaiser,)18 b(H.)h(Neidhardt,)g(J.)g(Rehber)o(g)89 147 y Fz(where)612 224 y Fo(e)635 231 y Fi(b)665 224 y Fp(:=)731 154 y Fl(\022)788 194 y Fp(1)788 254 y(0)833 154 y Fl(\023)927 224 y Fz(and)49 b Fo(e)1077 231 y Fi(a)1112 224 y Fp(:=)1177 154 y Fl(\022)1234 194 y Fp(0)1234 254 y(1)1280 154 y Fl(\023)1325 224 y Fo(:)397 b Fz(\(2.36\))89 335 y(The)16 b(functions)h(are)f(m)o(utually) e(orthogonal,)j(i.e.)j(one)d(has)489 397 y Fl(\020)525 439 y Fo(~)519 452 y( )553 431 y Fj(\000)582 452 y Fp(\()p Fm(\001)p Fo(;)8 b(\025;)g(\034)e Fp(\))p Fo(;)760 439 y(~)755 452 y( )789 431 y Fj(\000)817 452 y Fp(\()p Fm(\001)p Fo(;)i(\025)900 431 y Fj(0)912 452 y Fo(;)g(\034)961 431 y Fj(0)972 452 y Fp(\))991 397 y Fl(\021)1021 486 y Fi(L)1045 477 y Ff(2)1062 486 y Fk(\()1080 478 y(^)1076 486 y(\012\))1131 452 y Fp(=)14 b Fo(\016)r Fp(\()p Fo(\025)d Fm(\000)g Fo(\025)1343 431 y Fj(0)1355 452 y Fp(\))p Fo(\016)1396 459 y Fi(\034)t(\034)1436 450 y Fd(0)1448 452 y Fo(;)274 b Fz(\(2.37\))89 581 y(in)14 b(the)f(sense)h(of)g (distribution)g(for)g Fo(\025;)8 b(\025)815 563 y Fj(0)841 581 y Fm(2)14 b Fn(R)p Fz(,)d Fo(\034)s(;)d(\034)1025 563 y Fj(0)1050 581 y Fp(=)14 b Fo(a;)8 b(b)p Fz(.)19 b(Moreo)o(v)o(er,)13 b(elemen)o(ts)e(of)j(the)g(form)693 640 y Fl(Z)721 753 y Fg(R)755 708 y Fo(d\025)840 661 y Fl(X)831 767 y Fi(\034)t Fk(=)p Fi(a;b)936 695 y Fo(~)930 708 y( )964 687 y Fj(\000)993 708 y Fp(\()p Fm(\001)p Fo(;)8 b(\025;)g(\034)e Fp(\))r(^)-26 b Fo(g)1169 687 y Fi(\034)1190 708 y Fp(\()p Fo(\025)p Fp(\))480 b Fz(\(2.38\))89 865 y(where)13 b Fp(^)-26 b Fo(g)250 847 y Fi(\034)272 865 y Fp(\()p Fm(\001)p Fp(\))p Fz(,)11 b Fo(\034)20 b Fp(=)14 b Fo(a;)8 b(b)p Fz(,)j(are)g(smo)q(oth)g(functions)g(with)g (compact)f(supp)q(ort,)j(are)e(dense)g(in)g Fh(K)p Fz(.)20 b(W)l(e)89 930 y(note)e(that)f(the)g(generalized)f(eigenfunctions)969 917 y Fo(~)963 930 y( )997 912 y Fj(\000)1026 930 y Fp(\()p Fm(\001)p Fo(;)8 b(\025;)g(\034)e Fp(\))17 b Fz(are)g(usually)g(called) f(the)h(incoming)89 991 y(eigenfunctions.)45 b(Using)24 b(the)g(incoming)e(eigenfunctions)i(one)g(de\034nes)g(a)h (transformation)89 1051 y Fp(\010)124 1058 y Fj(\000)168 1051 y Fp(:)13 b Fh(K)h Fm(\000)-8 b(!)341 1038 y Fp(^)336 1051 y Fh(K)15 b Fp(=)e Fo(L)468 1033 y Fk(2)488 1051 y Fp(\()p Fn(R)p Fo(;)8 b Fn(C)598 1033 y Fk(2)621 1051 y Fp(\))635 1185 y(\(\010)689 1192 y Fj(\000)717 1185 y Fo(~)-22 b(g)r Fp(\)\()p Fo(\025)p Fp(\))14 b(=:)h(^)-25 b Fo(g)r Fp(\()p Fo(\025)p Fp(\))14 b(=)1066 1115 y Fl(\022)1127 1155 y Fp(^)-26 b Fo(g)1150 1137 y Fi(b)1167 1155 y Fp(\()p Fo(\025)p Fp(\))1125 1215 y(^)g Fo(g)1148 1197 y Fi(a)1169 1215 y Fp(\()p Fo(\025)p Fp(\))1256 1115 y Fl(\023)1301 1185 y Fo(;)421 b Fz(\(2.39\))89 1317 y(where)485 1389 y Fp(^)-26 b Fo(g)508 1369 y Fi(\034)530 1389 y Fp(\()p Fo(\025)p Fp(\))14 b(:=)675 1321 y Fl(Z)707 1426 y Fk(^)703 1434 y(\012)739 1389 y Fo(dx)814 1334 y Fl(\020)842 1389 y Fo(~)-22 b(g)r Fp(\()p Fo(x)p Fp(\))p Fo(;)963 1376 y(~)957 1389 y( )991 1369 y Fj(\000)1020 1389 y Fp(\()p Fo(x;)8 b(\025;)g(\034)e Fp(\))1185 1334 y Fl(\021)1222 1389 y Fo(;)57 b(\034)19 b Fp(=)14 b Fo(a;)8 b(b:)268 b Fz(\(2.40\))89 1498 y Fp(\010)124 1505 y Fj(\000)167 1498 y Fz(is)13 b(unitary)h(and)g(called)e(the)h(incoming)f(F)l(ourier) g(transformation.)21 b(The)13 b(in)o(v)o(erse)e(incoming)89 1558 y(F)l(ourier)16 b(transformation)g Fp(\010)625 1537 y Fj(\000)p Fk(1)625 1569 y Fj(\000)689 1558 y Fz(is)g(giv)o(en)f(b)o (y)361 1690 y Fp(\(\010)415 1669 y Fj(\000)p Fk(1)415 1702 y Fj(\000)464 1690 y Fp(^)-26 b Fo(g)r Fp(\)\()p Fo(x)p Fp(\))14 b(=)637 1622 y Fl(Z)665 1735 y Fg(R)699 1690 y Fo(d\025)785 1642 y Fl(X)775 1748 y Fi(\034)t Fk(=)p Fi(a;b)880 1676 y Fo(~)874 1690 y( )908 1669 y Fj(\000)937 1690 y Fp(\()p Fo(x;)8 b(\025;)g(\034)e Fp(\))r(^)-26 b Fo(g)1127 1669 y Fi(\034)1148 1690 y Fp(\()p Fo(\025)p Fp(\))p Fo(;)59 b Fp(^)-26 b Fo(g)16 b Fm(2)e Fo(L)1404 1669 y Fk(2)1424 1690 y Fp(\()p Fn(R)p Fo(;)8 b Fn(C)1534 1669 y Fk(2)1557 1690 y Fp(\))p Fo(:)146 b Fz(\(2.41\))89 1844 y(W)l(e)16 b(note)h(that)820 1904 y Fp(\010)855 1911 y Fj(\000)885 1904 y Fo(K)t Fp(\010)965 1883 y Fj(\000)p Fk(1)965 1916 y Fj(\000)1026 1904 y Fp(=)d Fo(M)611 b Fz(\(2.42\))89 1996 y(where)16 b Fo(M)22 b Fz(is)16 b(the)g(m)o (ultipli)o(cation)e(op)q(erator)j(b)o(y)f(the)g(indep)q(enden)o(t)f(v)m (ariable)h Fo(\025)h Fz(on)1669 1983 y Fp(^)1664 1996 y Fh(K)q Fz(,)e(i.e.)422 2098 y(dom)o Fp(\()p Fo(M)5 b Fp(\))42 b(:=)f Fm(f)r Fp(^)-26 b Fo(g)16 b Fm(2)e Fo(L)883 2080 y Fk(2)903 2098 y Fp(\()p Fn(R)p Fo(;)8 b Fn(C)1013 2080 y Fk(2)1036 2098 y Fp(\))13 b(:)h Fo(\025)r Fp(^)-26 b Fo(g)r Fp(\()p Fo(\025)p Fp(\))15 b Fm(2)f Fo(L)1310 2080 y Fk(2)1330 2098 y Fp(\()p Fn(R)o Fo(;)8 b Fn(C)1439 2080 y Fk(2)1462 2098 y Fp(\))p Fm(g)g Fo(;)422 2182 y Fp(\()p Fo(M)f Fp(^)-26 b Fo(g)s Fp(\)\()p Fo(\025)p Fp(\))42 b(:=)f Fo(\025)r Fp(^)-26 b Fo(g)s Fp(\()p Fo(\025)p Fp(\))p Fo(;)59 b Fp(^)-26 b Fo(g)16 b Fm(2)e Fz(dom)o Fp(\()p Fo(M)5 b Fp(\))p Fo(:)1736 2140 y Fz(\(2.43\))89 2287 y(The)15 b(represen)o(tation)f(\(2.43\))i(induced)e(b)o(y)g Fp(\010)926 2294 y Fj(\000)970 2287 y Fz(is)h(called)f(the)g(incoming)g (sp)q(ectral)g(represen)o(ta-)89 2347 y(tion)i(of)h Fo(K)t Fz(.)148 2431 y(Finally)l(,)12 b(w)o(e)g(note)g(that)h(eac)o(h)f(b)q (ounded)i(self-adjoin)o(t)e(op)q(erator)i Fo(G)f Fz(on)g Fh(K)p Fz(,)g(whic)o(h)f(comm)o(utes)89 2491 y(with)23 b Fo(K)t Fz(,)i(corresp)q(onds)g(to)e(a)h(measurable)e(family)f Fm(f)p Fo(G)p Fp(\()p Fo(\025)p Fp(\))p Fm(g)1252 2498 y Fi(\025)p Fj(2)p Fg(R)1346 2491 y Fz(of)j(t)o(w)o(o-b)o(y-t)o(w)o(o)f (matrices,)89 2551 y(whic)o(h)17 b(are)h(uniformly)e(b)q(ounded,)i (i.e.,)e Fo(G)p Fp(\()p Fm(\001)p Fp(\))h Fm(2)g Fo(L)1035 2533 y Fj(1)1072 2551 y Fp(\()p Fn(R)p Fo(;)8 b Fm(B)q Fp(\()p Fn(C)1235 2533 y Fk(2)1258 2551 y Fp(\)\))p Fz(,)18 b(suc)o(h)f(that)h(the)g(m)o(ultiplic)o(a-)89 2611 y(tion)e(op)q (erator)397 2599 y Fp(^)386 2611 y Fo(G)h Fz(on)f Fo(L)541 2593 y Fk(2)561 2611 y Fp(\()p Fn(R)p Fo(;)8 b Fn(C)671 2593 y Fk(2)694 2611 y Fp(\))16 b Fz(de\034ned)g(b)o(y)391 2721 y(dom)o Fp(\()513 2708 y(^)502 2721 y Fo(G)q Fp(\))41 b(:=)g Fm(f)r Fp(^)-26 b Fo(g)16 b Fm(2)e Fo(L)838 2702 y Fk(2)858 2721 y Fp(\()p Fn(R)p Fo(;)7 b Fn(C)968 2702 y Fk(2)990 2721 y Fp(\))14 b(:)g Fo(G)p Fp(\()p Fo(\025)p Fp(\))r(^)-26 b Fo(g)s Fp(\()p Fo(\025)p Fp(\))14 b Fm(2)g Fo(L)1341 2702 y Fk(2)1361 2721 y Fp(\()p Fn(R)p Fo(;)7 b Fn(C)1471 2702 y Fk(2)1493 2721 y Fp(\))p Fm(g)h Fo(;)391 2813 y Fp(\()421 2800 y(^)410 2813 y Fo(G)r Fp(^)-26 b Fo(g)s Fp(\)\()p Fo(\025)p Fp(\))42 b(:=)f Fo(G)p Fp(\()p Fo(\025)p Fp(\))r(^)-26 b Fo(g)s Fp(\()p Fo(\025)p Fp(\))p Fo(;)59 b Fp(^)-26 b Fo(g)16 b Fm(2)e Fz(dom)o Fp(\()1191 2797 y(^)1158 2813 y Fo(G)p Fp(\()p Fm(\001)p Fp(\)\))1736 2764 y Fz(\(2.44\))p eop %%Page: 11 13 11 12 bop 89 23 a Fs(Density)18 b(and)g(current)h(of)f(a)g(dissip)l(a)l (tive)g(Schr)o(\366dinger)h(opera)l(tor)187 b Fz(11)89 147 y(is)16 b(unitarily)f(equiv)m(alen)o(t)g(to)i Fo(G)p Fz(,)f(i.e.)823 260 y Fp(\010)858 267 y Fj(\000)888 260 y Fo(G)p Fp(\010)961 240 y Fj(\000)p Fk(1)961 273 y Fj(\000)1023 260 y Fp(=)1086 248 y(^)1075 260 y Fo(G:)609 b Fz(\(2.45\))89 380 y(Indeed,)23 b(if)f Fo(G)h Fz(comm)o(utes)c(with)k Fo(K)t Fz(,)g(then)945 367 y Fp(^)934 380 y Fo(G)g Fz(comm)o(ute)o(s)d (with)i Fo(M)5 b Fz(.)41 b(Applying)21 b(Theorem)89 440 y(VI)q(I.2.3)12 b(of)h([3])g(one)g(imm)o(ediatel)o(y)d(gets)j(that)938 428 y Fp(^)927 440 y Fo(G)h Fz(is)e(a)i(m)o(ultipli)o(cation)c(op)q (erator)k(of)f(t)o(yp)q(e)g(\(2.44\).)89 500 y(The)j(represen)o(tation) g(\(2.44\))h(is)f(called)f(the)h(incoming)f(sp)q(ectral)h(represen)o (tation)f(of)i Fo(G)p Fz(.)89 695 y Fx(3)77 b(Carrier)26 b(densit)n(y)89 829 y Fz(In)16 b(the)g(follo)o(wing)g(w)o(e)f(call)h (an)g(op)q(erator)i Fo(\045)c Fp(:)f Fh(K)h Fm(\000)-8 b(!)13 b Fh(K)k Fz(a)f(densit)o(y)g(matrix)e(if)i Fo(\045)g Fz(is)g(a)g(b)q(ounded,)89 889 y(non-negativ)o(e)i(op)q(erator.)26 b(The)18 b(op)q(erator)g Fo(\045)g Fz(is)g(called)e(a)i(steady)g(state) g(if)f Fo(\045)g Fz(comm)o(utes)e(with)89 950 y Fo(K)t Fz(.)26 b(Ob)o(viously)l(,)17 b(a)h(steady)g(state)g(do)q(es)h(not)f(c) o(hange)g(in)g(time.)24 b(If)17 b Fo(\045)h Fz(is)g(a)g(steady)g (state,)g(then)89 1010 y(there)j(is)h(a)g(measurable)e(matrix-v)m (alued)g(function)i Fo(\045)p Fp(\()p Fm(\001)p Fp(\))h Fm(2)h Fo(L)1284 992 y Fj(1)1321 1010 y Fp(\()p Fn(R)p Fo(;)8 b Fm(B)q Fp(\()p Fn(C)1484 992 y Fk(2)1507 1010 y Fp(\)\))21 b Fz(suc)o(h)h(that)g(the)89 1070 y(m)o(ultiplic)o(ation) 16 b(op)q(erator)25 b Fp(^)-29 b Fo(\045)19 b Fz(on)g Fo(L)750 1052 y Fk(2)770 1070 y Fp(\()p Fn(R)p Fo(;)8 b Fn(C)880 1052 y Fk(2)903 1070 y Fp(\))18 b Fz(generated)h(b)o(y)f Fo(\045)p Fp(\()p Fm(\001)p Fp(\))h Fz(is)f(unitarily)f(equiv)m(alen)o (t)g(to)89 1130 y Fo(\045)p Fz(,)f(i.e.)837 1193 y Fo(\045)e Fp(=)f(\010)962 1173 y Fj(\000)p Fk(1)962 1206 y Fj(\000)1014 1193 y Fp(^)-28 b Fo(\045)p Fp(\010)1070 1200 y Fj(\000)1100 1193 y Fo(;)647 b Fz(\(3.1\))89 1284 y(see)16 b(ab)q(o)o(v)o(e.)21 b(Ob)o(viously)l(,)15 b(the)h(measurable)f(function)h Fo(\045)p Fp(\()p Fm(\001)p Fp(\))g Fz(tak)o(es)g(the)g(form)689 1426 y Fo(\045)p Fp(\()p Fo(\025)p Fp(\))e(=)846 1355 y Fl(\022)905 1395 y Fo(\045)930 1377 y Fi(bb)963 1395 y Fp(\()p Fo(\025)p Fp(\))45 b Fo(\045)1099 1377 y Fi(ba)1135 1395 y Fp(\()p Fo(\025)p Fp(\))904 1455 y Fo(\045)929 1437 y Fi(ab)965 1455 y Fp(\()p Fo(\025)p Fp(\))d Fo(\045)1098 1437 y Fi(aa)1137 1455 y Fp(\()p Fo(\025)p Fp(\))1224 1355 y Fl(\023)1761 1426 y Fz(\(3.2\))89 1568 y(where)19 b Fo(\045)258 1549 y Fi(\034)t(\027)299 1568 y Fp(\()p Fm(\001)p Fp(\))f Fm(2)h Fo(L)454 1549 y Fj(1)491 1568 y Fp(\()p Fn(R)p Fp(\))p Fz(,)d Fo(\034)s(;)8 b(\027)21 b Fp(=)e Fo(a;)8 b(b)p Fz(.)28 b(Since)18 b Fo(\045)g Fm(\025)g Fp(0)i Fz(one)f(gets)g(that)g Fo(\045)p Fp(\()p Fo(\025)p Fp(\))g Fm(\025)f Fp(0)h Fz(a.e.)29 b(\(with)89 1628 y(resp)q(ect)16 b(to)h(the)f(Leb)q(esgue)h(measure\).)89 1771 y Fq(De\034nition)j(3.1)k Fz(A)12 b(b)q(ounded)i(self-adjoin)o(t)f (op)q(erator)h Fo(A)e Fz(on)i Fh(K)f Fz(is)g(called)e(an)j(observ)m (able.)20 b(W)l(e)89 1831 y(sa)o(y)c(the)g(observ)m(able)h Fo(A)135 1960 y Fz(\(i\))24 b(is)11 b(admissible)f(with)h(resp)q(ect)g (to)h Fo(\045)f Fz(if)g Fo(\045A)g Fz(is)g(a)h(n)o(uclear)e(op)q (erator)j(on)f Fh(K)p Fz(,)g(i.e.)18 b Fo(\045A)c Fm(2)g(L)1756 1967 y Fk(1)1776 1960 y Fp(\()p Fh(K)p Fp(\))p Fz(,)122 2067 y(\(ii\))23 b(is)15 b(admissible)f(with)h(resp)q(ect)g(to)g Fo(K)20 b Fz(if)14 b Fo(E)969 2074 y Fi(K)1004 2067 y Fp(\(\001\))p Fo(A)f Fm(2)h(L)1214 2074 y Fk(1)1234 2067 y Fp(\()p Fh(K)p Fp(\))h Fz(for)h(eac)o(h)f(b)q(ounded)h(in)o(terv)m (al)211 2127 y Fp(\001)e Fm(\022)f Fn(R)g Fz(where)j Fo(E)547 2134 y Fi(K)582 2127 y Fp(\()p Fm(\001)p Fp(\))g Fz(denotes)g(the)g(sp)q(ectral)g(measure)f(of)i Fo(K)t Fz(.)89 2271 y(If)d(the)g(observ)m(able)h Fo(A)f Fz(is)g(admissible)f (with)h(resp)q(ect)g(to)h Fo(\045)p Fz(,)f(then)h(its)f(exp)q(ectation) g(v)m(alue)g Fn(E)1763 2278 y Fi(\045)1786 2271 y Fp(\()p Fo(A)p Fp(\))89 2331 y Fz(with)i(resp)q(ect)g(to)h(the)f(densit)o(y)f (matrix)f Fo(\045)j Fz(is)f(de\034ned)g(b)o(y)796 2444 y Fn(E)826 2451 y Fi(\045)849 2444 y Fp(\()p Fo(A)p Fp(\))e(:=)f Fz(tr)p Fp(\()p Fo(\045A)p Fp(\))p Fo(:)606 b Fz(\(3.3\))89 2556 y(In)19 b(the)f(follo)o(wing)h(w)o(e)f(sho)o(w)h(that)g(the)g (admissibilit)o(y)c(of)k Fo(A)g Fz(with)f(resp)q(ect)h(to)g Fo(K)k Fz(leads)18 b(to)h(a)89 2617 y(certain)d(lo)q(calization)f(in)h (the)g(incoming)f(sp)q(ectral)h(represen)o(tation:)89 2746 y Fq(Prop)r(osition)k(3.2)k Fc(Assume)f Fo(m)15 b Fp(+)798 2726 y Fk(1)p 791 2734 32 2 v 791 2763 a Fi(m)852 2746 y Fm(2)24 b Fo(L)942 2728 y Fj(1)980 2746 y Fp(\([)p Fo(a;)8 b(b)p Fp(]\))p Fc(,)22 b Fo(V)36 b Fm(2)24 b Fo(L)1306 2728 y Fk(2)1326 2746 y Fp(\([)p Fo(a;)8 b(b)p Fp(]\))21 b Fc(and)j Fo(\024)1611 2753 y Fi(a)1632 2746 y Fo(;)8 b(\024)1682 2753 y Fi(b)1723 2746 y Fm(2)24 b Fn(C)1813 2753 y Fk(+)1846 2746 y Fc(.)89 2806 y(If)19 b(the)h(observable)i Fo(A)d Fc(is)g(admissible)i(with)e(r)n(esp)n(e)n (ct)g(to)h Fo(K)t Fc(,)g(then)h(ther)n(e)e(exists)h(a)g(me)n(asur)n (able)p eop %%Page: 12 14 12 13 bop 89 23 a Fz(12)692 b Fs(H.-Ch.)25 b(Kaiser,)18 b(H.)h(Neidhardt,)g(J.)g(Rehber)o(g)89 147 y Fc(matrix-value)n(d)14 b(function)h Fo(A)p Fp(\()p Fm(\001)p Fp(\))e Fm(2)h Fo(L)759 129 y Fk(1)759 160 y Fi(loc)805 147 y Fp(\()p Fn(R)p Fo(;)7 b Fm(B)r Fp(\()p Fn(C)968 129 y Fk(2)990 147 y Fp(\)\))p Fc(,)14 b(such)g(that)f Fo(A)p Fp(\()p Fo(\025)p Fp(\))h(=)g Fo(A)p Fp(\()p Fo(\025)p Fp(\))1531 129 y Fj(\003)1564 147 y Fc(for)e(a.e.)22 b Fo(\025)14 b Fm(2)g Fn(R)89 208 y Fc(and)575 282 y Fz(tr)o Fp(\()p Fo(\045AE)729 289 y Fi(K)763 282 y Fp(\(\001\)\))g(=)927 214 y Fl(Z)954 327 y Fk(\001)994 282 y Fo(d\025)h Fz(tr)1100 290 y Fg(C)1121 281 y Ff(2)1143 282 y Fp(\()p Fo(\045)p Fp(\()p Fo(\025)p Fp(\))p Fo(A)p Fp(\()p Fo(\025)p Fp(\)\))386 b Fz(\(3.4\))89 395 y Fc(for)21 b(any)h(b)n(ounde)n(d)g(Bor)n(el)f(set) i Fp(\001)e Fm(\022)h Fn(R)c Fc(and)k(any)g(ste)n(ady)f(state)h Fo(\045)g Fc(of)g Fo(K)t Fc(.)35 b(The)22 b(me)n(asur)n(able)89 455 y(function)d Fo(A)p Fp(\()p Fm(\001)p Fp(\))e Fc(is)g(uniquely)i (de\034ne)n(d)f(up)g(to)f(a)h(Bor)n(el)f(set)h(of)f(L)n(eb)n(esgue)h (me)n(asur)n(e)f(zer)n(o.)148 539 y(If)k(the)g(observable)i Fo(A)e Fc(is)g(in)g(addition)g(admissible)g(with)h(r)n(esp)n(e)n(ct)e (to)h(the)h(ste)n(ady)e(state)i Fo(\045)p Fc(,)89 599 y(then)c Fz(tr)235 608 y Fg(C)256 598 y Ff(2)279 599 y Fp(\()p Fo(\045)p Fp(\()p Fm(\001)p Fp(\))p Fo(A)p Fp(\()p Fm(\001)p Fp(\)\))13 b Fm(2)h Fo(L)576 581 y Fk(1)596 599 y Fp(\()p Fn(R)p Fp(\))g Fc(and)k(the)g(r)n(epr)n (esentation)652 736 y Fz(tr)p Fp(\()p Fo(\045A)p Fp(\))13 b(=)855 668 y Fl(Z)883 781 y Fg(R)917 736 y Fo(d\025)i Fz(tr)1023 744 y Fg(C)1043 735 y Ff(2)1066 736 y Fp(\()p Fo(\045)p Fp(\()p Fo(\025)p Fp(\))p Fo(A)p Fp(\()p Fo(\025)p Fp(\)\))463 b Fz(\(3.5\))89 872 y Fc(holds.)89 1000 y Fq(Pro)r(of.)23 b Fz(Notice)16 b(that)h(the)f(sp)q(ectral)h(measure)e Fo(E)1018 1007 y Fi(K)1053 1000 y Fp(\()p Fm(\001)p Fp(\))h Fz(of)h Fo(K)k Fz(is)c(absolutely)f(con)o(tin)o(uous)h(with)89 1060 y(resp)q(ect)f(to)h(Leb)q(esgue)g(measure.)i(Hence,)c(the)h(set)g (function)703 1173 y Fo(\026)732 1180 y Fi(\045;A)789 1173 y Fp(\(\001\))d(:=)h Fz(tr)p Fp(\()p Fo(\045AE)1102 1180 y Fi(K)1136 1173 y Fp(\(\001\)\))p Fo(;)513 b Fz(\(3.6\))89 1285 y(where)22 b Fp(\001)g Fz(is)g(b)q(ounded)h(Borel)e(set)i(of)f Fn(R)p Fz(,)e(is)i(in)g(fact)h(a)f(Leb)q(esgue)h(absolutely)f(con)o (tin)o(uous)89 1345 y(measure.)e(Let)390 1320 y Fi(d\026)429 1326 y Fb(\032;A)p 390 1334 90 2 v 416 1363 a Fi(d\025)501 1345 y Fz(denote)d(its)f(Radon-Nik)o(o)q(dym)f(deriv)m(ativ)o(e)f(and)j (de\034ne)808 1487 y Fo(T)837 1494 y Fi(A)865 1487 y Fp(\()t(^)-28 b Fo(\045)p Fp(\))14 b(:=)1012 1454 y Fo(d\026)1066 1461 y Fi(\045;A)p 1012 1476 112 2 v 1041 1521 a Fo(d\025)1128 1487 y(:)619 b Fz(\(3.7\))89 1617 y(Then)17 b Fo(T)246 1624 y Fi(A)274 1617 y Fp(\()p Fm(\001)p Fp(\))g Fz(maps)g Fo(L)505 1599 y Fj(1)542 1617 y Fp(\()p Fn(R)p Fo(;)8 b Fm(B)q Fp(\()p Fn(C)705 1599 y Fk(2)728 1617 y Fp(\)\))17 b Fz(con)o(tin)o(uously)f(in)o(to)h Fo(L)1200 1599 y Fk(1)1200 1630 y Fi(loc)1246 1617 y Fp(\()p Fn(R)p Fp(\))d Fz(\(the)j(latter)f(in)h(its)g(canonic)89 1677 y(F)l(rec)o(het)e(top)q (ology\))i(and,)g(additionally)l(,)e(one)h(has)444 1815 y(tr)p Fp(\()p Fo(\045AE)599 1822 y Fi(K)633 1815 y Fp(\(\001\)\))d(=) 796 1747 y Fl(Z)824 1860 y Fk(\001)864 1815 y Fo(d\025)936 1781 y(d\026)990 1788 y Fi(\045;A)p 936 1803 V 965 1849 a Fo(d\025)1052 1815 y Fp(\()p Fo(\025)p Fp(\))h(=)1184 1747 y Fl(Z)1212 1860 y Fk(\001)1252 1815 y Fo(d\025)g(T)1348 1822 y Fi(A)1376 1815 y Fp(\()t(^)-28 b Fo(\045)q Fp(\)\()p Fo(\025)p Fp(\))255 b Fz(\(3.8\))89 1953 y(for)13 b(an)o(y)f(b)q (ounded)h(Borel)f(set)g Fp(\001)h Fm(\032)h Fn(R)p Fz(.)j(It)12 b(is)g(not)h(hard)g(to)f(see)g(that)h(\(3.8\))g(implies)c(the)k (equalit)o(y)760 2066 y Fo(T)789 2073 y Fi(A)817 2066 y Fp(\()p Fo(\037)867 2073 y Fk(\001)903 2066 y Fp(^)-29 b Fo(\045)p Fp(\))14 b(=)g Fo(\037)1039 2073 y Fk(\001)1070 2066 y Fo(T)1099 2073 y Fi(A)1127 2066 y Fp(\()t(^)-28 b Fo(\045)p Fp(\))571 b Fz(\(3.9\))89 2178 y(for)17 b(an)o(y)f(b)q (ounded)h(Borel)e(set)h Fp(\001)e Fm(\022)f Fn(R)h Fz(where)i Fo(\037)992 2185 y Fk(\001)1039 2178 y Fz(denotes)g(its)g(indicator)g (function.)21 b(W)l(e)16 b(put)161 2317 y Fo(e)184 2324 y Fi(bb)230 2317 y Fp(:=)295 2247 y Fl(\022)353 2286 y Fp(1)41 b(0)353 2346 y(0)g(0)464 2247 y Fl(\023)509 2317 y Fo(;)21 b(e)567 2324 y Fi(ba)617 2317 y Fp(:=)682 2247 y Fl(\022)739 2286 y Fp(0)42 b(1)739 2346 y(0)g(0)850 2247 y Fl(\023)895 2317 y Fo(;)22 b(e)954 2324 y Fi(ab)1003 2317 y Fp(:=)1069 2247 y Fl(\022)1126 2286 y Fp(0)42 b(0)1126 2346 y(1)g(0)1237 2247 y Fl(\023)1282 2317 y Fo(;)22 b(e)1341 2324 y Fi(aa)1394 2317 y Fp(:=)1459 2247 y Fl(\022)1517 2286 y Fp(0)41 b(0)1517 2346 y(0)g(1)1628 2247 y Fl(\023)1736 2317 y Fz(\(3.10\))89 2458 y(and)17 b(de\034ne)f(for)g(an)o(y)g Fo(h)e Fm(2)g Fo(L)613 2440 y Fj(1)651 2458 y Fp(\()p Fn(R)p Fp(\))599 2571 y Fo(A)636 2578 y Fi(ij)666 2571 y Fp(\()p Fo(h)p Fp(\))g(:=)f Fo(T)840 2578 y Fi(A)868 2571 y Fp(\()p Fo(h)e Fm(\001)g Fo(e)974 2578 y Fi(j)r(i)1004 2571 y Fp(\))p Fo(;)105 b(i;)8 b(j)17 b Fp(=)c Fo(a;)8 b(b:)384 b Fz(\(3.11\))89 2683 y(Clearly)l(,)21 b(eac)o(h)g(of)g(the)g(mappings)g Fo(A)800 2690 y Fi(ij)829 2683 y Fp(\()p Fm(\001)p Fp(\))g Fz(then)g(maps)g Fo(L)1184 2665 y Fj(1)1221 2683 y Fp(\()p Fn(R)p Fp(\))d Fz(con)o(tin)o(uously)j (in)o(to)f Fo(L)1741 2665 y Fk(1)1741 2696 y Fi(loc)1787 2683 y Fp(\()p Fn(R)p Fp(\))89 2743 y Fz(and)d(\(3.11\))g(implies)747 2806 y Fo(A)784 2813 y Fi(ij)814 2806 y Fp(\()p Fo(\037)864 2813 y Fk(\001)895 2806 y Fo(h)p Fp(\))d(=)g Fo(\037)1039 2813 y Fk(\001)1070 2806 y Fo(A)1107 2813 y Fi(ij)1137 2806 y Fp(\()p Fo(h)p Fp(\))533 b Fz(\(3.12\))p eop %%Page: 13 15 13 14 bop 89 23 a Fs(Density)18 b(and)g(current)h(of)f(a)g(dissip)l(a)l (tive)g(Schr)o(\366dinger)h(opera)l(tor)187 b Fz(13)89 147 y(for)17 b(an)o(y)f Fo(h)e Fm(2)g Fo(L)378 129 y Fj(1)415 147 y Fp(\()p Fn(R)p Fp(\))f Fz(and)k(an)o(y)f(Borel)g(set)g Fp(\001)p Fz(.)21 b(T)l(aking)16 b(in)g(particular)g Fo(h)e Fm(\021)g Fp(1)p Fz(,)i(this)g(yields)756 233 y Fo(A)793 240 y Fi(ij)823 233 y Fp(\()p Fo(\037)873 240 y Fk(\001)904 233 y Fp(\))e(=)g Fo(\037)1020 240 y Fk(\001)1051 233 y Fo(A)1088 240 y Fi(ij)1118 233 y Fp(\(1\))p Fo(:)542 b Fz(\(3.13\))89 319 y(Ob)o(viously)l(,)17 b(b)o(y)h(the)g(linearit)o(y)f(of)h(the)h(mappings)e Fo(A)1083 326 y Fi(ij)1113 319 y Fp(\()p Fm(\001)p Fp(\))p Fz(,)h(this)h(last)f(equation)g(remains)f(true)89 379 y(if)22 b Fo(\037)171 386 y Fk(\001)225 379 y Fz(is)h(there)f(replaced) g(b)o(y)g(an)o(y)h(\(\034nite\))f(linear)g(com)o(bination)f(of)i (indicator)f(functions.)89 439 y(Because)11 b(the)h(set)g(of)g (\034nite)f(linear)h(com)o(binations)e(of)i(indicator)g(functions)g(is) g(dense)f(in)h Fo(L)1736 421 y Fj(1)1773 439 y Fp(\()p Fn(R)p Fp(\))p Fz(,)89 499 y(one)17 b(gets)f(for)h(all)e Fo(h)f Fm(2)g Fo(L)543 481 y Fj(1)581 499 y Fp(\()p Fn(R)p Fp(\))f Fz(the)j(equation)772 585 y Fo(A)809 592 y Fi(ij)839 585 y Fp(\()p Fo(h)p Fp(\))e(=)g Fo(h)d Fm(\001)g Fo(A)1072 592 y Fi(ij)1102 585 y Fp(\(1\))p Fo(:)558 b Fz(\(3.14\))89 671 y(Since)471 731 y Fo(\045)p Fp(\()p Fm(\001)p Fp(\))14 b(=)f Fo(\045)638 711 y Fi(bb)671 731 y Fp(\()p Fm(\001)p Fp(\))p Fo(e)746 738 y Fi(bb)789 731 y Fp(+)e Fo(\045)863 711 y Fi(ba)899 731 y Fp(\()p Fm(\001)p Fp(\))p Fo(e)974 738 y Fi(ba)1020 731 y Fp(+)g Fo(\045)1094 711 y Fi(ab)1130 731 y Fp(\()p Fm(\001)p Fp(\))p Fo(e)1205 738 y Fi(ab)1251 731 y Fp(+)g Fo(\045)1325 711 y Fi(aa)1365 731 y Fp(\()p Fm(\001)p Fp(\))p Fo(e)1440 738 y Fi(aa)1736 731 y Fz(\(3.15\))89 808 y(one)17 b(gets)251 915 y(tr)p Fp(\()p Fo(\045AE)406 922 y Fi(K)440 915 y Fp(\(\001\)\))d(=)603 847 y Fl(Z)631 960 y Fk(\001)671 915 y Fo(d\025)h(T)768 922 y Fi(A)796 915 y Fp(\()t(^)-28 b Fo(\045)p Fp(\)\()p Fo(\025)p Fp(\))14 b(=)759 b Fz(\(3.16\))471 997 y Fl(X)452 1103 y Fi(i;j)r Fk(=)p Fi(a;b)570 977 y Fl(Z)598 1089 y Fk(\001)637 1044 y Fo(d\025)15 b(A)742 1051 y Fi(ij)772 1044 y Fp(\()p Fo(\045)816 1024 y Fi(ij)846 1044 y Fp(\)\()p Fo(\025)p Fp(\))f(=)1016 997 y Fl(X)997 1103 y Fi(i;j)r Fk(=)p Fi(a;b)1115 977 y Fl(Z)1142 1089 y Fk(\001)1182 1044 y Fo(d\025)h(\045)1275 1024 y Fi(ij)1305 1044 y Fp(\()p Fo(\025)p Fp(\))p Fo(A)1408 1051 y Fi(j)r(i)1438 1044 y Fp(\(1\)\()p Fo(\025)p Fp(\))p Fo(:)89 1179 y Fz(Setting)625 1265 y Fo(A)p Fp(\()p Fm(\001)p Fp(\))e(:=)792 1195 y Fl(\022)851 1235 y Fo(A)888 1242 y Fi(bb)920 1235 y Fp(\(1\)\()p Fm(\001)p Fp(\))46 b Fo(A)1117 1242 y Fi(ba)1152 1235 y Fp(\(1\)\()p Fm(\001)p Fp(\))850 1295 y Fo(A)887 1302 y Fi(ab)922 1295 y Fp(\(1\)\()p Fm(\001)p Fp(\))c Fo(A)1115 1302 y Fi(aa)1154 1295 y Fp(\(1\)\()p Fm(\001)p Fp(\))1289 1195 y Fl(\023)1736 1265 y Fz(\(3.17\))89 1371 y(w)o(e)17 b(\034nally)f(obtain)h(\(3.4\))h(for)f(an)o(y)g(b)q(ounded)h(Borel)e (set)g Fp(\001)f Fm(\032)g Fn(R)f Fz(and)j(an)o(y)g(steady)g(state)g Fo(\045)h Fz(of)89 1431 y Fo(K)t Fz(.)148 1515 y(Assume)e(that)450 1502 y Fp(~)437 1515 y Fo(A)p Fp(\()p Fm(\001)p Fp(\))h Fz(ob)q(eys)h(also)g(the)f(conditions)g(of)h(the)f(prop)q(osition.)26 b(Setting)18 b Fo(G)p Fp(\()p Fo(\025)p Fp(\))e(:=)89 1575 y Fo(A)p Fp(\()p Fo(\025)p Fp(\))11 b Fm(\000)266 1563 y Fp(~)253 1575 y Fo(A)p Fp(\()p Fo(\025)p Fp(\))p Fz(,)16 b Fo(\025)e Fm(2)g Fn(R)p Fz(,)f(one)j(gets)h(that)738 1617 y Fl(Z)766 1730 y Fk(\001)806 1685 y Fz(tr)844 1693 y Fg(C)865 1684 y Ff(2)887 1685 y Fp(\()p Fo(\045)p Fp(\()p Fo(\025)p Fp(\))p Fo(G)p Fp(\()p Fo(\025)p Fp(\)\))e(=)f(0)525 b Fz(\(3.18\))89 1797 y(for)17 b(an)o(y)f(b)q(ounded)h(Borel)e(set)h Fp(\001)f Fz(and)i(an)o(y)f(steady)g(state)h Fo(\045)p Fz(.)k(Hence)15 b(tr)1419 1806 y Fg(C)1440 1796 y Ff(2)1462 1797 y Fp(\()p Fo(\045)p Fp(\()p Fo(\025)p Fp(\))p Fo(G)p Fp(\()p Fo(\025)p Fp(\)\))g(=)f(0)j Fz(for)89 1857 y(a.e.)26 b Fo(\025)18 b Fz(and)h(an)o(y)f(steady)g(state)g Fo(\045)g Fz(whic)o(h)f(imme)o(diately)d(yields)j Fo(G)p Fp(\()p Fo(\025)p Fp(\))h(=)e(0)i Fz(or)h Fo(A)p Fp(\()p Fo(\025)p Fp(\))d(=)1771 1845 y(~)1758 1857 y Fo(A)o Fp(\()p Fo(\025)p Fp(\))89 1918 y Fz(for)h(a.e.)j Fo(\025)15 b Fm(2)f Fn(R)p Fz(.)148 2001 y(If)26 b Fo(\045)f Fz(is)h(admissible)e(with)h(resp)q (ect)h(to)g Fo(A)p Fz(,)h(then)f Fm(j)p Fz(tr)p Fp(\()p Fo(\045AE)1296 2008 y Fi(K)1330 2001 y Fp(\(\001\))p Fm(j)j Fo(<)h Fm(k)p Fo(\045A)p Fm(k)1632 2008 y Fj(L)1656 2013 y Ff(1)1701 2001 y Fz(for)c(an)o(y)89 2062 y(Borel)d(set)h Fp(\001)j Fm(\022)g Fn(R)p Fz(.)42 b(By)24 b(\(3.4\))g(this)g(implies)d (that)k(tr)p Fp(\()p Fo(\045)p Fp(\()p Fm(\001)p Fp(\))p Fo(A)p Fp(\()p Fm(\001)p Fp(\)\))h Fm(2)i Fo(L)1500 2043 y Fk(1)1520 2062 y Fp(\()p Fn(R)p Fp(\))p Fz(.)41 b(Since)24 b(one)89 2122 y(has)17 b Fp(lim)244 2129 y Fk(\001)p Fj(")p Fg(R)325 2122 y Fz(tr)p Fp(\()p Fo(\045AE)480 2129 y Fi(K)514 2122 y Fp(\(\001\)\))d(=)f Fz(tr)p Fp(\()p Fo(\045A)p Fp(\))j Fz(w)o(e)g(obtain)h(from)e(\(3.4\))h(the)g(equalit)o (y)f(\(3.5\).)152 b Fa(\003)148 2206 y Fz(Prop)q(osition)21 b(3.2)f(sa)o(ys)g(that)g(the)f(a)o(v)o(eraging)h(pro)q(cedure)g(lo)q (calizes)e(with)i(resp)q(ect)f(to)h(the)89 2266 y(incoming)10 b(sp)q(ectral)i(represen)o(tation.)19 b(Indeed,)12 b(the)g(quan)o(tit)o (y)e(tr)p Fp(\()p Fo(\045)p Fp(\()p Fo(\025)p Fp(\))p Fo(A)p Fp(\()p Fo(\025)p Fp(\)\))j Fz(can)f(b)q(e)g(regarded)89 2326 y(as)17 b(the)f(lo)q(cal)h(a)o(v)o(erage)e(of)i(the)f(observ)m (able)h Fo(A)p Fp(\()p Fo(\025)p Fp(\))f Fz(with)g(resp)q(ect)g(to)h (the)f(densit)o(y)g(matrix)e Fo(\045)p Fp(\()p Fo(\025)p Fp(\))89 2386 y Fz(at)j(energy)f Fo(\025)e Fm(2)g Fn(R)p Fz(,)f(i.e.)672 2446 y Fn(E)703 2454 y Fi(\045)p Fk(\()p Fi(\025)p Fk(\))774 2446 y Fp(\()p Fo(A)p Fp(\()p Fo(\025)p Fp(\)\))g(:=)h Fz(tr)p Fp(\()p Fo(\045)p Fp(\()p Fo(\025)p Fp(\))p Fo(A)p Fp(\()p Fo(\025)p Fp(\)\))p Fo(:)458 b Fz(\(3.19\))89 2523 y(F)l(orm)o(ula)15 b(\(3.5\))h(has)h(the)f(meaning) f(that)h(the)g(total)g(a)o(v)o(erage)g Fn(E)1273 2530 y Fi(\045)1296 2523 y Fp(\()p Fo(A)p Fp(\))g Fz(is)g(the)g(sum)f(of)h (the)g(lo)q(cal)89 2583 y(a)o(v)o(erages)g Fn(E)313 2591 y Fi(\045)p Fk(\()p Fi(\025)p Fk(\))385 2583 y Fp(\()p Fo(A)p Fp(\()p Fo(\025)p Fp(\)\))p Fz(,)f(i.e.)686 2697 y Fn(E)716 2704 y Fi(\045)739 2697 y Fp(\()p Fo(A)p Fp(\))e(=)879 2629 y Fl(Z)907 2742 y Fg(R)941 2697 y Fo(d\025)i Fn(E)1039 2705 y Fi(\045)p Fk(\()p Fi(\025)p Fk(\))1110 2697 y Fp(\()p Fo(A)p Fp(\()p Fo(\025)p Fp(\)\))p Fo(:)471 b Fz(\(3.20\))89 2806 y(Prop)q(osition)18 b(3.2)e(giv)o(es)g(rise)f(to)i (the)f(follo)o(wing)p eop %%Page: 14 16 14 15 bop 89 23 a Fz(14)692 b Fs(H.-Ch.)25 b(Kaiser,)18 b(H.)h(Neidhardt,)g(J.)g(Rehber)o(g)89 147 y Fq(De\034nition)h(3.3)k Fz(Let)f Fo(A)h Fz(b)q(e)f(an)i(observ)m(able)e(whic)o(h)g(is)h (admissible)d(with)j(resp)q(ect)f(to)h Fo(K)t Fz(.)89 208 y(An)e(elemen)n(t)d Fo(A)p Fp(\()p Fm(\001)p Fp(\))k Fm(2)g Fo(L)561 189 y Fk(1)561 221 y Fi(loc)606 208 y Fp(\()p Fn(R)p Fo(;)8 b Fm(B)q Fp(\()p Fn(C)769 189 y Fk(2)792 208 y Fp(\)\))22 b Fz(is)f(called)g(a)h(lo)q(calizer)e(of)i Fo(A)g Fz(with)f(resp)q(ect)h(to)g Fo(K)j Fz(if)89 268 y Fo(A)p Fp(\()p Fo(\025)p Fp(\))192 250 y Fj(\003)226 268 y Fp(=)13 b Fo(A)p Fp(\()p Fo(\025)p Fp(\))k Fz(for)f(a.e.)21 b Fo(\025)14 b Fm(2)g Fn(R)g Fz(and)596 405 y(tr)p Fp(\()p Fo(\045AE)751 412 y Fi(K)785 405 y Fp(\(\001\)\))g(=)948 337 y Fl(Z)976 450 y Fk(\001)1016 405 y Fo(d\025)g Fz(tr)p Fp(\()p Fo(\045)p Fp(\()p Fo(\025)p Fp(\))p Fo(A)p Fp(\()p Fo(\025)p Fp(\)\))383 b Fz(\(3.21\))89 541 y(holds)17 b(for)f(an)o(y)g(b)q(ounded)h(Borel)f(subset)g Fp(\001)e Fm(\022)f Fn(R)h Fz(and)j(an)o(y)f(steady)g(state)g Fo(\045)p Fz(.)89 685 y(Prop)q(osition)j(3.2)g(sa)o(ys)f(that)g(an)g(observ)m (able)h(whic)o(h)e(is)g(admissible)f(with)i(resp)q(ect)g(to)g Fo(K)k Fz(has)89 745 y(alw)o(a)o(ys)16 b(a)h(unique)e(lo)q(calizer.)148 829 y(T)l(o)k(calculate)f(the)g(carrier)g(densit)o(y)g(w)o(e)g (consider)g(the)g(observ)m(able)h Fo(U)5 b Fp(\()p Fo(!)r Fp(\))p Fz(,)19 b Fo(!)h Fm(\022)e Fp(\012)p Fz(,)h(giv)o(en)89 889 y(b)o(y)471 953 y Fp(\()p Fo(U)5 b Fp(\()p Fo(!)r Fp(\))605 940 y Fo(~)598 953 y(f)19 b Fp(\)\()p Fo(x)p Fp(\))14 b(=)g(0)d Fm(\010)g Fo(\037)908 960 y Fi(!)933 953 y Fp(\()p Fo(x)p Fp(\))p Fo(f)5 b Fp(\()p Fo(x)p Fp(\))10 b Fm(\010)h Fp(0)p Fo(;)1256 940 y(~)1249 953 y(f)19 b Fm(2)c Fo(L)1373 932 y Fk(2)1392 953 y Fp(\()1416 940 y(^)1411 953 y(\012)q(\))p Fo(;)256 b Fz(\(3.22\))89 1043 y(for)19 b(an)o(y)f(Borel)f(subset)i Fo(!)h Fm(\022)d Fp(\012)p Fz(.)27 b(W)l(e)18 b(note)h(that)g(the)f(observ)m(able)g Fo(U)5 b Fp(\()p Fo(!)r Fp(\))19 b Fz(is)f(a)h(pro)s(jection)e(on)89 1104 y Fh(K)j Fz(with)f(ran)p Fp(\()p Fo(U)5 b Fp(\()p Fo(!)r Fp(\)\))20 b Fm(\022)e Fh(H)p Fz(.)31 b(Lo)q(osely)20 b(sp)q(eaking,)g(the)f(pro)s(jection)g(acts)g(in)g(fact)g(only)g(on)h (the)89 1164 y(subspace)d Fh(H)d Fm(\022)f Fh(K)p Fz(.)89 1293 y Fq(Lemma)18 b(3.4)24 b Fc(Assume)16 b Fo(m)9 b Fp(+)674 1274 y Fk(1)p 667 1282 32 2 v 667 1310 a Fi(m)717 1293 y Fm(2)14 b Fo(L)797 1275 y Fj(1)835 1293 y Fp(\([)p Fo(a;)8 b(b)p Fp(]\))p Fc(,)14 b Fo(V)25 b Fm(2)14 b Fo(L)1132 1275 y Fk(2)1152 1293 y Fp(\([)p Fo(a;)8 b(b)p Fp(]\))15 b Fc(and)h Fo(\024)1423 1300 y Fi(a)1444 1293 y Fo(;)8 b(\024)1494 1300 y Fi(b)1525 1293 y Fm(2)14 b Fn(C)1605 1300 y Fk(+)1638 1293 y Fc(.)22 b(Then)16 b(for)89 1353 y(any)j(Bor)n(el)f(set)h Fo(!)f Fm(\022)e Fp(\012)j Fc(the)g(observable)h Fo(U)5 b Fp(\()p Fo(!)r Fp(\))19 b Fc(is)f(admissible)h(with)g(r)n(esp)n(e)n(ct)f(to)h(the)g (minimal)89 1414 y(self-adjoint)h(dilation)f Fo(K)j Fc(of)c(the)h (maximal)g(dissip)n(ative)f(op)n(er)n(ator)f Fo(H)t Fc(.)25 b(If)18 b(the)h(ste)n(ady)f(state)h Fo(\045)89 1474 y Fc(satis\034es)f(the)g(c)n(ondition)597 1587 y Fo(C)635 1594 y Fk(^)-21 b Fi(\045)666 1587 y Fp(:=)14 b(sup)734 1628 y Fi(\025)p Fj(2)p Fg(R)813 1543 y Fm(p)p 855 1543 133 2 v 44 x Fo(\025)883 1572 y Fk(2)914 1587 y Fp(+)d(1)p Fm(k)p Fo(\045)p Fp(\()p Fo(\025)p Fp(\))p Fm(k)1128 1595 y Fj(B)q Fk(\()p Fg(C)1188 1586 y Ff(2)1208 1595 y Fk(\))1237 1587 y Fo(<)j Fm(1)p Fo(;)383 b Fz(\(3.23\))89 1729 y Fc(then)18 b(for)f(any)h(Bor)n(el)f(set)h Fo(!)e Fm(\022)e Fp(\012)j Fc(the)h(observable)i Fo(U)5 b Fp(\()p Fo(!)r Fp(\))17 b Fc(is)h(admissible)g(with)g(r)n(esp)n(e)n(ct)e(to)i Fo(\045)p Fc(.)89 1859 y Fq(Pro)r(of.)j Fz(The)c(relation)e(\(2.33\))i (implies)133 2004 y Fo(U)5 b Fp(\()p Fo(!)r Fp(\)\()p Fo(K)16 b Fm(\000)11 b Fo(z)r Fp(\))411 1984 y Fj(\000)p Fk(1)464 1991 y Fo(~)458 2004 y(f)19 b Fp(=)14 b(0)d Fm(\010)g Fo(\037)669 2011 y Fi(!)694 2004 y Fp(\()p Fo(H)k Fm(\000)c Fo(z)r Fp(\))862 1984 y Fj(\000)p Fk(1)909 2004 y Fo(f)16 b Fp(+)11 b Fo(i\037)1046 2011 y Fi(!)1071 2004 y Fo(T)1100 2011 y Fj(\003)1119 2004 y Fp(\()p 1138 1977 25 2 v Fo(z)r Fp(\))1182 1984 y Fj(\003)1210 1936 y Fl(Z)1260 1950 y Fk(0)1237 2049 y Fj(\0001)1311 2004 y Fo(dy)k(e)1398 1984 y Fj(\000)p Fi(iy)q(z)1476 2004 y Fo(f)1500 2011 y Fj(\000)1529 2004 y Fp(\()p Fo(y)r Fp(\))25 b Fm(\010)11 b Fp(0)44 b Fz(\(3.24\))89 2158 y(for)172 2145 y Fo(~)165 2158 y(f)23 b Fm(2)17 b Fh(K)h Fz(and)h Fo(z)g Fm(2)e Fn(C)535 2165 y Fk(+)567 2158 y Fz(.)27 b(By)17 b(Theorem)g(3.1)h(of)g([11)q(])f(one)i(gets)f(that)g Fp(\()p Fo(H)f Fm(\000)12 b Fo(z)r Fp(\))1597 2140 y Fj(\000)p Fk(1)1662 2158 y Fz(is)18 b(a)g(trace)89 2219 y(class)g(op)q(erator)h(for)e(eac)o(h)g Fo(z)h Fm(2)f Fn(C)713 2226 y Fk(+)745 2219 y Fz(.)25 b(Hence)17 b Fo(U)5 b Fp(\()p Fo(!)r Fp(\)\()p Fo(H)16 b Fm(\000)c Fo(z)r Fp(\))1209 2201 y Fj(\000)p Fk(1)1274 2219 y Fz(is)17 b(a)h(trace)f(class)h(op)q(erator)h(for)89 2279 y(eac)o(h)d Fo(!)g Fm(\022)d Fp(\012)p Fz(.)21 b(Since)15 b(the)h(op)q(erator)h Fo(T)803 2286 y Fj(\003)822 2279 y Fp(\()p 841 2251 V Fo(z)r Fp(\))885 2261 y Fj(\003)921 2279 y Fz(acts)f(from)f(the)g(t)o (w)o(o)h(dimensional)e(Hilb)q(ert)h(space)89 2339 y Fn(C)122 2321 y Fk(2)161 2339 y Fz(in)o(to)g Fh(H)g Fz(one)h(easily)f(gets)h (that)g(the)f(second)h(addend)g(of)g(\(3.24\))h(is)e(a)h(trace)f(class) h(op)q(erator,)89 2399 y(to)q(o.)29 b(Hence)17 b Fo(U)5 b Fp(\()p Fo(!)r Fp(\)\()p Fo(K)17 b Fm(\000)12 b Fo(z)r Fp(\))627 2381 y Fj(\000)p Fk(1)691 2399 y Fm(2)18 b(L)776 2406 y Fk(1)796 2399 y Fp(\()p Fh(K)p Fp(\))h Fz(for)f(eac)o(h)g(Borel) g(set)g Fo(!)h Fm(\022)f Fp(\012)g Fz(and)h Fo(z)g Fm(2)f Fn(C)1666 2406 y Fk(+)1698 2399 y Fz(.)28 b(Using)89 2459 y(the)16 b(represen)o(tation)375 2572 y Fo(U)5 b Fp(\()p Fo(!)r Fp(\))p Fo(E)519 2579 y Fi(K)554 2572 y Fp(\(\001\))13 b(=)h Fo(U)5 b Fp(\()p Fo(!)r Fp(\)\()p Fo(K)15 b Fm(\000)c Fo(z)r Fp(\))975 2552 y Fj(\000)p Fk(1)1022 2572 y Fp(\()p Fo(K)k Fm(\000)c Fo(z)r Fp(\))p Fo(E)1227 2579 y Fi(K)1261 2572 y Fp(\(\001\))p Fo(;)56 b(z)16 b Fm(2)e Fn(C)1529 2579 y Fk(+)1562 2572 y Fo(;)160 b Fz(\(3.25\))89 2685 y(w)o(e)13 b(\034nd)g(that)h Fo(U)5 b Fp(\()p Fo(!)r Fp(\))p Fo(E)499 2692 y Fi(K)534 2685 y Fp(\(\001\))12 b Fz(is)h(a)h(trace)f(class)g(op)q(erator)h(for)g(eac) o(h)e(Borel)g(sets)i Fo(!)i Fm(\022)d Fp(\012)h Fz(and)f(eac)o(h)89 2746 y(b)q(ounded)18 b(in)o(terv)m(al)d Fp(\001)f Fm(\022)h Fn(R)p Fz(.)k(Hence,)c(the)i(observ)m(able)f Fo(U)5 b Fp(\()p Fo(!)r Fp(\))18 b Fz(is)e(admissible)f(with)h(resp)q(ect)g(to) 89 2806 y Fo(K)21 b Fz(for)16 b(eac)o(h)g(Borel)f(set)h Fo(!)g Fm(\022)e Fp(\012)p Fz(.)p eop %%Page: 15 17 15 16 bop 89 23 a Fs(Density)18 b(and)g(current)h(of)f(a)g(dissip)l(a)l (tive)g(Schr)o(\366dinger)h(opera)l(tor)187 b Fz(15)148 147 y(Moreo)o(v)o(er,)15 b(taking)h(in)o(to)g(accoun)o(t)g(\(3.23\))h (and)g(the)f(represen)o(tation)620 245 y Fo(U)5 b Fp(\()p Fo(!)r Fp(\))j Fo(\045)15 b Fp(=)f Fo(U)5 b Fp(\()p Fo(!)r Fp(\)\()p Fo(K)15 b Fm(\000)c Fo(i)p Fp(\))1097 224 y Fj(\000)p Fk(1)1144 245 y Fp(\()p Fo(K)k Fm(\000)c Fo(i)p Fp(\))p Fo(\045)406 b Fz(\(3.26\))89 343 y(one)17 b(imme)o(diately)c (gets)k(that)g Fo(U)5 b Fp(\()p Fo(!)r Fp(\))j Fo(\045)17 b Fz(is)g(a)g(trace)f(class)h(op)q(erator)g(for)g(eac)o(h)f Fo(!)h Fm(\022)d Fp(\012)p Fz(,)j(b)q(ecause)89 403 y Fp(\()p Fo(K)8 b Fm(\000)t Fo(i)p Fp(\))p Fo(\045)k Fz(is)h(b)q (ounded.)21 b(Hence,)12 b Fo(\045)c(U)d Fp(\()p Fo(!)r Fp(\))13 b Fz(is)g(a)g(trace)f(class)h(op)q(erator)h(for)f(eac)o(h)f (Borel)g(set)h Fo(!)j Fm(\022)e Fp(\012)89 463 y Fz(whic)o(h)i(yields)f (that)h Fo(U)5 b Fp(\()p Fo(!)r Fp(\))j Fo(\045)18 b Fz(is)e(admissible)e(with)i(resp)q(ect)g(to)g Fo(\045)545 b Fa(\003)148 547 y Fz(Since)11 b Fo(U)5 b Fp(\()p Fo(!)r Fp(\))13 b Fz(is)f(admissible)d(with)j(resp)q(ect)g(to)g Fo(K)k Fz(for)d(an)o(y)e(Borel)h(set)f Fo(!)16 b Fm(\022)e Fp(\012)e Fz(b)o(y)g(Prop)q(osition)89 607 y(3.2)j(there)g(is)f(a)h (unique)f(lo)q(calizer)g Fo(U)5 b Fp(\()p Fo(!)r Fp(\)\()p Fm(\001)p Fp(\))14 b(:)f Fn(R)e Fm(\000)-8 b(!)13 b(B)r Fp(\()p Fn(C)1155 589 y Fk(2)1177 607 y Fp(\))p Fz(.)21 b(W)l(e)15 b(are)f(going)i(to)f(calculate)f(this)89 667 y(lo)q(calizer.)89 782 y Fq(Prop)r(osition)20 b(3.5)k Fc(Assume)f Fo(m)15 b Fp(+)798 762 y Fk(1)p 791 771 32 2 v 791 799 a Fi(m)852 782 y Fm(2)24 b Fo(L)942 764 y Fj(1)980 782 y Fp(\([)p Fo(a;)8 b(b)p Fp(]\))p Fc(,)22 b Fo(V)36 b Fm(2)24 b Fo(L)1306 764 y Fk(2)1326 782 y Fp(\([)p Fo(a;)8 b(b)p Fp(]\))21 b Fc(and)j Fo(\024)1611 789 y Fi(a)1632 782 y Fo(;)8 b(\024)1682 789 y Fi(b)1723 782 y Fm(2)24 b Fn(C)1813 789 y Fk(+)1846 782 y Fc(.)89 842 y(Then)f(for)e(any)h(Bor)n(el)h(set)f Fo(!)j Fm(\022)d Fp(\012)g Fc(the)h(lo)n(c)n(alizer)f(of)g Fo(U)5 b Fp(\()p Fo(!)r Fp(\)\()p Fm(\001)p Fp(\))22 b Fc(of)g(the)h(observable)g Fo(U)5 b Fp(\()p Fo(!)r Fp(\))23 b Fc(is)89 902 y(given)c(by)712 984 y Fo(U)5 b Fp(\()p Fo(!)r Fp(\)\()p Fo(\025)p Fp(\))14 b(=)952 916 y Fl(Z)980 1029 y Fi(!)1013 984 y Fo(dx)g(D)q Fp(\()p Fo(x;)8 b(\025)p Fp(\))499 b Fz(\(3.27\))89 1089 y Fc(wher)n(e)268 1209 y Fo(D)q Fp(\()p Fo(x;)8 b(\025)p Fp(\))15 b(:=)505 1138 y Fl(\022)652 1178 y Fm(j)p Fo( )700 1160 y Fj(\000)729 1178 y Fp(\()p Fo(x;)8 b(\025;)g(b)p Fp(\))p Fm(j)902 1160 y Fk(2)1052 1178 y Fo( )1086 1160 y Fj(\000)1115 1178 y Fp(\()p Fo(x;)g(\025;)g(a)p Fp(\))p 1279 1135 222 2 v Fo( )1313 1164 y Fj(\000)1341 1178 y Fp(\()p Fo(x;)g(\025;)g(b)p Fp(\))562 1243 y Fo( )596 1225 y Fj(\000)625 1243 y Fp(\()p Fo(x;)g(\025;)g(b)p Fp(\))p 784 1200 227 2 v Fo( )818 1229 y Fj(\000)847 1243 y Fp(\()p Fo(x;)g(\025;)g(a)p Fp(\))128 b Fm(j)p Fo( )1187 1225 y Fj(\000)1216 1243 y Fp(\()p Fo(x;)8 b(\025;)g(a)p Fp(\))p Fm(j)1394 1225 y Fk(2)1521 1138 y Fl(\023)1736 1209 y Fz(\(3.28\))89 1338 y Fo(x)14 b Fm(2)g Fp(\012)p Fc(,)j Fo(\025)e Fm(2)f Fn(R)p Fc(.)148 1421 y(If)j(the)h(ste)n(ady)f(state)h Fo(\045)g Fc(satis\034es)g(the)g (c)n(ondition)g(\(3.23\),)f(then)578 1543 y Fn(E)608 1550 y Fi(\045)631 1543 y Fp(\()p Fo(U)5 b Fp(\()p Fo(!)r Fp(\)\))14 b(=)843 1476 y Fl(Z)871 1588 y Fg(R)905 1543 y Fo(d\025)h 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b(Lemma)f(3.4)i(the)g(op)q(erator)97 2281 y Fp(^)89 2293 y Fo(U)122 2300 y Fk(\001)154 2293 y Fp(\()p Fo(!)r Fp(\))16 b Fz(is)g(n)o(uclear.)k(Hence,)15 b(w)o(e)h(\034nd)703 2391 y Fo(\026)732 2399 y Fi(\045;U)t Fk(\()p Fi(!)q Fk(\))840 2391 y Fp(\(\001\))d(=)h Fz(tr)p Fp(\()t(^)-28 b Fo(\045)1088 2378 y Fp(^)1080 2391 y Fo(U)1113 2398 y Fk(\001)1145 2391 y Fp(\()p Fo(!)r Fp(\)\))p Fo(:)488 b Fz(\(3.31\))89 2496 y(Let)21 b(us)g(calculate)f(the)h(k)o(ernel)e(of) 762 2483 y Fp(^)753 2496 y Fo(U)786 2503 y Fk(\001)818 2496 y Fp(\()p Fo(!)r Fp(\))p Fz(.)35 b(T)l(o)22 b(his)f(end)f(w)o(e)h (consider)f(the)h(scalar)g(pro)q(duct)89 2556 y Fp(\()116 2543 y(^)108 2556 y Fo(U)141 2563 y Fk(\001)173 2556 y Fp(\()p Fo(!)r Fp(\))r(^)-26 b Fo(g)r(;)300 2543 y Fp(^)290 2556 y Fo(f)5 b Fp(\))p Fz(,)18 b Fp(^)-26 b Fo(g)r(;)426 2543 y Fp(^)415 2556 y Fo(f)19 b Fm(2)14 b Fo(L)538 2538 y Fk(2)558 2556 y Fp(\(\001)p Fo(;)8 b Fn(C)673 2538 y Fk(2)695 2556 y Fp(\))p Fz(.)21 b(Using)c(\(2.41\))f (one)h(has)158 2654 y Fp(\()186 2641 y(^)177 2654 y Fo(U)210 2661 y Fk(\001)242 2654 y Fp(\()p Fo(!)r Fp(\))r(^)-26 b Fo(g)r(;)370 2641 y Fp(^)359 2654 y Fo(f)5 b Fp(\))14 b(=)1277 b Fz(\(3.32\))359 2693 y Fl(Z)387 2806 y Fi(!)421 2761 y Fo(dx)496 2693 y Fl(Z)523 2806 y Fk(\001)563 2761 y Fo(d\025)648 2713 y Fl(X)639 2819 y Fi(\034)t Fk(=)p Fi(a;b)738 2761 y Fo( )772 2740 y Fj(\000)801 2761 y Fp(\()p Fo(x;)8 b(\025;)g(\034)e Fp(\))r(^)-26 b Fo(g)991 2740 y Fi(\034)1012 2761 y Fp(\()p Fo(\025)p Fp(\))p 1078 2687 582 2 v -68 x Fl(Z)1106 2806 y Fk(\001)1146 2761 y Fo(d\026)1231 2713 y Fl(X)1223 2819 y Fi(\030)q Fk(=)p Fi(a;b)1319 2761 y Fo( )1353 2746 y Fj(\000)1382 2761 y Fp(\()p Fo(x;)8 b(\026;)g(\030)r Fp(\))1555 2747 y(^)1544 2761 y Fo(f)1573 2746 y Fi(\030)1593 2761 y Fp(\()p Fo(\026)p Fp(\))p Fo(:)p eop %%Page: 16 18 16 17 bop 89 23 a Fz(16)692 b Fs(H.-Ch.)25 b(Kaiser,)18 b(H.)h(Neidhardt,)g(J.)g(Rehber)o(g)89 147 y Fz(Setting)369 229 y Fo(U)407 205 y Fi(\034)t(\030)402 243 y Fk(\001)446 229 y Fp(\()p Fo(!)r Fp(\)\()p Fo(\026;)8 b(\025)p Fp(\))14 b(:=)712 161 y Fl(Z)740 274 y Fi(!)774 229 y Fo(dx)g( )875 208 y Fj(\000)904 229 y Fp(\()p Fo(x;)8 b(\025;)g(\030)r Fp(\))p 1065 186 229 2 v Fo( )1099 215 y Fj(\000)1128 229 y Fp(\()p Fo(x;)g(\026;)g(\034)e Fp(\))p Fo(;)56 b(\034)s(;)8 b(\030)17 b Fp(=)c Fo(a;)8 b(b;)154 b Fz(\(3.33\))89 333 y(w)o(e)16 b(get)162 438 y Fp(\()189 425 y(^)181 438 y Fo(U)214 445 y Fk(\001)246 438 y Fp(\()p Fo(!)r Fp(\))r(^)-26 b Fo(g)r(;)373 425 y Fp(^)363 438 y Fo(f)5 b Fp(\))14 b(=)1273 b Fz(\(3.34\))245 477 y Fl(Z)273 590 y Fk(\001)313 545 y Fo(d\026)389 475 y Fl(\022)426 477 y(Z)454 590 y Fk(\001)494 545 y Fo(d\025)569 475 y Fl(\022)629 515 y Fo(U)667 497 y Fi(bb)662 527 y Fk(\001)699 515 y Fp(\()p Fo(!)r Fp(\)\()p Fo(\026;)8 b(\025)p Fp(\))46 b Fo(U)970 497 y Fi(ba)965 527 y Fk(\001)1006 515 y Fp(\()p Fo(!)r Fp(\)\()p Fo(\026;)8 b(\025)p Fp(\))627 575 y Fo(U)665 557 y Fi(ab)660 587 y Fk(\001)701 575 y Fp(\()p Fo(!)r Fp(\)\()p Fo(\026;)g(\025)p Fp(\))42 b Fo(U)968 557 y Fi(aa)963 587 y Fk(\001)1008 575 y Fp(\()p Fo(!)r Fp(\)\()p Fo(\026;)8 b(\025)p Fp(\))1217 475 y Fl(\023)g(\022)1323 515 y Fp(^)-26 b Fo(g)1346 497 y Fi(b)1363 515 y Fp(\()p Fo(\025)p Fp(\))1321 575 y(^)g Fo(g)1344 557 y Fi(a)1365 575 y Fp(\()p Fo(\025)p Fp(\))1452 475 y Fl(\023)1497 545 y Fo(;)1519 475 y Fl(\022)1589 502 y Fp(^)1578 515 y Fo(f)1607 497 y Fi(b)1625 515 y Fp(\()p Fo(\026)p Fp(\))1587 567 y(^)1576 580 y Fo(f)1605 562 y Fi(a)1627 580 y Fp(\()p Fo(\026)p Fp(\))1715 475 y Fl(\023)o(\023)89 681 y Fz(what)17 b(sho)o(ws)g(that)467 802 y Fo(U)500 809 y Fk(\001)532 802 y Fp(\()p Fo(!)r Fp(\)\()p Fo(\026;)8 b(\025)p Fp(\))14 b(:=)799 732 y Fl(\022)858 772 y Fo(U)896 754 y Fi(bb)891 784 y Fk(\001)929 772 y Fp(\()p Fo(!)r Fp(\)\()p Fo(\026;)8 b(\025)p Fp(\))45 b Fo(U)1199 754 y Fi(ba)1194 784 y Fk(\001)1236 772 y Fp(\()p Fo(!)r Fp(\)\()p Fo(\026;)8 b(\025)p Fp(\))856 832 y Fo(U)894 814 y Fi(ab)889 845 y Fk(\001)930 832 y Fp(\()p Fo(!)r 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b(\025;)g(a)p Fp(\))p 1302 1076 V Fo( )1336 1105 y Fj(\000)1365 1119 y Fp(\()p Fo(x;)g(\026;)g(a)p Fp(\))1550 1014 y Fl(\023)1736 1084 y Fz(\(3.36\))89 1220 y(w)o(e)16 b(obtain)h(the)f(represen)o (tation)532 1346 y Fo(U)565 1353 y Fk(\001)597 1346 y Fp(\()p Fo(!)r Fp(\)\()p Fo(\026;)8 b(\025)p Fp(\))14 b(=)850 1278 y Fl(Z)878 1391 y Fi(!)911 1346 y Fo(dx)g(D)q Fp(\()p Fo(x;)8 b(\026;)g(\025)p Fp(\))p Fo(;)23 b(\026;)8 b(\025)14 b Fm(2)h Fp(\001)p Fo(:)317 b Fz(\(3.37\))89 1477 y(Since)24 b Fo(U)258 1484 y Fk(\001)289 1477 y Fp(\()p Fo(!)r Fp(\)\()p Fo(\026;)8 b(\025)p Fp(\))29 b(=)f Fo(U)604 1484 y Fk(\001)633 1475 y Fd(0)647 1477 y Fp(\()p Fo(!)r Fp(\)\()p Fo(\026;)8 b(\025)p Fp(\))25 b Fz(for)g Fo(\026;)8 b(\025)28 b Fm(2)h Fp(\001)e Fm(\022)h Fp(\001)1287 1459 y Fj(0)1323 1477 y Fz(it)c(mak)o(es)f(sense)h(to)h (de\034ne)89 1537 y Fo(U)5 b Fp(\()p Fo(!)r Fp(\)\()p Fo(\026;)j(\025)p Fp(\))21 b(:=)f(lim)475 1544 y Fk(\001)p Fj(")p Fg(R)556 1537 y Fo(U)589 1544 y Fk(\001)621 1537 y Fp(\()p Fo(!)r Fp(\)\()p Fo(\026;)8 b(\025)p Fp(\))p Fz(,)21 b Fo(\026;)8 b(\025)20 b Fm(2)h Fn(R)p Fz(.)29 b(Hence)18 b Fo(U)1259 1544 y Fk(\001)1291 1537 y Fp(\()p Fo(!)r Fp(\)\()p Fo(\026;)8 b(\025)p Fp(\))21 b(=)f Fo(U)5 b Fp(\()p Fo(!)r Fp(\)\()p Fo(\026;)j(\025)p Fp(\))21 b Fz(for)89 1597 y Fo(\026;)8 b(\025)15 b Fm(2)f Fp(\001)i Fz(and)528 1679 y Fo(U)5 b Fp(\()p Fo(!)r Fp(\)\()p Fo(\026;)j(\025)p Fp(\))15 b(=)820 1611 y Fl(Z)847 1724 y Fi(!)881 1679 y Fo(dx)f(D)q Fp(\()p Fo(x;)8 b(\026;)g(\025)p Fp(\))p Fo(;)57 b(\026;)8 b(\025)15 b Fm(2)f Fp(\001)p Fo(:)313 b Fz(\(3.38\))89 1787 y(Since)16 b(the)f(k)o(ernel)g(dep)q(ends)i(con)o (tin)o(uously)e(on)i Fo(\026)f Fz(and)h Fo(\025)g Fz(one)f(gets)h(that) 525 1913 y(tr)p Fp(\()t(^)-28 b Fo(\045)629 1901 y Fp(^)621 1913 y Fo(U)654 1920 y Fk(\001)686 1913 y Fp(\()p Fo(!)r Fp(\)\))13 b(=)840 1845 y Fl(Z)868 1958 y Fk(\001)908 1913 y Fo(d\025)h Fz(tr)1013 1922 y Fg(C)1034 1912 y Ff(2)1057 1913 y Fp(\()p Fo(\045)p Fp(\()p Fo(\025)p 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Fp(\)\))p Fz(.)21 b(By)12 b(\(3.39\))i(the)f (matrix-v)m(alued)f(function)h Fo(U)5 b Fp(\()p Fo(!)r Fp(\)\()p Fo(\025)p Fp(\))15 b(:=)e Fo(U)5 b Fp(\()p Fo(!)r Fp(\)\()p Fo(\025;)j(\025)p Fp(\))p Fz(,)15 b Fo(\025)f Fm(2)g Fn(R)p Fz(,)89 2285 y(is)19 b(the)f(unique)g(lo)q (calizer)f(of)i(the)f(observ)m(able)h Fo(U)5 b Fp(\()p Fo(!)r Fp(\))p Fz(.)28 b(By)18 b(condition)g(\(3.23\))h(the)g(observ)m (able)89 2345 y Fo(U)5 b Fp(\()p Fo(!)r Fp(\))17 b Fz(is)f(admissible)e (with)i(resp)q(ect)g(to)h Fo(\045)p Fz(.)k(Applying)15 b(Prop)q(osition)j(3.2)e(w)o(e)g(v)o(erify)f(\(3.29\).)148 2429 y(It)22 b(remains)f(to)i(v)o(erify)d(\(3.27\))j(and)g(\(3.28\).)40 b(F)l(rom)21 b(\(3.38\))i(and)g(\(3.39\))g(w)o(e)f(obtain)h(the)89 2489 y(represen)o(tation)459 2615 y(tr)o Fp(\()t(^)-28 b Fo(\045)563 2602 y Fp(^)554 2615 y Fo(U)587 2622 y Fk(\001)619 2615 y Fp(\()p Fo(!)r Fp(\)\))14 b(=)774 2547 y Fl(Z)801 2660 y Fk(\001)841 2615 y Fo(d\025)917 2547 y Fl(Z)945 2660 y Fi(!)978 2615 y Fo(dx)g Fz(tr)1083 2623 y Fg(C)1104 2614 y Ff(2)1126 2615 y Fp(\()p Fo(\045)p Fp(\()p Fo(\025)p Fp(\))h Fo(D)q Fp(\()p Fo(x;)8 b(\025;)g(\025)p Fp(\)\))p Fo(:)245 b Fz(\(3.40\))89 2746 y(Setting)21 b Fo(D)q Fp(\()p Fo(x;)8 b(\025)p Fp(\))22 b(:=)f Fo(D)q Fp(\()p Fo(x;)8 b(\025;)g(\025)p Fp(\))p Fz(,)23 b Fo(x)e Fm(2)h Fp(\012)p Fz(.)35 b Fo(\025)22 b Fm(2)g Fn(R)p Fz(,)c(w)o(e)i(immediatel)o(y)e(obtain)j(\(3.27\))g(and)89 2806 y(\(3.28\).)1595 b Fa(\003)p eop %%Page: 17 19 17 18 bop 89 23 a Fs(Density)18 b(and)g(current)h(of)f(a)g(dissip)l(a)l (tive)g(Schr)o(\366dinger)h(opera)l(tor)187 b Fz(17)148 147 y(W)l(e)16 b(set)662 208 y Fo(u)693 215 y Fk(^)-21 b Fi(\045)710 208 y Fp(\()p Fo(x;)8 b(\025)p Fp(\))14 b(:=)f Fz(tr)943 216 y Fg(C)964 206 y Ff(2)987 208 y Fp(\()p Fo(\045)p Fp(\()p Fo(\025)p Fp(\))h Fo(D)q Fp(\()p Fo(x;)8 b(\025)p Fp(\)\))449 b Fz(\(3.41\))89 290 y(for)18 b Fo(x)c Fm(2)i Fp(\012)h Fz(and)h Fo(\025)e Fm(2)f Fn(R)p Fz(.)21 b(A)16 b(simple)f(computation)h(sho)o(ws)i(that)g(the)f(eigen)o (v)m(alues)f(of)h Fo(D)q Fp(\()p Fo(x;)8 b(\025)p Fp(\))89 351 y Fz(are)16 b(equal)g(to)g Fm(j)p Fo( )406 333 y Fj(\000)435 351 y Fp(\()p Fo(x;)8 b(\025;)g(b)p Fp(\))p Fm(j)608 333 y Fk(2)643 351 y Fz(and)17 b Fm(j)p Fo( )786 333 y Fj(\000)815 351 y Fp(\()p Fo(x;)8 b(\025;)g(a)p Fp(\))p Fm(j)993 333 y Fk(2)1028 351 y Fz(whic)o(h)15 b(sho)o(ws)i(that)f(the)g(matrix)e Fo(D)q Fp(\()p Fo(x;)8 b(\025)p Fp(\))17 b Fz(is)89 411 y(non-negativ)o(e)i(for)h(eac)o(h)f Fo(x)g Fm(2)g Fp(\012)g Fz(and)h Fo(\025)g Fm(2)f Fn(R)p Fz(.)27 b(Since)19 b(for)g(a.e.)30 b Fo(\025)20 b Fm(2)f Fn(R)e Fz(the)i(matrix)e Fo(\045)p Fp(\()p Fo(\025)p Fp(\))j Fz(is)89 471 y(non-negativ)o(e,)f(to)q(o)h(one)e(gets)h(that)g Fo(u)814 478 y Fk(^)-21 b Fi(\045)831 471 y Fp(\()p Fo(x;)8 b(\025)p Fp(\))18 b Fm(\025)g Fp(0)h Fz(for)g Fo(x)e Fm(2)h Fp(\012)h Fz(and)g(a.e.)28 b Fo(\025)19 b Fm(2)f Fn(R)p Fz(.)25 b(This)19 b(fact)89 531 y(can)e(also)f(b)q(e)h(v)o (eri\034ed)d(taking)j(in)o(to)f(accoun)o(t)g(the)g(represen)o(tation) 409 658 y Fo(u)440 665 y Fk(^)-21 b Fi(\045)457 658 y Fp(\()p Fo(x;)8 b(\025)p Fp(\))14 b(=)639 587 y Fl(\034)676 658 y Fo(\045)701 637 y Fi(t)716 658 y Fp(\()p Fo(\025)p Fp(\))790 587 y Fl(\022)851 627 y Fo( )885 609 y Fj(\000)914 627 y Fp(\()p Fo(x;)8 b(\025;)g(b)p Fp(\))848 687 y Fo( )882 669 y Fj(\000)911 687 y Fp(\()p Fo(x;)g(\025;)g(a)p Fp(\))1095 587 y Fl(\023)1140 658 y Fo(;)1162 587 y Fl(\022)1222 627 y Fo( )1256 609 y Fj(\000)1285 627 y Fp(\()p Fo(x;)g(\025;)g(b)p Fp(\))1220 687 y Fo( )1254 669 y Fj(\000)1283 687 y Fp(\()p Fo(x;)g(\025;)g(a)p Fp(\))1467 587 y Fl(\023)o(\035)1736 658 y Fz(\(3.42\))89 788 y(where)24 b Fo(\045)263 770 y Fi(t)278 788 y Fp(\()p Fo(\025)p Fp(\))h Fz(is)f(the)g(transp)q(osed) i(matrix)d(to)h(\(3.2\).)46 b(Moreo)o(v)o(er,)25 b(if)f(condition)g (\(3.23\))h(is)89 848 y(satis\034ed,)16 b(then)g(from)f(Prop)q(osition) j(\(3.5\))e(w)o(e)g(obtain)h(the)f(represen)o(tation)627 972 y Fn(E)658 979 y Fi(\045)681 972 y Fp(\()p Fo(U)5 b Fp(\()p Fo(!)r Fp(\)\))14 b(=)893 905 y Fl(Z)920 1017 y Fg(R)955 972 y Fo(d\025)1031 905 y Fl(Z)1058 1017 y Fi(!)1092 972 y Fo(dx)g(u)1190 979 y Fk(^)-21 b Fi(\045)1207 972 y Fp(\()p Fo(x;)8 b(\025)p Fp(\))413 b Fz(\(3.43\))89 1096 y(for)17 b(Borel)e(sets)h Fo(!)g Fm(\022)e Fp(\012)p Fz(.)21 b(T)l(aking)c(in)o(to)f(accoun)o(t)g(F)l(ubini's)f(theorem)g(w) o(e)h(\034nd)g(that)715 1217 y Fn(E)745 1224 y Fi(\045)768 1217 y Fp(\()p Fo(U)5 b Fp(\()p Fo(!)r Fp(\)\))14 b(=)980 1149 y Fl(Z)1008 1262 y Fi(!)1041 1217 y Fo(dx)g(u)1139 1224 y Fk(^)-21 b Fi(\045)1156 1217 y Fp(\()p Fo(x)p Fp(\))p Fo(:)500 b Fz(\(3.44\))89 1341 y(where)582 1412 y Fo(u)613 1419 y Fk(^)-21 b Fi(\045)630 1412 y Fp(\()p Fo(x)p Fp(\))14 b(:=)775 1345 y Fl(Z)803 1457 y Fg(R)837 1412 y Fo(d\025)h(u)936 1419 y Fk(^)-21 b Fi(\045)953 1412 y Fp(\()p Fo(x;)8 b(\025)p Fp(\))14 b Fm(\025)f Fp(0)p Fo(;)57 b(x)14 b Fm(2)g Fp(\012)p Fo(;)368 b Fz(\(3.45\))89 1524 y(and)19 b Fo(u)217 1531 y Fk(^)-21 b Fi(\045)251 1524 y Fm(2)17 b Fo(L)334 1506 y Fk(1)354 1524 y Fp(\(\012\))p Fz(.)27 b(The)18 b(represen)o(tation)g(\(3.44\))h(sho)o(ws)g(that)f Fn(E)1313 1531 y Fi(\045)1336 1524 y Fp(\()p Fo(U)5 b Fp(\()p Fm(\001)p Fp(\)\))19 b Fz(de\034nes)f(a)g(measure)89 1584 y(on)h Fp(\012)g Fz(whic)o(h)g(is)f(absolutely)h(con)o(tin)o(uous) f(with)h(resp)q(ect)f(to)h(the)g(Leb)q(esgue)g(measure.)28 b(Since)89 1644 y(the)15 b(exp)q(ectation)f(v)m(alue)g Fn(E)588 1651 y Fi(\045)611 1644 y Fp(\()p Fo(U)5 b Fp(\()p Fo(!)r Fp(\)\))15 b Fz(has)g(the)g(meaning)e(of)i(the)g(n)o(um)o(b)q (er)e(of)i(carriers)f(in)g Fo(!)i Fm(\022)e Fp(\012)89 1704 y Fz(its)f(Radon-Nik)o(o)q(dym)e(deriv)m(ativ)o(e)g(can)i(b)q(e)g (in)o(terpreted)e(as)i(the)f(carrier)g(densit)o(y)g(of)h(the)f(system) 89 1765 y(describ)q(ed)k(b)o(y)g Fo(K)k Fz(and)d(m)o(utatis)d(m)o (utandis)h(b)o(y)h Fo(H)t Fz(.)89 1891 y Fq(De\034nition)k(3.6)k Fz(Assume)15 b Fo(m)c Fp(+)743 1871 y Fk(1)p 736 1880 32 2 v 736 1908 a Fi(m)787 1891 y Fm(2)k Fo(L)868 1873 y Fj(1)905 1891 y Fp(\([)p Fo(a;)8 b(b)p Fp(]\))p Fz(,)15 b Fo(V)26 b Fm(2)15 b Fo(L)1204 1873 y Fk(2)1224 1891 y Fp(\([)p Fo(a;)8 b(b)p Fp(]\))15 b Fz(and)i Fo(\024)1497 1898 y Fi(a)1518 1891 y Fo(;)8 b(\024)1568 1898 y Fi(b)1600 1891 y Fm(2)15 b Fn(C)1681 1898 y Fk(+)1713 1891 y Fz(.)23 b(Then)89 1951 y(the)11 b(matrix)e Fo(D)q Fp(\()p Fo(x;)f(\025)p Fp(\))p Fz(,)13 b Fo(x)g Fm(2)h Fp(\012)p Fz(,)e(and)g(the)f(v)m(alue)f Fo(u)974 1958 y Fk(^)-21 b Fi(\045)991 1951 y Fp(\()p Fo(x;)8 b(\025)p Fp(\))j Fz(are)g(called)f(carrier)h(densit)o(y)f (observ)m(able)89 2011 y(and)20 b(carrier)e(densit)o(y)g(at)i Fo(x)e Fm(2)h Fp(\012)h Fz(and)f(at)h(energy)f Fo(\025)g Fm(2)g Fn(R)d Fz(of)j(the)g(system)f(describ)q(ed)g(b)o(y)h Fo(H)t Fz(,)89 2072 y(resp)q(ectiv)o(ely)l(.)89 2198 y(This)j(de\034nition)g(is)g(justi\034ed)g(b)o(y)f(the)h(fact)g(that)g (b)o(y)g(\(3.41\))h(the)f(carrier)f(densit)o(y)g Fo(u)1728 2205 y Fk(^)-21 b Fi(\045)1745 2198 y Fp(\()p Fo(x;)8 b(\025)p Fp(\))89 2258 y Fz(is)20 b(the)h(exp)q(ectation)f(v)m(alue)g (of)h(the)f(carrier)f(densit)o(y)h(observ)m(able)g Fo(D)q Fp(\()p Fo(x;)8 b(\025)p Fp(\))p Fz(,)22 b(i.e.)33 b Fo(u)1669 2265 y Fk(^)-21 b Fi(\045)1686 2258 y Fp(\()p Fo(x;)8 b(\025)p Fp(\))21 b(=)89 2318 y Fn(E)119 2326 y Fi(\045)p Fk(\()p Fi(\025)p Fk(\))191 2318 y Fp(\()p Fo(D)q Fp(\()p Fo(x;)8 b(\025)p Fp(\)\))15 b Fz(at)g Fo(x)e Fm(2)h Fp(\012)h Fz(and)g(at)g Fo(\025)f Fm(2)g Fn(R)p Fz(.)k(Moreo)o(v)o(er,)13 b(w)o(e)h(note)h(that)g(\(3.45\))g (can)g(b)q(e)f(written)89 2379 y(as)568 2451 y Fo(u)599 2458 y Fk(^)-21 b Fi(\045)616 2451 y Fp(\()p Fo(x)p Fp(\))13 b(=)747 2383 y Fl(Z)775 2496 y Fg(R)809 2451 y Fo(d\025)i Fn(E)907 2458 y Fi(\045)p Fk(\()p Fi(\025)p Fk(\))978 2451 y Fp(\()p Fo(D)q Fp(\()p Fo(x;)8 b(\025)p Fp(\)\))p Fo(;)58 b(x)13 b Fm(2)h Fp(\012)p Fo(;)354 b Fz(\(3.46\))89 2557 y(i.e,)12 b(the)g(carrier)f(densit)o(y)g(at)i Fo(x)g Fm(2)h Fp(\012)f Fz(is)f(the)g(sum)f(of)i(exp)q(ectation)f(v)m(alues)g (of)g(the)g(carrier)g(densit)o(y)89 2617 y(observ)m(able)k(at)h Fo(x)d Fm(2)g Fp(\012)i Fz(o)o(v)o(er)f(all)h(energies.)148 2701 y(On)h(the)e(Hilb)q(ert)g(space)i Fh(K)d Fp(=)g Fo(L)744 2683 y Fk(2)764 2701 y Fp(\()788 2688 y(^)783 2701 y(\012\))i Fz(w)o(e)g(consider)g(the)g(m)o(ultipli)o(cation)e(op)q (erator)j Fo(M)5 b Fp(\()p Fo(h)p Fp(\))p Fz(,)533 2806 y Fp(\()p Fo(M)g Fp(\()p Fo(h)p Fp(\))677 2793 y Fo(~)670 2806 y(f)h Fp(\)\()p Fo(x)p Fp(\))13 b(=)h(0)d Fm(\010)g Fo(h)p Fp(\()p Fo(x)p Fp(\))p Fo(f)5 b Fp(\()p Fo(x)p Fp(\))11 b Fm(\010)g Fp(0)p Fo(;)57 b(x)13 b Fm(2)h Fp(\012)p Fo(;)319 b Fz(\(3.47\))p eop %%Page: 18 20 18 19 bop 89 23 a Fz(18)692 b Fs(H.-Ch.)25 b(Kaiser,)18 b(H.)h(Neidhardt,)g(J.)g(Rehber)o(g)89 147 y Fz(for)14 b(real)f(functions)h Fo(h)g Fm(2)g Fo(L)583 129 y Fj(1)620 147 y Fp(\(\012\))p Fz(.)21 b(W)l(e)13 b(note)h(that)g Fo(M)5 b Fp(\()p Fo(\037)1120 154 y Fi(!)1145 147 y Fp(\))14 b(=)g Fo(U)5 b Fp(\()p Fo(!)r Fp(\))14 b Fz(for)g(an)o(y)f(Borel)g(set) g Fo(!)j Fm(\022)e Fp(\012)p Fz(.)89 208 y(In)i(particular,)g(one)g (has)h Fo(M)5 b Fp(\()p Fo(\037)667 215 y Fk(\012)694 208 y Fp(\))14 b(=)g Fo(U)5 b Fp(\(\012\))14 b(=)g Fo(P)994 189 y Fe(K)987 220 y(H)1019 208 y Fz(.)22 b(Ob)o(viously)l(,)14 b(the)i(represen)o(tation)564 320 y Fo(\045)e(M)5 b Fp(\()p Fo(h)p Fp(\))14 b(=)g Fo(\045)g(U)5 b Fp(\(\012\))14 b Fo(M)5 b Fp(\()p Fo(h)p Fp(\))p Fo(;)57 b(h)14 b Fm(2)g Fo(L)1262 300 y Fj(1)1300 320 y Fp(\(\012\))p Fo(;)349 b Fz(\(3.48\))89 433 y(is)23 b(v)m(alid.)39 b(Since)22 b(the)g(observ)m(able)h Fo(U)5 b Fp(\(\012\))23 b Fz(is)f(admissible)f (with)h(resp)q(ect)g(to)h Fo(\045)g Fz(the)f(pro)q(duct)89 493 y Fo(\045)14 b(U)5 b Fp(\(\012\))16 b Fz(is)f(a)g(n)o(uclear)f(op)q (erator)j(on)e Fh(K)h Fz(whic)o(h)e(yields)g(that)i Fo(\045M)5 b Fp(\()p Fo(h)p Fp(\))15 b Fz(is)g(a)h(n)o(uclear)e(op)q(erator)i(on) 89 553 y Fh(K)p Fz(,)g(i.e.)k(the)c(observ)m(able)h Fo(M)5 b Fp(\()p Fo(h)p Fp(\))16 b Fz(is)g(admissible)e(with)j(resp)q(ect)e (to)i Fo(\045)p Fz(.)89 682 y Fq(Prop)r(osition)j(3.7)k Fc(Assume)c Fo(m)12 b Fp(+)789 662 y Fk(1)p 782 670 32 2 v 782 699 a Fi(m)836 682 y Fm(2)17 b Fo(L)919 664 y Fj(1)957 682 y Fp(\([)p Fo(a;)8 b(b)p Fp(]\))p Fc(,)18 b Fo(V)29 b Fm(2)17 b Fo(L)1265 664 y Fk(2)1285 682 y Fp(\([)p Fo(a;)8 b(b)p Fp(]\))18 b Fc(and)h Fo(\024)1562 689 y Fi(a)1583 682 y Fo(;)8 b(\024)1633 689 y Fi(b)1668 682 y Fm(2)17 b Fn(C)1751 689 y Fk(+)1784 682 y Fc(.)28 b(If)89 742 y(the)19 b(ste)n(ady)f(state)i Fo(\045)e Fc(satis\034es)i(the)f(c)n(ondition)g(\(3.23\),)f(then)i(the)f(c)n (arrier)e(density)i Fo(u)1678 749 y Fk(^)-21 b Fi(\045)1714 742 y Fc(de\034ne)n(d)89 802 y(by)18 b(\(3.45\))f(is)g(a)g(non-ne)n (gative)j Fo(L)712 784 y Fk(1)732 802 y Fc(-function)f(such)f(that)664 939 y Fz(tr)p Fp(\()p Fo(\045M)5 b Fp(\()p Fo(h)p Fp(\)\))14 b(=)949 871 y Fl(Z)976 984 y Fk(\012)1012 939 y Fo(dx)g(u)1110 946 y Fk(^)-21 b Fi(\045)1127 939 y Fp(\()p Fo(x)p Fp(\))p Fo(h)p Fp(\()p Fo(x)p Fp(\))449 b Fz(\(3.49\))89 1078 y Fc(for)17 b(r)n(e)n(al)g(functions)h Fo(h)c Fm(2)g Fo(L)593 1059 y Fj(1)631 1078 y Fp(\(\012\))p Fc(.)22 b(In)c(p)n(articular,)f(one)h(has)469 1190 y Fm(k)p Fo(u)525 1197 y Fk(^)-21 b Fi(\045)542 1190 y Fm(k)567 1198 y Fi(L)591 1189 y Ff(1)608 1198 y Fk(\()p Fi(!)q Fk(\))675 1190 y Fp(=)14 b Fz(tr)o Fp(\()p Fo(\045)8 b(U)d Fp(\()p Fo(!)r Fp(\)\))15 b Fm(\024)f Fo(C)1049 1197 y Fk(^)-21 b Fi(\045)1066 1190 y Fm(k)p Fp(\()p Fo(K)15 b Fm(\000)c Fo(i)p Fp(\))1252 1169 y Fj(\000)p Fk(1)1298 1190 y Fo(P)1336 1169 y Fe(K)1329 1202 y(H)1362 1190 y Fm(k)1387 1198 y Fj(L)1411 1203 y Ff(1)1428 1198 y Fk(\()p Fe(K)p Fk(\))1736 1190 y Fz(\(3.50\))89 1303 y Fc(for)17 b(e)n(ach)h(Bor)n(el)f(set)h Fo(!)e Fm(\022)e Fp(\012)p Fc(.)89 1431 y Fq(Pro)r(of.)24 b Fz(Setting)17 b Fo(!)h Fp(=)d(\012)j Fz(w)o(e)e(obtain)i(from)e (\(3.44\))i(that)g Fo(u)1209 1438 y Fk(^)-21 b Fi(\045)1241 1431 y Fm(2)16 b Fo(L)1323 1413 y Fk(1)1343 1431 y Fp(\(\012\))p Fz(.)24 b(W)l(e)17 b(c)o(ho)q(ose)h Fo(h)d Fp(=)h Fo(\037)1823 1438 y Fi(!)1847 1431 y Fz(,)89 1491 y Fo(!)g Fm(\022)e Fp(\012)p Fz(.)21 b(By)16 b(\(3.44\))h(w)o(e)e(get)629 1628 y(tr)p Fp(\()p Fo(\045M)5 b Fp(\()p Fo(\037)813 1635 y Fi(!)838 1628 y Fp(\)\))14 b(=)942 1560 y Fl(Z)970 1673 y Fk(\012)1005 1628 y Fo(dx)g(u)1103 1635 y Fk(^)-21 b Fi(\045)1120 1628 y Fp(\()p Fo(x)p Fp(\))p Fo(\037)1217 1635 y Fi(!)1242 1628 y Fp(\()p Fo(x)p Fp(\))p Fo(:)414 b Fz(\(3.51\))89 1764 y(By)16 b(linearit)o(y)e(this)i(equation)g (extends)g(to)664 1898 y(tr)p Fp(\()p Fo(\045M)5 b Fp(\()p Fo(h)p Fp(\)\))14 b(=)949 1830 y Fl(Z)976 1943 y Fk(\012)1012 1898 y Fo(dx)g(u)1110 1905 y Fk(^)-21 b Fi(\045)1127 1898 y Fp(\()p Fo(x)p Fp(\))p Fo(h)p Fp(\()p Fo(x)p Fp(\))449 b Fz(\(3.52\))89 2039 y(where)19 b Fo(h)g Fz(is)g(an)h(arbitrary)f (step)g(function)g(on)g Fp(\012)p Fz(.)30 b(Since)18 b Fo(u)1223 2046 y Fk(^)-21 b Fi(\045)1260 2039 y Fz(is)18 b(from)g Fo(L)1462 2021 y Fk(1)1482 2039 y Fp(\(\012\))i Fz(and)f Fo(\045U)5 b Fp(\(\012\))20 b Fz(is)89 2099 y(a)c(n)o(uclear)f(op)q(erator)i(b)q(oth)g(sides)e(of)h(\(3.52\))g (admit)f(a)h(con)o(tin)o(uation)f(to)h Fo(L)1474 2081 y Fj(1)1512 2099 y Fz(-functions)g(whic)o(h)89 2159 y(v)o(eri\034es)f (\(3.49\).)148 2243 y(It)h(remains)f(to)h(sho)o(w)h(the)f(estimate)f (\(3.50\).)21 b(F)l(rom)15 b(\(3.44\))i(w)o(e)f(immedi)o(ately)d(get) 384 2312 y Fl(Z)412 2425 y Fk(\012)448 2380 y Fo(dx)g(u)542 2387 y Fi(g)562 2380 y Fp(\()p Fo(x)p Fp(\))h(=)g Fz(tr)o Fp(\()p Fo(\045U)5 b Fp(\(\012\)\))15 b(=)f Fz(tr)p Fp(\()p Fo(\045)p Fp(\()p Fo(K)h Fm(\000)c Fo(i)p Fp(\)\()p Fo(K)j Fm(\000)d Fo(i)p Fp(\))1375 2359 y Fj(\000)p Fk(1)1422 2380 y Fo(U)5 b Fp(\(\012\)\))p Fo(:)170 b Fz(\(3.53\))89 2521 y(Since)16 b Fo(U)5 b Fp(\(\012\)\()p Fo(K)15 b Fm(\000)c Fo(i)p Fp(\))489 2503 y Fj(\000)p Fk(1)550 2521 y Fm(2)j(L)631 2528 y Fk(1)651 2521 y Fp(\()p Fh(K)p Fp(\))i Fz(and)h Fo(\045)p Fp(\()p Fo(K)f Fm(\000)10 b Fo(i)p Fp(\))16 b Fz(is)g(a)h(b)q(ounded)g(op)q(erator)h(whic)o(h)d (norm)h(can)89 2581 y(b)q(e)h(estimated)d(b)o(y)i Fo(C)484 2588 y Fk(^)-21 b Fi(\045)517 2581 y Fz(w)o(e)16 b(obtain)418 2693 y(tr)p Fp(\()p Fo(\045)p Fp(\()p Fo(K)f Fm(\000)c Fo(i)p Fp(\)\()p Fo(K)k Fm(\000)10 b Fo(i)p Fp(\))821 2673 y Fj(\000)p Fk(1)868 2693 y Fo(U)5 b Fp(\()p Fo(!)r Fp(\)\))14 b Fm(\024)g Fo(C)1100 2700 y Fk(^)-21 b Fi(\045)1117 2693 y Fm(k)p Fp(\()p Fo(K)15 b Fm(\000)c Fo(i)p Fp(\))1303 2673 y Fj(\000)p Fk(1)1350 2693 y Fo(P)1388 2673 y Fe(K)1381 2706 y(H)1413 2693 y Fm(k)1438 2701 y Fj(L)1462 2706 y Ff(1)1479 2701 y Fk(\()p Fe(K)p Fk(\))1736 2693 y Fz(\(3.54\))89 2806 y(whic)o(h)16 b(v)o(eri\034es)f(\(3.50\).)1292 b Fa(\003)p eop %%Page: 19 21 19 20 bop 89 23 a Fs(Density)18 b(and)g(current)h(of)f(a)g(dissip)l(a)l (tive)g(Schr)o(\366dinger)h(opera)l(tor)187 b Fz(19)148 147 y(Ob)o(viously)l(,)15 b(the)h(relation)g(\(3.49\))h(tak)o(es)e(the) h(form)743 264 y(tr)o Fp(\()p Fo(\045M)5 b Fp(\()p Fo(h)p Fp(\)\))15 b(=)f Fm(h)p Fo(u)1078 271 y Fk(^)-21 b Fi(\045)1095 264 y Fo(;)8 b(h)p Fm(i)1164 280 y Fi(L)1188 270 y Ff(1)1736 264 y Fz(\(3.55\))89 380 y(where)16 b(b)o(y)g Fm(h\001)p Fo(;)8 b(h)p Fm(i)400 395 y Fi(L)424 385 y Ff(1)444 380 y Fz(,)15 b Fo(h)f Fm(2)g Fo(L)595 362 y Fj(1)633 380 y Fp(\(\012\))p Fz(,)i(w)o(e)g(denote)g(the)g(linear)f(functionals)h (on)h Fo(L)1533 362 y Fk(1)1553 380 y Fp(\(\012\))p Fz(.)148 464 y(W)l(e)g(conclude)g(this)g(section)g(with)g(some)f(considerations) i(whic)o(h)e(w)o(e)h(need)g(in)g(the)g(follo)o(w-)89 524 y(ing)23 b(section.)42 b(Since)22 b Fm(f)p Fo(\045)p Fp(\()p Fo(\025)p Fp(\))p Fm(g)655 531 y Fi(\025)p Fj(2)p Fg(R)749 524 y Fz(is)h(a)g(measurable)f(family)f(of)i(non-negativ)o(e)g (self-adjoin)o(t)89 584 y(op)q(erators)c(there)f(is)f(a)i(measurable)d (family)g Fm(f)p Fo(V)11 b Fp(\()p Fo(\025)p Fp(\))p Fm(g)1088 591 y Fi(\025)p Fj(2)p Fg(R)1176 584 y Fz(of)18 b(unitary)g(op)q(erators)h(on)g Fn(C)1727 566 y Fk(2)1767 584 y Fz(suc)o(h)89 644 y(that)580 728 y Fo(\045)p Fp(\()p Fo(\025)p Fp(\))14 b(=)g Fo(V)d Fp(\()p Fo(\025)p Fp(\))850 657 y Fl(\022)908 697 y Fo(\026)937 704 y Fi(b)955 697 y Fp(\()p Fo(\025)p Fp(\))88 b(0)952 757 y(0)f Fo(\026)1092 764 y Fi(a)1113 757 y Fp(\()p Fo(\025)p Fp(\))1200 657 y Fl(\023)1245 728 y Fo(V)11 b Fp(\()p Fo(\025)p Fp(\))1350 707 y Fj(\003)1736 728 y Fz(\(3.56\))89 851 y(for)j(a.e.)19 b Fo(\025)c Fm(2)f Fn(R)c Fz(where)j Fo(\026)560 858 y Fi(b)577 851 y Fp(\()p Fo(\025)p Fp(\))h Fz(and)g Fo(\026)778 858 y Fi(a)799 851 y Fp(\()p Fo(\025)p Fp(\))f Fz(are)h(the)f (non-negativ)o(e)g(eigen)o(v)m(alues)f(of)h Fo(\045)p Fp(\()p Fo(\025)p Fp(\))p Fz(.)21 b(F)l(rom)89 911 y(\(3.56\))c(w)o(e)f (get)g(that)557 1057 y Fo(\045)582 1036 y Fi(t)597 1057 y Fp(\()p Fo(\025)p Fp(\))f(=)e Fo(V)769 1036 y Fi(t)784 1057 y Fp(\()p Fo(\025)p Fp(\))850 1036 y Fj(\003)878 987 y Fl(\022)935 1026 y Fo(\026)964 1033 y Fi(b)982 1026 y Fp(\()p Fo(\025)p Fp(\))88 b(0)980 1087 y(0)e Fo(\026)1119 1094 y Fi(a)1140 1087 y Fp(\()p Fo(\025)p Fp(\))1227 987 y Fl(\023)1272 1057 y Fo(V)1311 1036 y Fi(t)1326 1057 y Fp(\()p Fo(\025)p Fp(\))344 b Fz(\(3.57\))89 1203 y(for)18 b(a.e.)23 b Fo(\025)16 b Fm(2)f Fn(R)f Fz(where)j Fo(V)588 1185 y Fi(t)603 1203 y Fp(\()p Fo(\025)p Fp(\))g Fz(is)g(the)g(transp)q(osed)i(matrix)c(to)i Fo(V)12 b Fp(\()p Fo(\025)p Fp(\))p Fz(.)24 b(Inserting)16 b(\(3.57\))i(in)o (to)89 1264 y(\(3.42\))f(w)o(e)f(obtain)h(the)f(expression)146 1380 y Fo(u)177 1387 y Fk(^)-21 b Fi(\045)194 1380 y Fp(\()p Fo(x;)8 b(\025)p Fp(\))14 b(=)1374 b Fz(\(3.58\))288 1413 y Fl(\034)q(\022)383 1453 y Fo(\026)412 1460 y Fi(b)430 1453 y Fp(\()p Fo(\025)p Fp(\))88 b(0)427 1513 y(0)f Fo(\026)567 1520 y Fi(a)588 1513 y Fp(\()p Fo(\025)p Fp(\))675 1413 y Fl(\023)720 1483 y Fo(V)759 1463 y Fi(t)774 1483 y Fp(\()p Fo(\025)p Fp(\))848 1413 y Fl(\022)908 1453 y Fo( )942 1435 y Fj(\000)971 1453 y Fp(\()p Fo(x;)8 b(\025;)g(b)p Fp(\))906 1513 y Fo( )940 1495 y Fj(\000)969 1513 y Fp(\()p Fo(x;)g(\025;)g(a)p Fp(\))1153 1413 y Fl(\023)1198 1483 y Fo(;)g(V)1259 1463 y Fi(t)1274 1483 y Fp(\()p Fo(\025)p Fp(\))1348 1413 y Fl(\022)1409 1453 y Fo(\036)1438 1435 y Fj(\000)1467 1453 y Fp(\()p Fo(x;)g(\025;)g(b)p Fp(\))1406 1513 y Fo(\036)1435 1495 y Fj(\000)1465 1513 y Fp(\()p Fo(x;)g(\025;)g(a)p Fp(\))1649 1413 y Fl(\023)o(\035)1731 1483 y Fo(:)89 1629 y Fz(Let)17 b(us)f(in)o(tro)q(duce)g(the)g(unit)g (v)o(ectors)f Fo(e)831 1636 y Fi(\034)852 1629 y Fp(\()p Fo(\025)p Fp(\))p Fz(,)696 1745 y Fo(e)719 1752 y Fi(\034)741 1745 y Fp(\()p Fo(\025)p Fp(\))f(=)g Fo(V)d Fp(\()p Fo(\025)p Fp(\))p Fo(e)1001 1752 y Fi(\034)1022 1745 y Fo(;)57 b(\034)20 b Fp(=)13 b Fo(b;)8 b(a)482 b Fz(\(3.59\))89 1861 y(whic)o(h)16 b(p)q(erform)f(an)i(orthonormal)f(basis)g(in)g Fn(C)969 1843 y Fk(2)992 1861 y Fz(.)21 b(W)l(e)16 b(set)155 1965 y Fo(~)149 1978 y( )q Fp(\()p Fo(x;)8 b(\025;)g(e)324 1985 y Fi(\034)345 1978 y Fp(\()p Fo(\025)p Fp(\)\))15 b(:=)e Fo( )542 1985 y Fj(\000)571 1978 y Fp(\()p Fo(x;)8 b(\025;)g(e)713 1985 y Fi(\034)734 1978 y Fp(\()p Fo(\025)p Fp(\)\))k Fm(\010)e Fo( )r Fp(\()p Fo(x;)e(\025;)g(e)1056 1985 y Fi(\034)1077 1978 y Fp(\()p Fo(\025)p Fp(\)\))j Fm(\010)g Fo( )1255 1985 y Fk(+)1284 1978 y Fp(\()p Fo(x;)d(\025;)g(e) 1426 1985 y Fi(\034)1447 1978 y Fp(\()p Fo(\025)p Fp(\)\))15 b(=)151 b Fz(\(3.60\))391 2039 y Fp(1)p 355 2061 96 2 v 355 2071 a Fm(p)p 397 2071 54 2 v 41 x Fp(2)p Fo(\031)455 2073 y(e)478 2052 y Fi(ix\025)533 2073 y Fo(e)556 2080 y Fi(\034)577 2073 y Fp(\()p Fo(\025)p Fp(\))11 b Fm(\010)744 2039 y Fp(1)p 709 2061 96 2 v 709 2071 a Fm(p)p 750 2071 54 2 v 750 2112 a Fp(2)p Fo(\031)809 2073 y Fp(\()p Fo(T)857 2080 y Fj(\003)877 2073 y Fp(\()p Fo(\025)p Fp(\))943 2052 y Fj(\003)963 2073 y Fo(e)986 2080 y Fi(\034)1007 2073 y Fp(\()p Fo(\025)p Fp(\)\)\()p Fo(x)p Fp(\))g Fm(\010)1259 2039 y Fp(1)p 1224 2061 96 2 v 1224 2071 a Fm(p)p 1265 2071 54 2 v 1265 2112 a Fp(2)p Fo(\031)1324 2073 y(e)1347 2052 y Fi(ix\025)1401 2073 y Fp(\002)1439 2080 y Fi(H)1473 2073 y Fp(\()p Fo(\025)p Fp(\))1539 2052 y Fj(\003)1559 2073 y Fo(e)1582 2080 y Fi(\034)1603 2073 y Fp(\()p Fo(\025)p Fp(\))p Fo(:)89 2228 y Fz(Ob)o(viously)l(,)j(the)h(system)f Fm(f)602 2215 y Fo(~)596 2228 y( )q Fp(\()p Fo(x;)8 b(\025;)g(e)771 2235 y Fi(\034)792 2228 y Fp(\()p Fo(\025)p Fp(\)\))p Fm(g)902 2235 y Fi(\034)t Fk(=)p Fi(b;a)1010 2228 y Fz(p)q(erforms)15 b(an)g(orthonormal)h(basis)f(of)h(gener-)89 2288 y(alized)f (eigenfunctions.)21 b(Moreo)o(v)o(er,)14 b(a)j(straigh)o(tforw)o(ard)g (computation)e(sho)o(ws)j(that)509 2431 y Fo(V)548 2410 y Fi(t)563 2431 y Fp(\()p Fo(\025)p Fp(\))637 2361 y Fl(\022)698 2400 y Fo( )732 2382 y Fj(\000)761 2400 y Fp(\()p Fo(x;)8 b(\025;)g(b)p Fp(\))695 2460 y Fo( )729 2442 y Fj(\000)758 2460 y Fp(\()p Fo(x;)g(\025;)g(a)p Fp(\))943 2361 y Fl(\023)993 2431 y Fp(=)1045 2361 y Fl(\022)1104 2400 y Fo( )r Fp(\()p Fo(x;)g(\025;)g(e)1280 2407 y Fi(b)1296 2400 y Fp(\()p Fo(\025)p Fp(\)\))1102 2460 y Fo( )r Fp(\()p Fo(x;)g(\025;)g(e)1278 2467 y Fi(a)1298 2460 y Fp(\()p Fo(\025)p Fp(\)\))1404 2361 y Fl(\023)1736 2431 y Fz(\(3.61\))89 2573 y(whic)o(h)16 b(leads)g(to)389 2689 y Fo(u)420 2696 y Fk(^)-21 b Fi(\045)437 2689 y Fp(\()p Fo(x;)8 b(\025)p Fp(\))14 b(=)f Fo(\026)647 2696 y Fi(b)665 2689 y Fp(\()p Fo(\025)p Fp(\))p Fm(j)p Fo( )r Fp(\()p Fo(x;)8 b(\025;)g(e)921 2696 y Fi(b)937 2689 y Fp(\()p Fo(\025)p Fp(\)\))p Fm(j)1036 2669 y Fk(2)1067 2689 y Fp(+)j Fo(\026)1145 2696 y Fi(a)1166 2689 y Fp(\()p Fo(\025)p Fp(\))p Fm(j)p Fo( )r Fp(\()p Fo(x;)d(\025;)g(e)1422 2696 y Fi(a)1442 2689 y Fp(\()p Fo(\025)p Fp(\)\))p Fm(j)1541 2669 y Fk(2)1736 2689 y Fz(\(3.62\))89 2806 y(for)17 b Fo(x)c Fm(2)h Fp(\012)j Fz(and)g Fo(\025)d Fm(2)g Fn(R)p Fz(.)p eop %%Page: 20 22 20 21 bop 89 23 a Fz(20)692 b Fs(H.-Ch.)25 b(Kaiser,)18 b(H.)h(Neidhardt,)g(J.)g(Rehber)o(g)89 155 y Fx(4)77 b(Curren)n(t)26 b(densit)n(y)89 289 y Fz(In)13 b(accordance)g(with)g (\(1.11\))h(the)e(curren)o(t)h(densit)o(y)f Fo(j)1074 296 y Fk(^)-21 b Fi(\045)1091 289 y Fp(\()p Fo(x;)8 b(\025)p Fp(\))13 b Fz(for)g(the)g(energy)g Fo(\025)h Fm(2)g Fn(R)c Fz(is)j(de\034ned)89 350 y(b)o(y)337 424 y Fo(j)360 431 y Fk(^)-21 b Fi(\045)377 424 y Fp(\()p Fo(x;)8 b(\025)p Fp(\))41 b(:=)g Fo(\026)656 431 y Fi(b)674 424 y Fp(\()p Fo(\025)p Fp(\))p Fm(=)p Fz(m)825 368 y Fl(\020)890 404 y Fk(1)p 860 412 79 2 v 860 441 a Fi(m)p Fk(\()p Fi(x)p Fk(\))943 424 y Fo( )r Fp(\()p Fo(x;)8 b(\025;)g(e)1119 431 y Fi(b)1136 424 y Fp(\()p Fo(\025)p Fp(\)\))1221 406 y Fj(0)p 1246 380 278 2 v 1246 424 a Fo( )r Fp(\()p Fo(x;)g(\025;)g(e)1422 431 y Fi(b)1439 424 y Fp(\()p Fo(\025)p Fp(\)\))1524 368 y Fl(\021)1562 424 y Fp(+)627 523 y Fo(\026)656 530 y Fi(a)678 523 y Fp(\()p Fo(\025)p Fp(\))p Fm(=)p Fz(m)829 467 y Fl(\020)894 503 y Fk(1)p 864 511 79 2 v 864 540 a Fi(m)p Fk(\()p Fi(x)p Fk(\))947 523 y Fo( )r Fp(\()p Fo(x;)g(\025;)g(e)1123 530 y Fi(a)1143 523 y Fp(\()p Fo(\025)p Fp(\)\))1228 505 y Fj(0)p 1254 479 282 2 v 1254 523 a Fo( )r Fp(\()p Fo(x;)g(\025;)g(e)1430 530 y Fi(a)1449 523 y Fp(\()p Fo(\025)p Fp(\)\))1535 467 y Fl(\021)1621 473 y Fo(;)126 b Fz(\(4.1\))89 627 y Fo(x)16 b Fm(2)h Fp(\012)p Fz(,)g(where)h(the)f(eigenfunctions)g Fo( )r Fp(\()p Fo(x;)8 b(\025;)g(e)973 634 y Fi(\034)994 627 y Fp(\()p Fo(\025)p Fp(\)\))p Fz(,)18 b Fo(\034)k Fp(=)16 b Fo(a;)8 b(b)p Fz(,)17 b(are)g(de\034ned)h(b)o(y)f(\(3.61\).) 26 b(W)l(e)89 687 y(are)15 b(going)i(to)e(relate)g(the)g(curren)o(t)f (densit)o(y)g(to)i(the)f(c)o(haracteristic)e(function)i(of)h(the)f (maximal)89 747 y(dissipativ)o(e)g(op)q(erator)j Fo(H)t Fz(.)89 877 y Fq(Prop)r(osition)i(4.1)k Fc(Assume)f Fo(m)15 b Fp(+)798 857 y Fk(1)p 791 865 32 2 v 791 894 a Fi(m)852 877 y Fm(2)24 b Fo(L)942 858 y Fj(1)980 877 y Fp(\([)p Fo(a;)8 b(b)p Fp(]\))p Fc(,)22 b Fo(V)36 b Fm(2)24 b Fo(L)1306 858 y Fk(2)1326 877 y Fp(\([)p Fo(a;)8 b(b)p Fp(]\))21 b Fc(and)j Fo(\024)1611 884 y Fi(a)1632 877 y Fo(;)8 b(\024)1682 884 y Fi(b)1723 877 y Fm(2)24 b Fn(C)1813 884 y Fk(+)1846 877 y Fc(.)89 937 y(F)l(urther,)17 b(let)i Fo(\045)e Fc(b)n(e)h(a)g(ste)n(ady)f(state.)23 b(Then)18 b(the)g(curr)n(ent)f(density)h Fo(j)1346 944 y Fk(^)-21 b Fi(\045)1363 937 y Fp(\()p Fo(x;)8 b(\025)p Fp(\))p Fc(,)18 b Fo(x)13 b Fm(2)h Fp(\012)p Fc(,)k Fo(\025)c Fm(2)h Fn(R)p Fc(,)f(is)89 997 y(indep)n(endent)19 b(fr)n(om)e Fo(x)g Fc(and)g(admits)g(the)h(r)n(epr)n(esentation)731 1110 y Fo(j)754 1117 y Fk(^)-21 b Fi(\045)771 1110 y Fp(\()p Fo(\025)p Fp(\))15 b(=)e Fz(tr)941 1118 y Fg(C)962 1109 y Ff(2)985 1110 y Fp(\()p Fo(\045)p Fp(\()p Fo(\025)p Fp(\))p Fo(C)t Fp(\()p Fo(\025)p Fp(\)\))542 b Fz(\(4.2\))89 1223 y Fc(wher)n(e)428 1296 y Fo(C)t Fp(\()p Fo(\025)p Fp(\))13 b(:=)h Fm(\000)678 1262 y Fp(1)p 656 1284 71 2 v 656 1330 a(2)p Fo(\031)r(i)745 1262 y(\013)776 1269 y Fi(a)797 1262 y Fo(\013)828 1269 y Fi(b)p 736 1284 120 2 v 736 1294 V 736 1337 a Fo(W)7 b Fp(\()p Fo(\025)p Fp(\))860 1296 y Fo(E)s Fp(\002)937 1303 y Fi(H)971 1296 y Fp(\()p Fo(\025)p Fp(\))1037 1275 y Fj(\003)1070 1296 y Fp(=)1150 1262 y(1)p 1127 1284 71 2 v 1127 1330 a(2)p Fo(\031)r(i)1217 1262 y(\013)1248 1269 y Fi(b)1265 1262 y Fo(\013)1296 1269 y Fi(a)p 1208 1284 120 2 v 1208 1330 a Fo(W)g Fp(\()p Fo(\025)p Fp(\))1332 1296 y(\002)1370 1303 y Fi(H)1403 1296 y Fp(\()p Fo(\025)p Fp(\))p Fo(E)s(;)239 b Fz(\(4.3\))89 1424 y Fo(\025)14 b Fm(2)h Fn(R)o Fc(.)20 b(Mor)n(e)n(over,)c(if)h Fz(tr)566 1432 y Fg(C)587 1423 y Ff(2)610 1424 y Fp(\()p Fo(\045)p Fp(\()p Fm(\001)p Fp(\)\))c Fm(2)h Fo(L)818 1406 y Fk(1)838 1424 y Fp(\()p Fn(R)p Fp(\))p Fc(,)h(then)j(the)g(total)g(curr)n(ent)f Fo(j)1439 1431 y Fk(^)-21 b Fi(\045)1457 1424 y Fc(,)790 1563 y Fo(j)813 1570 y Fk(^)g Fi(\045)844 1563 y Fp(:=)910 1495 y Fl(Z)937 1607 y Fg(R)972 1563 y Fo(d\025)14 b(j)1062 1570 y Fk(^)-21 b Fi(\045)1080 1563 y Fp(\()p Fo(\025)p Fp(\))p Fo(;)601 b Fz(\(4.4\))89 1699 y Fc(is)17 b(\034nite)i(and)f (satis\034es)g(the)g(estimate)702 1833 y Fm(j)p Fo(j)739 1840 y Fk(^)-21 b Fi(\045)756 1833 y Fm(j)13 b(\024)856 1799 y Fp(1)p 841 1822 54 2 v 841 1867 a(2)p Fo(\031)908 1765 y Fl(Z)936 1878 y Fg(R)970 1833 y Fo(d\025)i 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Fo( )r Fp(\()p Fo(x;)g(\025;)g(e)1548 2203 y Fi(b)1564 2196 y Fp(\()p Fo(\025)p Fp(\)\))1370 2256 y Fo( )r Fp(\()p Fo(x;)g(\025;)g(e)1546 2263 y Fi(a)1566 2256 y Fp(\()p Fo(\025)p Fp(\)\))1672 2156 y Fl(\023)1709 2141 y(+!)1797 2226 y Fo(:)89 2383 y Fz(T)l(aking)17 b(in)o(to)f(accoun)o(t)g (\(3.57\))h(and)f(\(3.61\))h(w)o(e)f(get)225 2539 y Fo(j)248 2546 y Fk(^)-21 b Fi(\045)266 2539 y Fp(\()p Fo(x;)8 b(\025)p Fp(\))14 b(=)f Fm(=)p Fz(m)532 2454 y Fl( )q(*)612 2539 y Fo(\045)637 2519 y Fi(t)652 2539 y Fp(\()p Fo(\025)p Fp(\))726 2454 y Fl( )824 2482 y Fk(1)p 794 2490 V 794 2519 a Fi(m)p Fk(\()p Fi(x)p Fk(\))878 2502 y Fo( )912 2483 y Fj(\000)941 2502 y Fp(\()p Fo(x;)8 b(\025;)g(b)p Fp(\))1100 2483 y Fj(0)822 2550 y Fk(1)p 792 2558 V 792 2586 a Fi(m)p Fk(\()p Fi(x)p Fk(\))875 2569 y Fo( )909 2551 y Fj(\000)938 2569 y Fp(\()p Fo(x;)g(\025;)g(a)p Fp(\))1102 2551 y Fj(0)1134 2454 y Fl(!)1182 2539 y Fo(;)1204 2469 y Fl(\022)1263 2509 y Fo( )1297 2491 y Fj(\000)1326 2509 y Fp(\()p Fo(x;)g(\025;)g(b)p Fp(\))1261 2569 y Fo( )1295 2551 y Fj(\000)1324 2569 y Fp(\()p Fo(x;)g(\025;)g(a)p Fp(\))1508 2469 y Fl(\023)1545 2454 y(+!)1761 2539 y Fz(\(4.7\))89 2693 y(whic)o(h)16 b(can)g(b)q(e)g(expressed)g(b)o(y)681 2806 y Fo(j)704 2813 y Fk(^)-21 b Fi(\045)722 2806 y Fp(\()p Fo(x;)8 b(\025)p Fp(\))14 b(=)f Fz(tr)941 2814 y Fg(C)962 2805 y Ff(2)985 2806 y Fp(\()p Fo(\045)p Fp(\()p Fo(\025)p Fp(\))p Fo(C)t Fp(\()p Fo(x;)8 b(\025)p Fp(\)\))492 b Fz(\(4.8\))p eop %%Page: 21 23 21 22 bop 89 23 a Fs(Density)18 b(and)g(current)h(of)f(a)g(dissip)l(a)l (tive)g(Schr)o(\366dinger)h(opera)l(tor)187 b Fz(21)89 147 y(where)147 249 y Fo(C)t Fp(\()p Fo(x;)8 b(\025)p Fp(\))13 b(:=)1394 b Fz(\(4.9\))289 365 y Fm(=)p Fz(m)374 280 y Fl( )q( )512 307 y Fk(1)p 481 315 79 2 v 481 344 a Fi(m)p Fk(\()p Fi(x)p Fk(\))565 327 y Fo( )599 309 y Fj(\000)628 327 y Fp(\()p Fo(x;)8 b(\025;)g(b)p Fp(\))787 309 y Fj(0)p 812 284 222 2 v 812 327 a Fo( )846 312 y Fj(\000)875 327 y Fp(\()p Fo(x;)g(\025;)g(b)p 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556 222 2 v Fo( )569 585 y Fj(\000)597 599 y Fp(\()p Fo(x;)h(\025;)g(b)p Fp(\))p Fo(;)g( )812 581 y Fj(\000)841 599 y Fp(\()p Fo(x;)g(\025;)g(b)p Fp(\)\))46 b Fo(W)7 b Fp(\()p 1137 556 V Fo( )1171 585 y Fj(\000)1200 599 y Fp(\()p Fo(x;)h(\025;)g(b)p Fp(\))o Fo(;)g( )1414 581 y Fj(\000)1443 599 y Fp(\()p Fo(x;)g(\025;)g(a)p Fp(\)\))460 664 y Fo(W)f Fp(\()p 532 621 227 2 v Fo( )566 650 y Fj(\000)595 664 y Fp(\()p Fo(x;)h(\025;)g(a)p Fp(\))o Fo(;)g( )814 646 y Fj(\000)843 664 y Fp(\()p Fo(x;)g(\025;)g(b)p Fp(\)\))41 b Fo(W)7 b Fp(\()p 1134 621 V Fo( )1168 650 y Fj(\000)1197 664 y Fp(\()p Fo(x;)h(\025;)g(a)p Fp(\))p Fo(;)g( )1417 646 y Fj(\000)1445 664 y Fp(\()p Fo(x;)g(\025;)g(a)p Fp(\)\))1649 559 y Fl(\023)1736 629 y Fz(\(4.10\))89 765 y(where)16 b Fo(W)7 b Fp(\()p Fm(\001)p Fo(;)h Fm(\001)p Fp(\))16 b Fz(is)g(the)g(W)l(ronskian)g(de\034ned)g(b)o(y)g(\(2.5\).)21 b(Since)263 896 y Fo(W)7 b Fp(\()p 335 852 222 2 v Fo( )369 881 y Fj(\000)398 896 y Fp(\()p Fo(x;)h(\025;)g(b)p Fp(\))o Fo(;)g( 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Fz(\(4.12\))258 1154 y Fo(W)f Fp(\()p 330 1110 222 2 v Fo( )364 1139 y Fj(\000)393 1154 y Fp(\()p Fo(x;)h(\025;)g(b)p Fp(\))p Fo(;)g( )608 1133 y Fj(\000)636 1154 y Fp(\()p Fo(x;)g(\025;)g(a)p Fp(\)\))41 b(=)h Fm(\000)998 1120 y Fp(1)p 984 1142 54 2 v 984 1188 a(2)p Fo(\031)1081 1120 y(\013)1112 1127 y Fi(b)1129 1120 y Fo(\013)1160 1127 y Fi(a)p 1047 1142 167 2 v 1047 1188 a Fm(j)p Fo(W)7 b Fp(\()p Fo(\025)p Fp(\))p Fm(j)1194 1174 y Fk(2)1227 1113 y Fl(\010)1256 1154 y Fo(W)g Fp(\()p Fo(\025)p Fp(\))k Fm(\000)g Fo(i\013)1484 1133 y Fk(2)1484 1166 y Fi(a)1505 1154 y Fo(v)1529 1161 y Fi(b)1546 1154 y Fp(\()p Fo(a;)d(\025)p Fp(\))1660 1113 y Fl(\011)1736 1154 y Fz(\(4.13\))253 1290 y Fo(W)f Fp(\()p 325 1247 227 2 v Fo( )359 1276 y Fj(\000)388 1290 y Fp(\()p Fo(x;)h(\025;)g(a)p Fp(\))p Fo(;)g( )608 1270 y Fj(\000)636 1290 y Fp(\()p Fo(x;)g(\025;)g(a)p Fp(\)\))41 b(=)963 1256 y Fo(i)p 945 1279 54 2 v 945 1324 a Fp(2)p Fo(\031)1040 1256 y(\013)1071 1238 y Fk(2)1071 1269 y Fi(a)1092 1256 y Fo(\013)1123 1238 y Fk(2)1123 1269 y Fi(b)p 1008 1279 167 2 v 1008 1324 a Fm(j)p Fo(W)7 b Fp(\()p Fo(\025)p Fp(\))p Fm(j)1155 1310 y Fk(2)1736 1290 y Fz(\(4.14\))89 1416 y(w)o(e)16 b(\034nd)140 1534 y Fo(C)t Fp(\()p Fo(x;)8 b(\025)p Fp(\))14 b(=)388 1500 y(1)p 365 1523 71 2 v 365 1568 a(2)p Fo(\031)r(i)479 1500 y(\013)510 1507 y Fi(b)527 1500 y Fo(\013)558 1507 y Fi(a)p 446 1523 167 2 v 446 1568 a Fm(j)p Fo(W)7 b Fp(\()p Fo(\025)p Fp(\))p Fm(j)593 1554 y Fk(2)626 1464 y Fl(\022)806 1504 y Fm(\000)p Fo(i\013)893 1511 y Fi(b)910 1504 y Fo(\013)941 1511 y Fi(a)1126 1504 y Fm(\000)p Fp(\()p Fo(W)g Fp(\()p Fo(\025)p Fp(\))k Fm(\000)g Fo(i\013)1412 1485 y Fk(2)1412 1516 y Fi(a)1432 1504 y Fo(v)1456 1511 y Fi(b)1473 1504 y Fp(\()p Fo(a;)d(\025)p Fp(\)\))683 1564 y Fo(W)f Fp(\()p Fo(\025)p Fp(\))k Fm(\000)g Fo(i\013)911 1546 y Fk(2)911 1577 y Fi(b)931 1564 y Fo(v)955 1571 y Fi(a)975 1564 y Fp(\()p Fo(b;)d(\025)p Fp(\))223 b Fo(i\013)1355 1571 y Fi(a)1376 1564 y Fo(\013)1407 1571 y Fi(b)1626 1464 y Fl(\023)1671 1534 y Fo(:)51 b Fz(\(4.15\))89 1661 y(This)17 b(yields)140 1762 y Fo(C)t Fp(\()p Fo(x;)8 b(\025)p Fp(\))14 b(=)1389 b Fz(\(4.16\))247 1864 y Fm(\000)314 1830 y Fp(1)p 291 1853 71 2 v 291 1898 a(2)p Fo(\031)r(i)405 1830 y(\013)436 1837 y Fi(b)453 1830 y Fo(\013)484 1837 y Fi(a)p 371 1853 167 2 v 371 1898 a Fm(j)p Fo(W)7 b Fp(\()p Fo(\025)p Fp(\))p Fm(j)518 1884 y Fk(2)551 1794 y Fl(\022)628 1834 y Fp(0)61 b(1)609 1894 y Fm(\000)p Fp(1)41 b(0)758 1794 y Fl(\023)8 b(\022)861 1834 y Fo(W)f Fp(\()p Fo(\025)p Fp(\))k Fm(\000)g Fo(i\013)1089 1816 y Fk(2)1089 1847 y Fi(b)1109 1834 y Fo(v)1133 1841 y Fi(a)1153 1834 y Fp(\()p Fo(b;)d(\025)p Fp(\))185 b Fo(i\013)1495 1841 y Fi(b)1512 1834 y Fo(\013)1543 1841 y Fi(a)1003 1894 y Fo(i\013)1051 1901 y Fi(b)1068 1894 y Fo(\013)1099 1901 y Fi(a)1303 1894 y Fo(W)7 b Fp(\()p Fo(\025)p Fp(\))12 b Fm(\000)f Fo(i\013)1532 1876 y Fk(2)1532 1906 y Fi(a)1552 1894 y Fo(v)1576 1901 y Fi(b)1593 1894 y Fp(\()p Fo(a;)d(\025)p Fp(\))1727 1794 y Fl(\023)1772 1864 y Fo(:)89 1994 y Fz(F)l(rom)15 b(\(2.21\))i(w)o(e)f(obtain)241 2125 y Fp(\002)279 2132 y Fi(H)313 2125 y Fp(\()p Fo(\025)p Fp(\))379 2104 y Fj(\003)413 2125 y Fp(=)517 2091 y(1)p 469 2113 120 2 v 469 2159 a Fo(W)7 b Fp(\()p Fo(\025)p Fp(\))602 2054 y Fl(\022)659 2094 y Fo(W)g Fp(\()p Fo(\025)p Fp(\))k Fm(\000)g Fo(i\013)887 2076 y Fk(2)887 2107 y Fi(b)907 2094 y Fo(v)931 2101 y Fi(a)951 2094 y Fp(\()p Fo(b;)d(\025)p Fp(\))185 b Fo(i\013)1293 2101 y Fi(b)1310 2094 y Fo(\013)1341 2101 y Fi(a)801 2154 y Fo(i\013)849 2161 y Fi(b)866 2154 y Fo(\013)897 2161 y Fi(a)1102 2154 y Fo(W)7 b Fp(\()p Fo(\025)p Fp(\))k Fm(\000)g Fo(i\013)1330 2136 y Fk(2)1330 2167 y Fi(a)1350 2154 y Fo(v)1374 2161 y Fi(b)1391 2154 y Fp(\()p Fo(a;)d(\025)p Fp(\))1526 2054 y Fl(\023)1571 2125 y Fo(:)151 b Fz(\(4.17\))89 2252 y(Hence,)15 b(one)h(has)642 2331 y Fo(C)t Fp(\()p Fo(x;)8 b(\025)p Fp(\))14 b(=)f Fm(\000)929 2298 y Fp(1)p 906 2320 71 2 v 906 2366 a(2)p Fo(\031)r(i)996 2298 y(\013)1027 2305 y Fi(b)1044 2298 y Fo(\013)1075 2305 y Fi(a)p 987 2320 120 2 v 987 2330 V 987 2373 a Fo(W)7 b Fp(\()p Fo(\025)p Fp(\))1111 2331 y Fo(E)s Fp(\002)1188 2338 y Fi(H)1221 2331 y Fp(\()p Fo(\025)p Fp(\))1287 2311 y Fj(\003)1736 2331 y Fz(\(4.18\))89 2447 y(where)783 2524 y Fo(E)17 b Fp(:=)901 2454 y Fl(\022)978 2494 y Fp(0)61 b(1)959 2554 y Fm(\000)p Fp(1)41 b(0)1108 2454 y Fl(\023)1153 2524 y Fo(:)569 b Fz(\(4.19\))89 2637 y(This)15 b(sho)o(ws)h(that)f Fo(C)t Fp(\()p Fo(x;)8 b(\025)p Fp(\))14 b Fz(is)h(actually)f(indep)q (enden)o(t)g(of)h Fo(x)e Fm(2)h Fp(\012)p Fz(.)21 b(By)15 b(\(4.8\))g(this)f(leads)h(to)g(the)89 2697 y(represen)o(tation)532 2776 y Fo(j)555 2783 y Fk(^)-21 b Fi(\045)572 2776 y Fp(\()p Fo(x;)8 b(\025)p Fp(\))14 b(=)g Fm(\000)821 2743 y Fp(1)p 798 2765 71 2 v 798 2811 a(2)p Fo(\031)r(i)888 2743 y(\013)919 2750 y Fi(b)936 2743 y Fo(\013)967 2750 y Fi(a)p 878 2765 120 2 v 878 2775 V 878 2818 a Fo(W)7 b Fp(\()p Fo(\025)p Fp(\))1002 2776 y Fz(tr)1040 2785 y Fg(C)1061 2775 y Ff(2)1092 2776 y Fp(\()p Fo(\045)p Fp(\()p Fo(\025)p Fp(\))p Fo(E)s Fp(\002)1279 2783 y Fi(H)1313 2776 y Fp(\()p Fo(\025)p Fp(\))1379 2756 y Fj(\003)1399 2776 y Fp(\))318 b Fz(\(4.20\))p eop %%Page: 22 24 22 23 bop 89 23 a Fz(22)692 b Fs(H.-Ch.)25 b(Kaiser,)18 b(H.)h(Neidhardt,)g(J.)g(Rehber)o(g)89 147 y Fz(whic)o(h)e(sho)o(ws)h (that)g(the)g(curren)o(t)e(densit)o(y)h(is)g(also)h(indep)q(enden)o(t)f (of)h Fo(x)e Fm(2)g Fp(\012)p Fz(.)26 b(Th)o(us)17 b(it)h(mak)o(es)89 208 y(sense)13 b(to)h(denote)f Fo(C)t Fp(\()p Fo(x;)8 b(\025)p Fp(\))13 b Fz(and)h Fo(j)705 215 y Fk(^)-21 b Fi(\045)722 208 y Fp(\()p Fo(x;)8 b(\025)p Fp(\))13 b Fz(b)o(y)g Fo(C)t Fp(\()p Fo(\025)p Fp(\))g Fz(and)h Fo(j)1149 215 y Fk(^)-21 b Fi(\045)1166 208 y Fp(\()p Fo(\025)p Fp(\))p Fz(,)14 b(resp)q(ectiv)o(ely)l(.)k(Hence,)12 b(w)o(e)h(ha)o(v)o(e)89 268 y(pro)o(v)o(ed)j(one)g(part)h(of)f(the)g (assertion)h(\(4.2\))f(and)h(\(4.3\).)22 b(T)l(aking)16 b(in)o(to)g(accoun)o(t)g(the)g(iden)o(tit)o(y)659 376 y Fp(1)p 612 398 120 2 v 612 408 V 612 452 a Fo(W)7 b Fp(\()p Fo(\025)p Fp(\))736 410 y Fo(E)s Fp(\002)813 417 y Fi(H)847 410 y Fp(\()p Fo(\025)p Fp(\))913 389 y Fj(\003)947 410 y Fp(=)13 b Fm(\000)1089 376 y Fp(1)p 1042 398 V 1042 444 a Fo(W)7 b Fp(\()p Fo(\025)p Fp(\))1166 410 y(\002)1204 417 y Fi(H)1238 410 y Fp(\()p Fo(\025)p Fp(\))p Fo(E)396 b Fz(\(4.21\))89 562 y(w)o(e)16 b(v)o(erify)e(the)i (other)h(part.)148 646 y(It)d(remains)g(to)h(sho)o(w)g(\(4.4\))g(and)g (\(4.5\).)21 b(T)l(o)15 b(this)g(end)g(w)o(e)f(note)h(that)g(from)e (\(4.17\))j(one)f(gets)411 795 y Fo(J)5 b Fp(\002)481 802 y Fi(H)514 795 y Fp(\()p Fo(\025)p Fp(\))580 775 y Fj(\003)614 795 y Fp(=)14 b Fo(J)h Fp(+)813 761 y Fo(i)p 762 784 V 762 829 a(W)7 b Fp(\()p Fo(\025)p Fp(\))895 725 y Fl(\022)1024 765 y Fo(\013)1055 772 y Fi(b)1072 765 y Fo(\013)1103 772 y Fi(a)1237 765 y Fm(\000)p Fo(\013)1307 747 y Fk(2)1307 777 y Fi(a)1327 765 y Fo(v)1351 772 y Fi(b)1368 765 y Fp(\()p Fo(a;)h(\025)p Fp(\))952 825 y Fm(\000)p Fo(\013)1022 807 y Fk(2)1022 838 y Fi(b)1042 825 y Fo(v)1066 832 y Fi(a)1086 825 y Fp(\()p Fo(b;)g(\025)p Fp(\))114 b Fo(\013)1340 832 y Fi(b)1358 825 y Fo(\013)1389 832 y Fi(a)1503 725 y Fl(\023)1736 795 y Fz(\(4.22\))89 941 y(where)806 1028 y Fo(J)19 b Fp(:=)917 958 y Fl(\022)974 997 y Fp(0)42 b(1)974 1057 y(1)g(0)1085 958 y Fl(\023)1130 1028 y Fo(:)592 b Fz(\(4.23\))89 1153 y(Hence)15 b(w)o(e)h(\034nd)703 1223 y(tr)741 1231 y Fg(C)762 1222 y Ff(2)784 1223 y Fp(\()p Fo(J)5 b Fp(\002)873 1230 y Fi(H)906 1223 y Fp(\()p Fo(\025)p Fp(\))972 1202 y Fj(\003)992 1223 y Fp(\))14 b(=)g(2)p Fo(i)1133 1189 y(\013)1164 1196 y Fi(b)1181 1189 y Fo(\013)1212 1196 y Fi(a)p 1123 1211 V 1123 1257 a Fo(W)7 b Fp(\()p Fo(\025)p Fp(\))1736 1223 y Fz(\(4.24\))89 1346 y(whic)o(h)16 b(yields)653 1402 y Fp(1)p 630 1424 71 2 v 630 1470 a(2)p Fo(\031)r(i)720 1402 y(\013)751 1409 y Fi(b)768 1402 y Fo(\013)799 1409 y Fi(a)p 711 1424 120 2 v 711 1470 a Fo(W)7 b Fp(\()p Fo(\025)p Fp(\))849 1436 y(=)13 b Fm(\000)959 1402 y Fp(1)p 944 1424 54 2 v 944 1470 a(4)p Fo(\031)1003 1436 y Fz(tr)1041 1444 y Fg(C)1062 1435 y Ff(2)1084 1436 y Fp(\()p Fo(J)5 b Fp(\002)1173 1443 y Fi(H)1206 1436 y Fp(\()p Fo(\025)p Fp(\))1272 1415 y Fj(\003)1292 1436 y Fp(\))p Fo(:)411 b Fz(\(4.25\))89 1563 y(Using)16 b(the)g(de\034nition)g(\(4.3\))g(w)o (e)g(\034nally)g(obtain)476 1705 y Fo(j)499 1712 y Fk(^)-21 b Fi(\045)516 1705 y Fp(\()p Fo(\025)p Fp(\))14 b(=)g Fm(\000)707 1671 y Fp(1)p 692 1693 V 692 1739 a(4)p Fo(\031)751 1705 y Fz(tr)789 1713 y Fg(C)810 1704 y Ff(2)832 1705 y Fp(\()p Fo(J)5 b Fp(\002)921 1712 y Fi(H)954 1705 y Fp(\()p Fo(\025)p Fp(\))1020 1684 y Fj(\003)1040 1705 y Fp(\))14 b Fz(tr)1111 1713 y Fg(C)1132 1704 y Ff(2)1154 1705 y Fp(\()p Fo(\045)p Fp(\()p Fo(\025)p Fp(\)\002)1302 1712 y Fi(H)1336 1705 y Fp(\()p Fo(\025)p Fp(\))p Fo(E)s Fp(\))p Fo(:)262 b Fz(\(4.26\))89 1837 y(This)17 b(leads)f(to)g(the)g (estimate)549 1967 y Fm(j)p Fo(j)586 1974 y Fk(^)-21 b Fi(\045)603 1967 y Fp(\()p Fo(\025)p Fp(\))p Fm(j)14 b(\024)770 1933 y Fp(1)p 755 1955 V 755 2001 a(4)p Fo(\031)814 1967 y Fm(k)p Fp(\002)877 1974 y Fi(H)910 1967 y Fp(\()p Fo(\025)p Fp(\))976 1946 y Fj(\003)996 1967 y Fm(k)1021 1975 y Fj(L)1045 1980 y Ff(1)1063 1975 y Fk(\()p Fg(C)1097 1966 y Ff(2)1118 1975 y Fk(\))1134 1967 y Fm(k)p Fo(\045)p Fp(\()p Fo(\025)p Fp(\))p Fm(k)1275 1975 y Fj(L)1299 1980 y Ff(1)1317 1975 y Fk(\()p Fg(C)1351 1966 y Ff(2)1372 1975 y Fk(\))1387 1967 y Fo(:)335 b Fz(\(4.27\))89 2102 y(Since)16 b Fm(k)p Fp(\002)280 2109 y Fi(H)313 2102 y Fp(\()p Fo(\025)p Fp(\))379 2084 y Fj(\003)399 2102 y Fm(k)424 2110 y Fj(L)448 2115 y Ff(1)466 2110 y Fk(\()p Fg(C)500 2101 y Ff(2)521 2110 y Fk(\))550 2102 y Fm(\024)e Fp(2)i Fz(and)h Fm(k)p Fo(\045)p Fp(\()p Fo(\025)p Fp(\))p Fm(k)879 2110 y Fj(L)903 2115 y Ff(1)921 2110 y Fk(\()p Fg(C)956 2101 y Ff(2)976 2110 y Fk(\))1006 2102 y Fp(=)d Fz(tr)1095 2110 y Fg(C)1116 2101 y Ff(2)1139 2102 y Fp(\()p Fo(\045)p Fp(\()p Fo(\025)p Fp(\)\))j Fz(w)o(e)e(\034nd)737 2248 y Fm(j)p Fo(j)774 2255 y Fk(^)-21 b Fi(\045)791 2248 y Fp(\()p Fo(\025)p Fp(\))p Fm(j)15 b(\024)958 2215 y Fp(1)p 943 2237 V 943 2283 a(2)p Fo(\031)1002 2248 y Fz(tr)1040 2257 y Fg(C)1061 2247 y Ff(2)1083 2248 y Fp(\()p Fo(\045)p Fp(\()p Fo(\025)p Fp(\)\))524 b Fz(\(4.28\))89 2384 y(whic)o(h)16 b(imme)o(diatel)o(y)d(pro)o(v)o(es)j(\(4.5\).)1051 b Fa(\003)148 2468 y Fz(In)20 b(corresp)q(ondence)g(to)g(the)g(carrier) f(densit)o(y)g(it)g(seems)g(to)h(b)q(e)g(useful)f(to)i(in)o(tro)q(duce) e(the)89 2528 y(follo)o(wing)d(de\034nition.)89 2685 y Fq(De\034nition)k(4.2)k Fz(Assume)15 b Fo(m)c Fp(+)743 2666 y Fk(1)p 736 2674 32 2 v 736 2703 a Fi(m)787 2685 y Fm(2)k Fo(L)868 2667 y Fj(1)905 2685 y Fp(\([)p Fo(a;)8 b(b)p Fp(]\))p Fz(,)15 b Fo(V)26 b Fm(2)15 b Fo(L)1204 2667 y Fk(2)1224 2685 y Fp(\([)p Fo(a;)8 b(b)p Fp(]\))15 b Fz(and)i Fo(\024)1497 2692 y Fi(a)1518 2685 y Fo(;)8 b(\024)1568 2692 y Fi(b)1600 2685 y Fm(2)15 b Fn(C)1681 2692 y Fk(+)1713 2685 y Fz(.)23 b(Then)89 2746 y(the)17 b(matrix)f Fo(C)t Fp(\()p Fo(\025)p Fp(\))h Fz(and)h(the)g(v)m(alue)f Fo(j)s(\045)p Fp(\()p Fo(\025)p Fp(\))h Fz(are)f(called)g(the)g(curren) o(t)f(densit)o(y)h(observ)m(able)g(and)89 2806 y(the)f(curren)o(t)f (densit)o(y)h(at)g(energy)g Fo(\025)e Fm(2)g Fn(R)g Fz(of)i(the)g (system)f(describ)q(ed)h(b)o(y)f Fo(H)t Fz(,)h(resp)q(ectiv)o(ely)l(.)p eop %%Page: 23 25 23 24 bop 89 23 a Fs(Density)18 b(and)g(current)h(of)f(a)g(dissip)l(a)l (tive)g(Schr)o(\366dinger)h(opera)l(tor)187 b Fz(23)89 147 y(The)15 b(de\034nition)e(is)i(again)g(justi\034ed)f(b)o(y)g(the)g (fact)g(that)h(the)f(curren)o(t)f(densit)o(y)h(at)h(energy)f Fo(\025)g Fm(2)g Fn(R)89 208 y Fz(is)23 b(the)f(exp)q(ectation)h(v)m (alue)f(of)h(the)g(curren)o(t)f(densit)o(y)f(observ)m(able)i(at)h (energy)e Fo(\025)j Fm(2)h Fn(R)p Fz(,)21 b(i.e.)89 268 y Fo(j)112 275 y Fk(^)-21 b Fi(\045)129 268 y Fp(\()p Fo(\025)p Fp(\))15 b(=)e Fn(E)292 275 y Fi(\045)p Fk(\()p Fi(\025)p Fk(\))363 268 y Fp(\()p Fo(C)t Fp(\()p Fo(\025)p Fp(\)\))j Fz(for)g Fo(\025)f Fm(2)f Fn(R)p Fz(.)k(Using)e(this)g (notation)h(form)o(ula)e(\(4.4\))h(tak)o(es)g(the)g(form)728 415 y Fo(j)751 422 y Fk(^)-21 b Fi(\045)782 415 y Fp(=)834 347 y Fl(Z)862 460 y Fg(R)896 415 y Fo(d\025)15 b Fn(E)994 423 y Fi(\045)p Fk(\()p Fi(\025)p Fk(\))1065 415 y Fp(\()p Fo(C)t Fp(\()p Fo(\025)p Fp(\)\))p Fo(:)514 b Fz(\(4.29\))89 557 y(In)16 b(the)g(follo)o(wing)g(corollary)g(w)o(e)g(consider)g(the)g (case)h(that)f(the)h(steady)f(state)h Fo(\045)f Fz(is)g(a)h(function)89 618 y(of)g Fo(K)t Fz(,)f(i.e.)867 687 y Fo(\045)e Fp(=)f Fo(f)5 b Fp(\()p Fo(K)t Fp(\))p Fo(;)653 b Fz(\(4.30\))89 785 y(where,)18 b(of)g(course,)g Fo(f)5 b Fp(\()p Fm(\001)p Fp(\))17 b Fm(2)g Fo(L)649 767 y Fj(1)686 785 y Fp(\()p Fn(R)p Fp(\))e Fz(and)k Fo(f)5 b Fp(\()p Fo(\025)p Fp(\))17 b Fm(\025)f Fp(0)j Fz(for)f(a.e.)26 b Fo(\025)17 b Fm(2)g Fn(R)e Fz(and.)27 b(In)17 b(this)h(case)g(the)89 845 y(densit)o(y)d(matrix)g Fo(\045)h Fz(b)q(elongs)h(to)g(the)f(bicomm)n (utan)o(t)e(of)i Fo(K)t Fz(.)89 983 y Fq(Corollary)k(4.3)k Fc(Assume)18 b Fo(m)12 b Fp(+)732 963 y Fk(1)p 725 971 32 2 v 725 1000 a Fi(m)776 983 y Fm(2)j Fo(L)857 965 y Fj(1)895 983 y Fp(\([)p Fo(a;)8 b(b)p Fp(]\))p Fc(,)16 b Fo(V)26 b Fm(2)15 b Fo(L)1196 965 y Fk(2)1216 983 y Fp(\([)p Fo(a;)8 b(b)p Fp(]\))16 b Fc(and)i Fo(\024)1490 990 y Fi(a)1511 983 y Fo(;)8 b(\024)1561 990 y Fi(b)1593 983 y Fm(2)15 b Fn(C)1674 990 y Fk(+)1707 983 y Fc(.)24 b(If)17 b(the)89 1043 y(ste)n(ady)j(state)g Fo(\045)g Fc(is)g(given)i(by)e(\(4.30\))f(with)i(a)f(non-ne)n(gative)i(function)f Fo(f)5 b Fp(\()p Fm(\001)p Fp(\))19 b Fm(2)g Fo(L)1623 1025 y Fj(1)1660 1043 y Fp(\()p Fn(R)p Fp(\))p Fc(,)f(then)89 1103 y Fo(j)112 1110 y Fk(^)-21 b Fi(\045)129 1103 y Fp(\()p Fo(\025)p Fp(\))15 b(=)e(0)18 b Fc(for)f(a.e.)23 b Fo(\025)14 b Fm(2)g Fn(R)p Fc(.)89 1241 y Fq(Pro)r(of.)21 b Fz(In)16 b(this)g(case)h(one)f(has)711 1360 y Fo(\045)p Fp(\()p Fo(\025)p Fp(\))f(=)f Fo(f)5 b Fp(\()p Fo(\025)p Fp(\))p Fo(I)986 1368 y Fg(C)1007 1359 y Ff(2)1029 1360 y Fo(;)57 b(\025)14 b Fm(2)g Fn(R)p Fo(:)494 b Fz(\(4.31\))89 1479 y(whic)o(h)16 b(giv)o(es)381 1618 y Fo(j)404 1625 y Fk(^)-21 b Fi(\045)421 1618 y Fp(\()p Fo(\025)p Fp(\))15 b(=)e Fo(f)5 b Fp(\()p Fo(\025)p Fp(\))p Fz(tr)q Fp(\()p Fo(C)t Fp(\()p Fo(\025)p Fp(\)\))14 b(=)f Fm(\000)962 1584 y Fp(1)p 939 1607 71 2 v 939 1652 a(2)p Fo(\031)r(i)1015 1618 y(f)5 b Fp(\()p Fo(\025)p Fp(\))1125 1584 y Fo(\013)1156 1591 y Fi(b)1173 1584 y Fo(\013)1204 1591 y Fi(a)p 1115 1607 120 2 v 1115 1617 V 1115 1660 a Fo(W)i Fp(\()p Fo(\025)p Fp(\))1240 1618 y Fz(tr)1277 1626 y Fg(C)1298 1617 y Ff(2)1321 1618 y Fp(\()p Fo(E)s Fp(\002)1417 1625 y Fi(H)1450 1618 y Fp(\()p Fo(\025)p Fp(\))1516 1598 y Fj(\003)1536 1618 y Fp(\))p Fo(:)167 b Fz(\(4.32\))89 1773 y(By)16 b(\(4.17\))h(w)o(e)e(immedi)o(ately)e(get)j(that)h(tr)881 1781 y Fg(C)902 1772 y Ff(2)924 1773 y Fp(\()p Fo(E)s Fp(\002)1020 1780 y Fi(H)1054 1773 y Fp(\()p Fo(\025)p Fp(\))1120 1755 y Fj(\003)1140 1773 y Fp(\))d(=)f(0)k Fz(for)g Fo(\025)d Fm(2)g Fn(R)p Fz(.)340 b Fa(\003)148 1857 y Fz(If)16 b(the)g(steady)g(state)h Fo(\045)f Fz(has)h(the)f(form) 680 2002 y Fo(\045)p Fp(\()p Fo(\025)p Fp(\))e(=)837 1932 y Fl(\022)894 1972 y Fo(\045)919 1954 y Fi(bb)952 1972 y Fp(\()p Fo(\025)p Fp(\))95 b(0)944 2032 y(0)d Fo(\045)1085 2014 y Fi(aa)1124 2032 y Fp(\()p Fo(\025)p Fp(\))1212 1932 y Fl(\023)1257 2002 y Fo(;)465 b Fz(\(4.33\))89 2148 y(then)16 b(the)g(curren)o(t)g(densit)o(y)f(is)h(giv)o(en)f(b)o(y) 566 2294 y Fo(j)589 2301 y Fk(^)-21 b Fi(\045)606 2294 y Fp(\()p Fo(\025)p Fp(\))14 b(=)g Fm(\000)796 2260 y Fp(1)p 782 2282 54 2 v 782 2328 a(2)p Fo(\031)877 2260 y(\013)908 2242 y Fk(2)908 2272 y Fi(a)929 2260 y Fo(\013)960 2242 y Fk(2)960 2273 y Fi(b)p 845 2282 167 2 v 845 2328 a Fm(j)p Fo(W)7 b Fp(\()p Fo(\025)p Fp(\))p Fm(j)992 2313 y Fk(2)1017 2294 y Fp(\()p Fo(\045)1061 2273 y Fi(bb)1093 2294 y Fp(\()p Fo(\025)p Fp(\))12 b Fm(\000)f Fo(\045)1246 2273 y Fi(aa)1285 2294 y Fp(\()p Fo(\025)p Fp(\)\))p Fo(:)352 b Fz(\(4.34\))89 2445 y(This)21 b(curren)o(t)f(densit)o(y)f (is)i(di\033eren)o(t)f(from)f(zero)i(if)f Fo(\045)1103 2427 y Fi(bb)1135 2445 y Fp(\()p Fo(\025)p Fp(\))i Fm(6)p Fp(=)g Fo(\045)1308 2427 y Fi(aa)1347 2445 y Fp(\()p Fo(\025)p Fp(\))f Fz(at)g(least)g(for)g(a)g(set)g(of)89 2505 y(p)q(ositiv)o(e)14 b(Leb)q(esgue)g(measure.)20 b(So)14 b(a)h(curren)o(t)e(densit)o(y)g(di\033eren)o(t)h(from)f(zero)g (arises)i(only)f(if)f(w)o(e)89 2565 y(ha)o(v)o(e)h(a)h(o)q(ccupation)g (disparit)o(y)f(b)q(et)o(w)o(een)f(the)i(t)o(w)o(o)f(eigenstates)h Fo( )1326 2547 y Fj(\000)1355 2565 y Fp(\()p Fo(x;)8 b(\025;)g(b)p Fp(\))14 b Fz(and)h Fo( )1655 2547 y Fj(\000)1684 2565 y Fp(\()p Fo(x;)8 b(\025;)g(a)p Fp(\))p Fz(.)89 2625 y(This)21 b(is)f(the)g(case)g(if)g(the)g(steady)g(state)h Fo(\045)f Fz(b)q(elongs)h(to)g(the)f(comm)o(utan)o(t)d(of)k Fo(K)j Fz(but)d(not)f(to)89 2685 y(the)i(bicomm)o(utan)o(t.)37 b(In)22 b(other)g(w)o(ords,)i(the)e(densit)o(y)g(matrix)e(\(1.8\))j (used)g(for)f(self-adjoin)o(t)89 2746 y(b)q(oundary)15 b(conditions)f(and)g(generalized)f(b)o(y)g(\(4.30\))i(to)f(the)g (dissipativ)o(e)e(case)i(leads)g(to)g(a)g(zero)89 2806 y(curren)o(t)h(densit)o(y)l(.)p eop %%Page: 24 26 24 25 bop 89 23 a Fz(24)692 b Fs(H.-Ch.)25 b(Kaiser,)18 b(H.)h(Neidhardt,)g(J.)g(Rehber)o(g)89 155 y Fx(5)77 b(Remarks)89 288 y Fz(1.)39 b(The)22 b(carrier)f(densit)o(y)g Fo(u)637 295 y Fk(^)-21 b Fi(\045)654 288 y Fp(\()p Fm(\001)p Fp(\))21 b Fz(p)q(erformed)g(b)o(y)g(\(3.41\))i(and)g(\(3.44\))f(is)g (a)g(straigh)o(tforw)o(ard)89 349 y(generalization)15 b(of)g(the)g(corresp)q(onding)h(de\034nition)e(\(1.5\))h(of)h(the)f (carrier)f(densit)o(y)g(in)g(the)h(self-)89 409 y(adjoin)o(t)h(case.)22 b(Indeed,)15 b(this)h(corresp)q(ondence)g(relies)f(on)i(the)f (replacemen)o(ts)743 508 y Fo(l)42 b Fm( )-8 b(!)41 b(f)p Fo(a;)8 b(b;)g(\025)p Fm(g)p Fo(;)606 598 y Fm(f)p Fo( )663 605 y Fi(l)676 598 y Fm(g)701 580 y Fj(1)701 611 y Fi(l)p Fk(=1)800 598 y Fm( )-8 b(!)41 b(f)964 585 y Fo(~)958 598 y( )q Fp(\()p Fm(\001)p Fo(;)8 b(\025;)g(\034)e Fp(\))p Fm(g)1167 605 y Fi(\025)p Fj(2)p Fg(R)-7 b Fi(;\034)s Fk(=)p Fi(a;b)1337 598 y Fo(;)648 645 y Fl(P)700 658 y Fj(1)700 696 y Fi(l)p Fk(=1)800 682 y Fm( )f(!)933 642 y Fl(R)956 699 y Fi(\025)p Fj(2)p Fg(R)1035 682 y Fo(d\025)1111 645 y Fl(P)1163 696 y Fi(\034)t Fk(=)p Fi(a;b)720 766 y Fo(\045)745 773 y Fi(l)800 766 y Fm( )g(!)41 b Fo(\045)p Fp(\()p Fo(\025)p Fp(\))1761 637 y Fz(\(5.1\))89 868 y(The)18 b(same)e(holds)i(for)g(the)f(curren)o(t)g(densit)o(y)g (de\034ned)g(b)o(y)g(\(4.1\))h(whic)o(h)f(is)g(a)h(straigh)o(tforw)o (ard)89 928 y(generalization)i(of)h(\(1.11\).)35 b Fc(However,)24 b(in)e(c)n(ontr)n(ast)f(to)g(the)h(self-adjoint)h(c)n(ase)f(the)g(curr) n(ent)89 989 y(density)c(now)g(is)f(not)h(ne)n(c)n(essarily)g(zer)n(o!) 148 1072 y Fz(2.)23 b(There)16 b(is)h(consensus)g(in)g(the)f(con)o (viction)f(that)j(scattering)e(states)h(are)g(resp)q(onsible)g(for)89 1133 y(the)12 b(curren)o(t.)19 b(This)12 b(usually)g(leads)g(to)h(a)f (a)h(relation)f(b)q(et)o(w)o(een)f(curren)o(t)g(densit)o(y)g(and)i (scattering)89 1193 y(matrix.)31 b(A)o(ctually)l(,)19 b(the)h(same)f(tak)o(es)h(place)g(here.)32 b(F)l(orm)o(ulae)18 b(\(4.2\))j(and)f(\(4.3\))h(relate)e(the)89 1253 y(curren)o(t)g(densit) o(y)g Fo(j)457 1260 y Fk(^)-21 b Fi(\045)474 1253 y Fp(\()p Fo(\025)p Fp(\))20 b Fz(with)f(the)h(c)o(haracteristic)e(function)h Fp(\002)1297 1260 y Fi(H)1329 1251 y Fd(\003)1349 1253 y Fp(\()p Fo(\025)p Fp(\))h Fz(of)g Fo(H)1538 1235 y Fj(\003)1558 1253 y Fz(.)31 b(It)19 b(turns)h(out)89 1313 y(that)j(the)f(c)o(haracteristic)f(function)h Fp(\002)832 1320 y Fi(H)864 1311 y Fd(\003)884 1313 y Fp(\()p Fo(\025)p Fp(\))h Fz(of)f Fo(H)1078 1295 y Fj(\003)1121 1313 y Fz(can)g(b)q(e)h(regarded)g(as)g(the)f(scattering)89 1373 y(matrix)15 b(of)j(an)f(asso)q(ciated)h(scattering)f(system.)22 b(Indeed,)16 b(with)h(the)g(self-adjoin)o(t)f(dilation)h Fo(K)89 1434 y Fz(one)g(can)f(asso)q(ciate)h(a)g(so-called)f (Lax-Phillips)f(scattering)i(theory)f([2)o(,)g(16)q(].)21 b(T)l(o)c(this)f(end)g(one)89 1494 y(in)o(tro)q(duces)g(the)g(Hilb)q (ert)f(space)h Fh(K)740 1501 y Fk(0)760 1494 y Fz(,)619 1597 y Fh(K)652 1604 y Fk(0)686 1597 y Fp(:=)d Fo(L)784 1576 y Fk(2)804 1597 y Fp(\()p Fn(R)p Fo(;)7 b Fn(C)914 1576 y Fk(2)936 1597 y Fp(\))14 b(=)g Fm(D)1059 1604 y Fj(\000)1100 1597 y Fm(\010)d(D)1188 1604 y Fk(+)1232 1597 y Fm(\022)i Fh(K)p Fo(:)430 b Fz(\(5.2\))89 1700 y(and)17 b(the)f(iden)o(ti\034cation)f(op)q(erators)j Fo(J)805 1707 y Fj(\006)848 1700 y Fp(:)13 b Fh(K)908 1707 y Fk(0)942 1700 y Fm(\000)-8 b(!)13 b Fh(K)p Fz(,)567 1794 y Fo(~)560 1807 y(f)19 b Fp(=)14 b Fo(J)682 1814 y Fj(\000)712 1807 y Fo(f)47 b Fp(:=)41 b Fo(P)914 1785 y Fe(K)937 1790 y Ff(0)907 1820 y Fj(D)934 1824 y Fd(\000)962 1807 y Fo(f)17 b Fm(\010)11 b Fp(0)g Fm(\010)g Fp(0)p Fo(;)57 b(f)19 b Fm(2)14 b Fh(K)1356 1814 y Fk(0)1376 1807 y Fo(;)567 1884 y(~)560 1897 y(f)19 b Fp(=)14 b Fo(J)682 1904 y Fk(+)712 1897 y Fo(f)47 b Fp(:=)41 b(0)11 b Fm(\010)g Fp(0)g Fm(\010)g Fo(P)1084 1875 y Fe(K)1107 1880 y Ff(0)1077 1911 y Fj(D)1104 1915 y Ff(+)1132 1897 y Fo(f)s(;)57 b(f)19 b Fm(2)14 b Fh(K)1353 1904 y Fk(0)1373 1897 y Fo(:)1761 1851 y Fz(\(5.3\))89 2000 y(The)f(subspaces)h Fm(D)444 2007 y Fj(\000)487 2000 y Fz(and)g Fm(D)617 2007 y Fk(+)660 2000 y Fz(are)f(called)f(incoming)g(and)h(outgoing)i (subspaces,)f(resp)q(ectiv)o(ely)l(.)89 2060 y(On)i(the)g(Hilb)q(ert)f (space)i Fh(K)586 2067 y Fk(0)622 2060 y Fz(one)f(de\034nes)g(the)g (self-adjoin)o(t)g(op)q(erator)i Fo(K)1444 2067 y Fk(0)1464 2060 y Fz(,)738 2185 y Fp(\()p Fo(K)798 2192 y Fk(0)819 2185 y Fo(f)5 b Fp(\)\()p Fo(x)p Fp(\))13 b(=)h Fm(\000)p Fo(i)1072 2152 y(d)p 1059 2174 54 2 v 1059 2220 a(dx)1117 2185 y(f)5 b Fp(\()p Fo(x)p Fp(\))549 b Fz(\(5.4\))89 2306 y(with)16 b(the)g(domain)g(dom)o Fp(\()p Fo(K)610 2313 y Fk(0)630 2306 y Fp(\))e(:=)f Fo(W)781 2288 y Fk(1)p Fi(;)p Fk(2)828 2306 y Fp(\()p Fn(R)p Fo(;)8 b Fn(C)938 2288 y Fk(2)960 2306 y Fp(\))p Fz(.)22 b(The)16 b(Lax-Phillips)g(w)o(a) o(v)o(e)f(op)q(erators)672 2409 y Fo(W)718 2416 y Fj(\006)761 2409 y Fp(=)f Fo(s)d Fm(\000)33 b Fp(lim)897 2439 y Fi(t)p Fj(!\0061)1016 2409 y Fo(e)1039 2389 y Fi(itK)1098 2409 y Fo(J)1125 2416 y Fj(\006)1154 2409 y Fo(e)1177 2389 y Fj(\000)p Fi(itK)1259 2394 y Ff(0)1761 2409 y Fz(\(5.5\))89 2532 y(exist)11 b(and)g(are)h(complete,)d(i.e.)18 b(ran)q Fp(\()p Fo(W)799 2539 y Fj(\006)828 2532 y Fp(\))c(=)g Fh(K)p Fz(.)20 b(The)11 b(corresp)q(onding)h(Lax-Phillips)f(scattering) 89 2592 y(op)q(erator)22 b Fo(S)j Fp(=)d Fo(W)458 2574 y Fj(\003)451 2605 y Fk(+)480 2592 y Fo(W)526 2599 y Fj(\000)577 2592 y Fp(:)g Fh(K)646 2599 y Fk(0)687 2592 y Fm(\000)-8 b(!)794 2579 y Fp(^)789 2592 y Fh(K)q Fz(,)21 b(is)g(unitary)g(and)g(comm)o(utes)d(with)j(the)f(self-adjoin)o(t)89 2660 y(op)q(erator)e Fo(K)327 2667 y Fk(0)347 2660 y Fz(.)j(By)16 b Fm(F)i Fp(:)c Fh(K)574 2667 y Fk(0)607 2660 y Fm(\000)-8 b(!)706 2647 y Fp(^)701 2660 y Fh(K)17 b Fz(w)o(e)f(denote)g(the)g(F)l(ourier)g(transform)335 2784 y Fp(\()p Fm(F)5 b Fo(f)g Fp(\)\()p Fo(\025)p Fp(\))14 b(=)585 2771 y(^)575 2784 y Fo(f)5 b Fp(\()p Fo(\025)p Fp(\))14 b(:=)790 2751 y(1)p 755 2773 96 2 v 755 2783 a Fm(p)p 796 2783 54 2 v 796 2824 a Fp(2)p Fo(\031)863 2717 y Fl(Z)891 2829 y Fg(R)925 2784 y Fo(dx)g(e)1015 2764 y Fj(\000)p Fi(i\025x)1097 2784 y Fo(f)5 b Fp(\()p Fo(x)p Fp(\))p Fo(;)56 b(f)20 b Fm(2)14 b Fh(K)1386 2791 y Fk(0)1406 2784 y Fo(;)56 b(\025)15 b Fm(2)f Fn(R)p Fo(:)142 b Fz(\(5.6\))p eop %%Page: 25 27 25 26 bop 89 23 a Fs(Density)18 b(and)g(current)h(of)f(a)g(dissip)l(a)l (tive)g(Schr)o(\366dinger)h(opera)l(tor)187 b Fz(25)89 147 y(W)l(e)22 b(note)h(that)f Fm(F)5 b Fo(K)487 154 y Fk(0)507 147 y Fm(F)548 129 y Fj(\000)p Fk(1)620 147 y Fp(=)24 b Fo(M)j Fz(where)22 b Fo(M)28 b Fz(is)22 b(de\034ned)g(b)o (y)g(\(2.43\).)40 b(Since)22 b Fo(S)j Fz(comm)o(utes)89 208 y(with)c Fo(K)246 215 y Fk(0)288 208 y Fz(the)g(op)q(erator)586 195 y Fp(^)578 208 y Fo(S)k Fp(=)d Fm(F)5 b Fo(S)s Fm(F)808 189 y Fj(\000)p Fk(1)877 208 y Fp(:)918 195 y(^)913 208 y Fh(K)23 b Fm(\000)-9 b(!)1076 195 y Fp(^)1071 208 y Fh(K)22 b Fz(comm)o(ute)o(s)d(with)i Fo(M)5 b Fz(.)36 b(Hence)1737 195 y Fp(^)1728 208 y Fo(S)24 b Fz(is)d(a)89 268 y(m)o(ultiplic)o(ation)14 b(op)q(erator)j(giv)o(en)f(b)o(y)560 368 y Fp(\()587 355 y(^)579 368 y Fo(S)622 354 y Fp(^)611 368 y Fo(f)6 b Fp(\)\()p Fo(\025)p Fp(\))14 b(=)g Fo(S)s Fp(\()p Fo(\025)p Fp(\))902 354 y(^)891 368 y Fo(f)5 b Fp(\()p Fo(\025)p Fp(\))p Fo(;)1068 354 y Fp(^)1057 368 y Fo(f)20 b Fm(2)1152 355 y Fp(^)1148 368 y Fh(K)p Fo(;)57 b(\025)14 b Fm(2)g Fn(R)p Fo(;)367 b Fz(\(5.7\))89 467 y(where)21 b Fm(f)p Fo(S)s Fp(\()p Fo(\025)p Fp(\))p Fm(g)384 474 y Fi(\025)p Fj(2)p Fg(R)476 467 y Fz(is)h(a)g(measurable)e (family)f(of)j(unitary)f(op)q(erators)i(whic)o(h)e(is)h(called)e(the)89 528 y(Lax-Phillips)c(scattering)g(matrix.)k(A)15 b(rather)i(in)o(v)o (olv)o(ed)d(computation)h(sho)o(ws)i(that)815 627 y Fo(S)s Fp(\()p Fo(\025)p Fp(\))d(=)f(\002)1017 634 y Fi(H)1049 625 y Fd(\003)1069 627 y Fp(\()p Fo(\025)p Fp(\))626 b Fz(\(5.8\))89 727 y(for)17 b(a.e.)j Fo(\025)15 b Fm(2)f Fn(R)p Fz(,)f(see)i(for)i(example)d([1].)21 b(By)15 b(the)h(w)o(a)o(y)g (one)g(has)640 827 y Fp(\010)675 834 y Fj(\000)719 827 y Fp(=)e Fm(F)5 b Fo(W)865 807 y Fj(\003)858 839 y Fk(+)936 827 y Fz(and)49 b Fp(\010)1098 834 y Fk(+)1142 827 y Fp(=)13 b Fm(F)5 b Fo(W)1287 807 y Fj(\003)1280 839 y(\000)1761 827 y Fz(\(5.9\))89 927 y(where)18 b(is)f Fp(\010)317 934 y Fj(\000)365 927 y Fz(is)g(the)h(incoming)e(F)l(ourier)h (transformation,)g(cf.)26 b(\(2.41\))18 b(and)g(\(2.42\),)h(and)f Fp(\010)1831 934 y Fk(+)89 987 y Fz(is)e(the)g(so-called)g(outgoing)i (F)l(ourier)d(transform)h(whic)o(h)g(w)o(as)g(in)o(tro)q(duced)g(in)g ([12].)148 1071 y(3.)22 b(As)16 b(men)o(tioned)e(ab)q(o)o(v)o(e)i(the)g (quan)o(tit)o(y)f Fo(N)973 1078 y Fi(\045)993 1071 y Fp(\()p Fo(!)r Fp(\))p Fz(,)615 1197 y Fo(N)654 1204 y Fi(\045)674 1197 y Fp(\()p Fo(!)r Fp(\))f(:=)f Fm(k)p Fo(u)879 1204 y Fk(^)-21 b Fi(\045)896 1197 y Fm(k)921 1205 y Fi(L)945 1195 y Ff(1)962 1205 y Fk(\()p Fi(!)q Fk(\))1028 1197 y Fp(=)1080 1129 y Fl(Z)1108 1242 y Fi(!)1141 1197 y Fo(dx)14 b(u)1239 1204 y Fk(^)-21 b Fi(\045)1256 1197 y Fp(\()p Fo(x)p Fp(\))p Fo(;)400 b Fz(\(5.10\))89 1323 y(has)19 b(the)g(meaning)e(of)i(the)f(n)o(um)o(b)q(er)f(of)h (carriers)g(on)h(the)g(Borel)e(set)h(subset)h Fo(!)h Fm(\022)e Fp(\012)f(=)h([)p Fo(a;)8 b(b)p Fp(])p Fz(.)89 1383 y(By)16 b(\(3.44\))h(one)f(has)h(the)f(represen)o(tation)766 1483 y Fo(N)805 1490 y Fi(\045)825 1483 y Fp(\()p Fo(!)r Fp(\))e(=)g Fz(tr)p Fp(\()p Fo(\045U)5 b Fp(\()p Fo(!)r Fp(\)\))p Fo(:)552 b Fz(\(5.11\))89 1583 y(Ob)o(viously)14 b Fo(N)19 b Fp(:=)13 b Fo(N)479 1590 y Fi(\045)499 1583 y Fp(\(\012\))i Fz(is)g(the)f(total)h(n)o(um)o(b)q(er)e(of)h(carriers)h (on)g(the)f(in)o(terv)m(al)f Fp([)p Fo(a;)8 b(b)p Fp(])p Fz(.)19 b(W)l(e)c(note)89 1643 y(that)i(under)f(the)f (condition\(3.23\))i(b)o(y)e(Prop)q(osition)i(3.7)g(the)f(total)g(n)o (um)o(b)q(er)e(of)i(carriers)g(in)g Fp(\012)89 1703 y Fz(is)g(alw)o(a)o(ys)g(\034nite.)148 1787 y(In)g(accordance)g(with)g ([15)q(])g(the)g(time)e(ev)o(olution)h(of)h(the)g(densit)o(y)g(matrix)e Fo(\045)j Fz(is)f(giv)o(en)f(b)o(y)695 1887 y Fo(\045)p Fp(\()p Fo(t)p Fp(\))e(=)h Fo(e)864 1866 y Fj(\000)p Fi(itK)950 1887 y Fo(\045e)998 1866 y Fi(itK)1057 1887 y Fo(;)56 b(t)14 b Fm(2)g Fn(R)p Fo(:)477 b Fz(\(5.12\))89 1986 y(Hence,)15 b(the)h(n)o(um)o(b)q(er)e(of)j(particles)e(at)i(time)d Fo(t)f Fm(2)h Fn(R)f Fz(on)k Fo(!)h Fz(is)e(giv)o(en)g(b)o(y)619 2086 y Fo(N)658 2094 y Fi(\045)p Fk(\()p Fi(t)p Fk(\))718 2086 y Fp(\()p Fo(!)r Fp(\))e(:=)g Fz(tr)o Fp(\()p Fo(\045)p Fp(\()p Fo(t)p Fp(\))p Fo(U)5 b Fp(\()p Fo(!)r Fp(\)\))p Fo(;)57 b(t)14 b Fm(2)g Fn(R)p Fo(:)401 b Fz(\(5.13\))89 2186 y(Since)16 b Fo(\045)g Fz(comm)o(utes)d(with)j Fo(K)21 b Fz(one)16 b(has)h Fo(\045)p Fp(\()p Fo(t)p Fp(\))c(=)h Fo(\045)j Fz(and)f Fo(N)1157 2194 y Fi(\045)p Fk(\()p Fi(t)p Fk(\))1218 2186 y Fp(\()p Fo(!)r Fp(\))e(=)g Fo(N)1393 2193 y Fi(\045)1413 2186 y Fp(\()p Fo(!)r Fp(\))i Fz(or)631 2281 y Fo(d)p 622 2303 43 2 v 622 2348 a(dt)670 2314 y(N)709 2322 y Fi(\045)p Fk(\()p Fi(t)p Fk(\))770 2314 y Fp(\()p Fo(!)r Fp(\))e(=)f(0)p Fo(;)57 b(!)16 b Fm(\022)e Fp(\012)p Fo(;)57 b(t)13 b Fm(2)h Fn(R)p Fo(:)400 b Fz(\(5.14\))89 2427 y(If)16 b(the)g(condition)610 2487 y Fo(C)648 2494 y Fk(^)-21 b Fi(\045)679 2487 y Fp(:=)13 b(sup)747 2528 y Fi(\025)p Fj(2)p Fg(R)818 2487 y Fp(\(1)e(+)g Fo(\025)949 2466 y Fk(2)969 2487 y Fp(\))p Fm(k)p Fo(\045)p Fp(\()p Fo(\025)p Fp(\))p Fm(k)1129 2495 y Fj(B)q Fk(\()p Fg(C)1188 2486 y Ff(2)1209 2495 y Fk(\))1238 2487 y Fo(<)j Fm(1)396 b Fz(\(5.15\))89 2596 y(is)16 b(satis\034ed,)g(then)152 2684 y Fo(d)p 143 2706 V 143 2752 a(dt)191 2718 y(N)230 2725 y Fi(\045)p Fk(\()p Fi(t)p Fk(\))290 2718 y Fp(\()p Fo(!)r Fp(\))e(=)1324 b Fz(\(5.16\))280 2806 y Fm(\000)p Fo(i)p Fz(tr)p Fp(\()p Fo(K)t(\045)p Fp(\()p Fo(t)p Fp(\))p Fo(U)5 b Fp(\()p Fo(!)r Fp(\)\))11 b(+)g Fo(i)p Fz(tr)o Fp(\()p Fo(\045)p Fp(\()p Fo(t)p Fp(\))p Fo(K)t(U)5 b Fp(\()p Fo(!)r Fp(\)\))14 b(=)g Fm(\000)p Fo(i)p Fz(tr)o Fp(\()p Fo(K)t(\045U)5 b Fp(\()p Fo(!)r Fp(\)\))12 b(+)f Fo(i)p Fz(tr)p Fp(\()p Fo(\045K)t(U)5 b Fp(\()p Fo(!)r Fp(\)\))p Fo(:)p eop %%Page: 26 28 26 27 bop 89 23 a Fz(26)692 b Fs(H.-Ch.)25 b(Kaiser,)18 b(H.)h(Neidhardt,)g(J.)g(Rehber)o(g)89 147 y Fz(Hence)130 240 y Fo(d)p 121 262 43 2 v 121 308 a(dt)169 274 y(N)208 281 y Fi(\045)p Fk(\()p Fi(t)p Fk(\))269 274 y Fp(\()p Fo(!)r Fp(\))14 b(=)1345 b Fz(\(5.17\))258 392 y Fm(\000)p Fo(i)322 324 y Fl(Z)350 437 y Fg(R)384 392 y Fo(d\025)460 324 y Fl(Z)488 437 y Fi(!)521 392 y Fo(dx)14 b(\025)g Fz(tr)668 401 y Fg(C)689 391 y Ff(2)712 392 y Fp(\()p Fo(\045)p Fp(\()p Fo(\025)p Fp(\))p Fo(D)q Fp(\()p Fo(x;)8 b(\025)p Fp(\)\))k(+)f Fo(i)1084 324 y Fl(Z)1112 437 y Fg(R)1146 392 y Fo(d\025)1222 324 y Fl(Z)1250 437 y Fi(!)1283 392 y Fo(dx)j(\025)g Fz(tr)1430 401 y Fg(C)1451 391 y Ff(2)1474 392 y Fp(\()p Fo(\045)p Fp(\()p Fo(\025)p Fp(\))p Fo(D)q Fp(\()p Fo(x;)8 b(\025)p Fp(\)\))p Fo(:)89 532 y Fz(By)16 b(form)o(ula)e(\(3.42\))j(w)o(e)f(\034nd)271 603 y Fl(Z)299 716 y Fi(!)332 671 y Fo(dx)e(\025)p Fp(\()p Fz(tr)484 679 y Fg(C)505 669 y Ff(2)528 671 y Fp(\()p Fo(\045)p Fp(\()p Fo(\025)p Fp(\))p Fo(D)q Fp(\()p Fo(x;)8 b(\025)p Fp(\)\))15 b(=)869 b Fz(\(5.18\))472 737 y Fl(Z)500 850 y Fi(!)533 805 y Fo(dx)14 b(\025)636 735 y Fl(\034)674 805 y Fo(\045)699 784 y Fi(t)714 805 y Fp(\()p Fo(\025)p Fp(\))788 735 y Fl(\022)849 774 y Fo( )883 756 y Fj(\000)912 774 y Fp(\()p Fo(x;)8 b(\025;)g(b)p Fp(\))846 834 y Fo( )880 816 y Fj(\000)909 834 y Fp(\()p Fo(x;)g(\025;)g(a)p Fp(\))1094 735 y Fl(\023)1139 805 y Fo(;)1161 735 y Fl(\022)1220 774 y Fo( )1254 756 y Fj(\000)1283 774 y Fp(\()p Fo(x;)g(\025;)g(b)p Fp(\))1218 834 y Fo( )1252 816 y Fj(\000)1281 834 y Fp(\()p Fo(x;)g(\025;)g(a)p Fp(\))1465 735 y Fl(\023\035)1547 805 y Fo(:)89 949 y Fz(Since)16 b Fo(l)q Fp(\()p Fo( )286 931 y Fj(\000)314 949 y Fp(\()p Fo(x;)8 b(\025;)g(\034)e Fp(\)\))13 b(=)h Fo(\025 )625 931 y Fj(\000)655 949 y Fp(\()p Fo(x;)8 b(\025;)g(\034)e Fp(\))16 b Fz(one)g(gets)258 1019 y Fl(Z)286 1132 y Fi(!)320 1087 y Fo(dx)e(\025)p Fp(\()p Fz(tr)472 1095 y Fg(C)493 1086 y Ff(2)515 1087 y Fp(\()p Fo(\045)p Fp(\()p Fo(\025)p Fp(\))p Fo(D)q Fp(\()p Fo(x;)8 b(\025)p Fp(\)\))15 b(=)882 b Fz(\(5.19\))460 1153 y Fl(Z)487 1266 y Fi(!)521 1221 y Fo(dx)596 1151 y Fl(\034)633 1221 y Fo(\045)658 1200 y Fi(t)673 1221 y Fp(\()p Fo(\025)p Fp(\))747 1151 y Fl(\022)808 1191 y Fo(l)q Fp(\()p Fo( )877 1172 y Fj(\000)905 1191 y Fp(\()p Fo(x;)8 b(\025;)g(b)p Fp(\)\))805 1251 y Fo(l)q Fp(\()p Fo( )874 1233 y Fj(\000)903 1251 y Fp(\()p Fo(x;)g(\025;)g(a)p Fp(\)\))1106 1151 y Fl(\023)1151 1221 y Fo(;)1173 1151 y Fl(\022)1233 1191 y Fo( )1267 1172 y Fj(\000)1296 1191 y Fp(\()p Fo(x;)g(\025;)g(b)p Fp(\))1230 1251 y Fo( )1264 1233 y Fj(\000)1293 1251 y Fp(\()p Fo(x;)g(\025;)g(a)p Fp(\))1478 1151 y Fl(\023)o(\035)1560 1221 y Fo(:)89 1361 y Fz(Hence)211 1454 y Fo(d)p 203 1476 V 203 1521 a(dt)250 1487 y(N)289 1495 y Fi(\045)p Fk(\()p Fi(t)p Fk(\))350 1487 y Fp(\()p Fo(!)r Fp(\))14 b(=)1264 b Fz(\(5.20\))340 1611 y Fm(\000)p Fo(i)404 1543 y Fl(Z)431 1656 y Fg(R)465 1611 y Fo(d\025)541 1543 y Fl(Z)569 1656 y Fi(!)602 1611 y Fo(dx)678 1540 y Fl(\034)715 1611 y Fo(\045)740 1590 y Fi(t)755 1611 y Fp(\()p Fo(\025)p Fp(\))829 1540 y Fl(\022)890 1580 y Fo(l)q Fp(\()p Fo( )959 1562 y Fj(\000)987 1580 y Fp(\()p Fo(x;)8 b(\025;)g(b)p Fp(\)\))887 1640 y Fo(l)q Fp(\()p Fo( )956 1622 y Fj(\000)985 1640 y Fp(\()p Fo(x;)g(\025;)g(a)p Fp(\)\))1188 1540 y Fl(\023)1233 1611 y Fo(;)1255 1540 y Fl(\022)1315 1580 y Fo( )1349 1562 y Fj(\000)1378 1580 y Fp(\()p Fo(x;)g(\025;)g(b)p Fp(\))1312 1640 y Fo( )1346 1622 y Fj(\000)1375 1640 y Fp(\()p Fo(x;)g(\025;)g(a)p Fp(\))1559 1540 y Fl(\023\035)1644 1611 y Fp(+)340 1748 y Fo(i)365 1680 y Fl(Z)392 1793 y Fg(R)427 1748 y Fo(d\025)503 1680 y Fl(Z)530 1793 y Fi(!)564 1748 y Fo(dx)639 1677 y Fl(\034)676 1748 y Fo(\045)701 1727 y Fi(t)716 1748 y Fp(\()p Fo(\025)p Fp(\))790 1677 y Fl(\022)851 1717 y Fo( )885 1699 y Fj(\000)914 1717 y Fp(\()p Fo(x;)g(\025;)g(b)p Fp(\))848 1777 y Fo( )882 1759 y Fj(\000)911 1777 y Fp(\()p Fo(x;)g(\025;)g(a)p Fp(\))1096 1677 y Fl(\023)1141 1748 y Fo(;)1163 1677 y Fl(\022)1222 1717 y Fo(l)q Fp(\()p Fo( )1291 1699 y Fj(\000)1320 1717 y Fp(\()p Fo(x;)g(\025;)g(b)p Fp(\)\))1220 1777 y Fo(l)q Fp(\()p Fo( )1289 1759 y Fj(\000)1317 1777 y Fp(\()p Fo(x;)g(\025;)g(a)p Fp(\)\))1521 1677 y Fl(\023)o(\035)89 1891 y Fz(Let)19 b Fo(!)h Fp(=)e([)p Fo(c;)8 b(d)p Fp(])18 b Fm(\022)f Fp([)p Fo(a;)8 b(b)p Fp(])p Fz(.)27 b(In)o(tegrating)19 b(b)o(y)f(parts)h(and)h(taking)e(in)o(to)h(accoun)o(t)f(form)o(ula)g (\(4.7\))89 1951 y(w)o(e)e(get)558 2003 y Fo(d)p 549 2025 V 549 2071 a(dt)597 2037 y(N)636 2045 y Fi(\045)p Fk(\()p Fi(t)p Fk(\))696 2037 y Fp(\()p Fo(!)r Fp(\))e(=)832 1969 y Fl(Z)860 2082 y Fg(R)894 2037 y Fo(d\025)23 b Fm(f)p Fo(j)1018 2044 y Fk(^)-21 b Fi(\045)1035 2037 y Fp(\()p Fo(c;)8 b(\025)p Fp(\))k Fm(\000)e Fo(j)1228 2044 y Fk(^)-21 b Fi(\045)1246 2037 y Fp(\()p Fo(d;)8 b(\025)p Fp(\))p Fm(g)g Fo(:)330 b Fz(\(5.21\))89 2155 y(The)16 b(total)h(curren)o(t)e Fo(j)498 2162 y Fk(^)-21 b Fi(\045)516 2155 y Fp(\()p Fo(x)p Fp(\))15 b Fz(at)i(the)f(p)q(oin)o (t)g Fo(x)e Fm(2)g Fp([)p Fo(a;)8 b(b)p Fp(])15 b Fz(is)h(de\034ned)f (b)o(y)733 2295 y Fo(j)756 2302 y Fk(^)-21 b Fi(\045)773 2295 y Fp(\()p Fo(x)p Fp(\))13 b(:=)918 2227 y Fl(Z)945 2340 y Fg(R)980 2295 y Fo(d\025)h(j)1070 2302 y Fk(^)-21 b Fi(\045)1088 2295 y Fp(\()p Fo(x;)8 b(\025)p Fp(\))p Fo(:)518 b Fz(\(5.22\))89 2432 y(This)17 b(yields)e(This)h(yields)713 2484 y Fo(d)p 704 2507 V 704 2552 a(dt)752 2518 y(N)791 2526 y Fi(\045)p Fk(\()p Fi(t)p Fk(\))851 2518 y Fp(\()p Fo(!)r Fp(\))e(=)g Fo(j)1010 2525 y Fk(^)-21 b Fi(\045)1027 2518 y Fp(\()p Fo(c)p Fp(\))11 b Fm(\000)g Fo(j)1170 2525 y Fk(^)-21 b Fi(\045)1188 2518 y Fp(\()p Fo(d)p Fp(\))485 b Fz(\(5.23\))89 2625 y(whic)o(h)13 b(sho)o(ws)i(that)g(the)e (c)o(hange)h(of)h(the)e(n)o(um)o(b)q(er)f(of)j(carriers)e(in)h(the)f (set)h Fp([)p Fo(c;)8 b(d)p Fp(])14 b Fz(is)f(equal)h(to)g(the)89 2685 y(di\033erence)k(b)q(et)o(w)o(een)g(the)g(incoming)g(curren)o(t)g Fo(j)994 2692 y Fk(^)-21 b Fi(\045)1011 2685 y Fp(\()p Fo(c)p Fp(\))19 b Fz(at)g(p)q(oin)o(t)g Fo(c)g Fz(and)g(the)g(outgoing) h(curren)o(t)89 2746 y Fo(j)112 2753 y Fk(^)-21 b Fi(\045)129 2746 y Fp(\()p Fo(d)p Fp(\))23 b Fz(at)f(p)q(oin)o(t)g Fo(d)g Fz(whic)o(h)f(v)o(ery)f(w)o(ell)h(corresp)q(onds)i(to)f(the)f (ph)o(ysical)g(in)o(tuition.)37 b(Since)20 b(b)o(y)89 2806 y(Prop)q(osition)f(4.1)g(the)f(curren)o(t)f(densit)o(y)g Fo(j)884 2813 y Fk(^)-21 b Fi(\045)901 2806 y Fp(\()p Fo(x;)8 b(\025)p Fp(\))19 b Fz(do)q(es)f(not)h(dep)q(end)f(on)h Fo(x)e Fm(2)g Fp([)p Fo(a;)8 b(b)p Fp(])16 b Fz(one)j(gets)p eop %%Page: 27 29 27 28 bop 89 23 a Fs(Density)18 b(and)g(current)h(of)f(a)g(dissip)l(a)l (tive)g(Schr)o(\366dinger)h(opera)l(tor)187 b Fz(27)89 147 y Fo(j)112 154 y Fk(^)-21 b Fi(\045)129 147 y Fp(\()p Fo(d)p Fp(\))16 b(=)e Fo(j)283 154 y Fk(^)-21 b Fi(\045)301 147 y Fp(\()p Fo(c)p Fp(\))16 b Fz(whic)o(h)g(again)i(v)o(eri\034es)d (\(5.14\).)24 b(The)17 b(relation)f(\(5.23\))i(is)e(the)h(in)o(tegral)f (form)g(of)89 208 y(the)g(so-called)g(con)o(tin)o(uit)o(y)e(equation)j (whic)o(h)e(has)i(the)f(di\033eren)o(tial)f(form)506 303 y Fo(@)p 497 325 47 2 v 497 371 a(@)s(t)548 337 y(u)579 344 y Fk(^)-21 b Fi(\045)p Fk(\()p Fi(t)p Fk(\))637 337 y Fp(\()p Fo(x)p Fp(\))10 b(+)781 303 y Fo(@)p 767 325 57 2 v 767 371 a(@)s(x)828 337 y(j)851 344 y Fk(^)-21 b Fi(\045)q Fk(\()p Fi(t)p Fk(\))909 337 y Fp(\()p Fo(x)p Fp(\))13 b(=)h(0)p Fo(;)57 b(t)14 b Fm(2)g Fn(R)p Fo(;)53 b(x)14 b Fm(2)g Fp(\012)p Fo(;)278 b Fz(\(5.24\))89 459 y(where)21 b Fo(u)266 467 y Fk(^)-21 b Fi(\045)p Fk(\()p Fi(t)p Fk(\))323 459 y Fp(\()p Fo(x)p Fp(\))20 b Fz(is)h(the)g(carrier) f(densit)o(y)g(at)h(time)e Fo(t)i Fm(2)h Fn(R)c Fz(and)j Fo(j)1334 467 y Fi(\045)p Fk(\()p Fi(t)p Fk(\))1395 459 y Fp(\()p Fo(x)p Fp(\))f Fz(is)h(curren)o(t)f(densit)o(y)89 519 y(at)h(time)e Fo(t)h Fz(giv)o(en)f(b)o(y)h(\(5.22\).)35 b(Since)20 b Fo(u)846 527 y Fk(^)-21 b Fi(\045)p Fk(\()p Fi(t)p Fk(\))903 519 y Fp(\()p Fo(x)p Fp(\))21 b(=)g Fo(u)1080 526 y Fk(^)-21 b Fi(\045)1097 519 y Fp(\()p Fo(x)p Fp(\))20 b Fz(is)h(indep)q(enden)o(t)e(of)i Fo(t)g Fz(and)g Fo(j)1735 527 y Fi(\045)p Fk(\()p Fi(t)p Fk(\))1795 519 y Fp(\()p Fo(x)p Fp(\))89 579 y Fz(is)g(indep)q(enden)o(t)f(of)h Fo(x)f Fz(the)h(con)o(tin)o(uit)o(y)e(equation)i(\(5.24\))g(ob)o (viously)f(holds)h(in)g(the)f(presen)o(t)89 640 y(situation.)148 723 y(4.)i(On)15 b(the)g(basis)h(of)f(the)g(considerations)h(ab)q(o)o (v)o(e)f(w)o(e)g(are)g(able)g(to)h(giv)o(e)e(a)i(mathematicall)o(y)89 784 y(rigourous)23 b(foundation)f(of)g(dissipativ)o(e)f(Sc)o(hr\366)q (dinger-P)o(oisson)h(systems.)36 b(T)l(o)22 b(this)g(end)f(w)o(e)89 844 y(consider)c(di\033eren)o(t)e(sp)q(ecies)h Fm(\006)h Fz(of)g(particles)f(\(holes)g(and)h(electrons\))f(and)h(assume)f(that)h (that)89 904 y(for)k(these)g(sp)q(ecies)f(the)h(e\033ectiv)o(e)e (masses)h Fo(m)954 911 y Fj(\006)983 904 y Fz(,)i(external)d(p)q(oten)o (tials)i Fo(V)1480 883 y Fj(\006)1469 916 y Fk(0)1531 904 y Fz(and)g(co)q(e\036cien)o(ts)89 964 y Fo(\024)117 946 y Fj(\006)117 977 y Fi(a)147 964 y Fo(;)8 b(\024)197 944 y Fj(\006)197 978 y Fi(b)240 964 y Fm(2)14 b Fn(C)320 971 y Fk(+)364 964 y Fz(are)d(giv)o(en.)18 b(F)l(or)11 b(eac)o(h)g(sp)q(ecies)f(this)h(leads)g(to)g(di\033eren)o(t)f (dissipativ)o(e)g(Sc)o(hr\366)q(dinger)89 1024 y(op)q(erators)17 b Fo(H)348 1006 y Fj(\006)378 1024 y Fp(\()p Fo(V)425 1031 y Fj(\006)455 1024 y Fp(\))f Fz(de\034ned)f(in)g(accordance)h (with)f(\(1.12\),)h(\(2.1\))g(and)g(\(2.2\).)22 b(The)15 b(p)q(oten)o(tial)89 1085 y Fo(V)29 b Fz(en)o(tering)16 b(in)o(to)h(the)g(de\034nition)g(of)h(the)f(Sc)o(hr\366)q(dinger)g(op)q (erators)i(is)e(di\033eren)o(t)f(for)i(di\033eren)o(t)89 1145 y(sp)q(ecies.)j(It)16 b(tak)o(es)g(the)g(form)786 1205 y Fo(V)814 1212 y Fj(\006)858 1205 y Fp(:=)d Fo(V)962 1184 y Fj(\006)951 1217 y Fk(0)1003 1205 y Fm(\006)e Fo(')p Fp(\()p Fo(u)p Fp(\))p Fo(:)571 b Fz(\(5.25\))89 1291 y(where)17 b(the)h(electrostatic)e(p)q(oten)o(tial)i Fo(')f Fz(is)h(a)g(solution)g(of)g(the)f(P)o(oisson)i(equation)e (\(1.1\))h(with)89 1351 y(b)q(oundary)f(condition)f(\(1.3\).)21 b(The)16 b(carrier)g(densities)f Fo(u)1138 1333 y Fj(\006)1183 1351 y Fz(en)o(tering)g(in)o(to)h(P)o(oisson's)g(equation)89 1411 y(are)f(obtained)f(from)f(the)h(dissipativ)o(e)g(Sc)o(hr\366)q (dinger)g(op)q(erators)i Fo(H)1324 1393 y Fj(\006)1353 1411 y Fp(\()p Fo(V)1400 1418 y Fj(\006)1430 1411 y Fp(\))f Fz(in)f(accordance)g(with)89 1471 y(Section)k(3.)27 b(T)l(o)19 b(this)f(end)g(w)o(e)g(assume)f(that)i(the)f(families)e(of)i(matrices)e Fm(f)p Fo(\045)1514 1478 y Fj(\006)1544 1471 y Fp(\()p Fo(\025)p Fp(\))p Fm(g)1635 1478 y Fi(\025)p Fj(2)p Fg(R)1705 1471 y Fz(,)j(whic)o(h)89 1532 y(ob)q(ey)570 1592 y Fo(C)608 1599 y Fk(^)-21 b Fi(\045)623 1603 y Fd(\006)665 1592 y Fp(:=)13 b(sup)733 1633 y Fi(\025)p Fj(2)p Fg(R)812 1548 y Fm(p)p 853 1548 133 2 v 853 1592 a Fo(\025)881 1577 y Fk(2)912 1592 y Fp(+)e(1)q Fm(k)p Fo(\045)1036 1599 y Fj(\006)1065 1592 y Fp(\()p Fo(\025)p Fp(\))p Fm(k)1156 1600 y Fj(B)q Fk(\()p Fg(C)1215 1591 y Ff(2)1236 1600 y Fk(\))1265 1592 y Fo(<)j Fm(1)p Fo(;)355 b Fz(\(5.26\))89 1710 y(are)22 b(giv)o(en)f(and)h(w)o(e)g(de\034ne)f(the)h(carrier)f (densities)g Fo(u)1115 1692 y Fj(\006)1144 1710 y Fp(\()p Fo(x)p Fp(\))i(:=)g Fo(u)1336 1690 y Fj(\006)1339 1724 y Fk(^)-21 b Fi(\045)1354 1728 y Fd(\006)1382 1710 y Fp(\()p Fo(V)1429 1717 y Fj(\006)1459 1710 y Fp(\)\()p Fo(x)p Fp(\))21 b Fz(in)h(accordance)89 1771 y(with)c(Section)f(3,)h (i.e.,)e(if)i Fo(D)612 1753 y Fj(\006)642 1771 y Fp(\()p Fo(V)689 1778 y Fj(\006)719 1771 y Fp(\)\()p Fo(x)p Fp(\))f Fz(are)h(the)g(carrier)f(densit)o(y)g(observ)m(ables)h(at)g Fo(x)e Fm(2)h Fp(\012)h Fz(and)89 1831 y(at)f(energy)f Fo(\025)e Fm(2)g Fn(R)p Fz(,)f(then)j(the)g(carrier)f(densities)g(are)i (computed)e(b)o(y)513 1959 y Fo(u)541 1939 y Fj(\006)544 1973 y Fk(^)-21 b Fi(\045)559 1977 y Fd(\006)587 1959 y Fp(\()p Fo(V)634 1966 y Fj(\006)664 1959 y Fp(\)\()p Fo(x)p Fp(\))13 b(=)814 1892 y Fl(Z)841 2004 y Fg(R)876 1959 y Fo(d\025)h(u)971 1939 y Fj(\006)974 1973 y Fk(^)-21 b Fi(\045)989 1977 y Fd(\006)1017 1959 y Fp(\()p Fo(V)1064 1966 y Fj(\006)1094 1959 y Fp(\)\()p Fo(x;)8 b(\025)p Fp(\))p Fo(;)57 b(x)13 b Fm(2)i Fp(\012)p Fo(;)298 b Fz(\(5.27\))89 2090 y(where)273 2197 y Fo(u)301 2176 y Fj(\006)304 2211 y Fk(^)-21 b Fi(\045)319 2215 y Fd(\006)347 2197 y Fp(\()p Fo(V)394 2204 y Fj(\006)424 2197 y Fp(\)\()p Fo(x;)8 b(\025)p Fp(\))14 b(=)g Fz(tr)663 2205 y Fg(C)684 2196 y Ff(2)706 2197 y Fp(\()p Fo(\045)750 2204 y Fj(\006)780 2197 y Fp(\()p Fo(\025)p Fp(\))p Fo(D)887 2176 y Fj(\006)917 2197 y Fp(\()p Fo(V)964 2204 y Fj(\006)994 2197 y Fp(\)\()p Fo(x;)8 b(\025)p Fp(\)\))p Fo(;)57 b(x)13 b Fm(2)h Fp(\012)p Fo(;)57 b(\025)15 b Fm(2)f Fn(R)p Fo(:)180 b Fz(\(5.28\))89 2310 y(Moreo)o(v)o(er,)15 b(if)g(tr)399 2318 y Fg(C)420 2309 y Ff(2)443 2310 y Fp(\()p Fo(\045)487 2317 y Fj(\006)516 2310 y Fp(\()p Fm(\001)p Fp(\)\))f Fm(2)g Fo(L)681 2292 y Fk(1)701 2310 y Fp(\()p Fn(R)p Fp(\))f Fz(is)j(v)m(alid,)f(then)i (the)f(curren)o(t)f(densities)g Fo(j)1557 2290 y Fj(\006)1557 2324 y Fk(^)-21 b Fi(\045)1572 2328 y Fd(\006)1601 2310 y Fz(,)730 2451 y Fo(j)753 2430 y Fj(\006)753 2465 y Fk(^)g Fi(\045)768 2469 y Fd(\006)810 2451 y Fp(=)862 2383 y Fl(Z)890 2496 y Fg(R)924 2451 y Fo(d\025)15 b(j)1015 2430 y Fj(\006)1015 2465 y Fk(^)-21 b Fi(\045)1030 2469 y Fd(\006)1058 2451 y Fp(\()p Fo(V)1105 2458 y Fj(\006)1135 2451 y Fp(\)\()p Fo(\025)p Fp(\))516 b Fz(\(5.29\))89 2581 y(are)19 b(also)g(w)o(ell-de\034ned)e(and)i(\034nite,)f(cf.)28 b(Section)18 b(4.)28 b(The)19 b(so)g(describ)q(ed)f(system)f(is)h (called)f(a)89 2641 y(dissipativ)o(e)g(Sc)o(hr\366)q(dinger-P)o(oisson) h(system.)25 b(W)l(e)17 b(note)h(that)h(the)e(total)h(n)o(um)o(b)q(er)e (of)i(carriers)89 2701 y Fo(N)133 2683 y Fj(\006)163 2701 y Fp(\()p Fo(V)210 2708 y Fj(\006)240 2701 y Fp(\))e Fz(is)g(giv)o(en)f(b)o(y)674 2786 y Fo(N)718 2765 y Fj(\006)748 2786 y Fp(\()p Fo(V)795 2793 y Fj(\006)825 2786 y Fp(\))f(=)910 2718 y Fl(Z)937 2831 y Fk(\012)973 2786 y Fo(dx)g(u)1068 2765 y Fj(\006)1071 2800 y Fk(^)-21 b Fi(\045)1086 2804 y Fd(\006)1114 2786 y Fp(\()p Fo(V)1161 2793 y Fj(\006)1191 2786 y Fp(\)\()p Fo(x)p Fp(\))460 b Fz(\(5.30\))p eop %%Page: 28 30 28 29 bop 89 23 a Fz(28)692 b Fs(H.-Ch.)25 b(Kaiser,)18 b(H.)h(Neidhardt,)g(J.)g(Rehber)o(g)89 147 y Fz(and)e(is)f(not)h (\034xed.)148 231 y(In)e(a)h(forthcoming)e(pap)q(er)h([13)q(])f(w)o(e)h (sho)o(w)h(that)f(under)g(suitable)g(conditions)g(on)h Fo(\017)p Fp(\()p Fm(\001)p Fp(\))p Fz(,)e Fo(C)t Fp(\()p Fm(\001)p Fp(\))p Fz(,)89 291 y Fo(m)132 298 y Fj(\006)161 291 y Fp(\()p Fm(\001)p Fp(\))p Fz(,)23 b Fo(V)289 271 y Fj(\006)278 304 y Fk(0)319 291 y Fp(\()p Fm(\001)p Fp(\))p Fz(,)f Fo(\024)435 271 y Fj(\006)435 305 y Fi(b)465 291 y Fz(,)h Fo(\024)530 273 y Fj(\006)530 304 y Fi(a)581 291 y Fz(and)g Fo(\045)707 298 y Fj(\006)758 291 y Fz(this)f (dissipativ)o(e)f(Sc)o(hr\366)q(dinger-P)o(oisson)i(system)d(alw)o(a)o (ys)89 352 y(admits)15 b(a)i(self-consisten)o(t)e(solution.)89 556 y Fx(A)n(c)n(kno)n(wledgemen)n(t)89 692 y Fz(The)23 b(authors)g(thank)g(M.)f(Baro)h(for)f(carefully)f(reading)i(the)f(man)o (uscript.)39 b(The)22 b(\034nancial)89 752 y(supp)q(ort)c(of)e(DF)o(G)g (is)g(grateful)h(ac)o(kno)o(wledged.)89 957 y Fx(References)114 1097 y Fz([1])23 b(A)o(dam)o(y)o(an,)f(V.M.;)h(Neidhardt,)g(H.:)33 b(On)22 b(the)g(absolutely)g(con)o(tin)o(uous)h(subspace)g(for)189 1157 y(non-selfadjoin)o(t)17 b(op)q(erators.)g Fc(Math.)g(Nachr.)f Fq(210)p Fz(,)g(5-42)i(\(2000\).)114 1273 y([2])23 b(Baumg\344rtel,)g (H.;)h(W)l(ollen)o(b)q(erg,)f(M.:)34 b Fc(Mathematic)n(al)23 b(sc)n(attering)h(the)n(ory)p Fz(,)f(Ak)m(ademie)189 1333 y(V)l(erlag,)16 b(Berlin)e(1983.)114 1449 y([3])23 b(Birman,)i(M.S.;)j(Solom)o(y)o(ak,)c(M.Z.:)38 b Fc(Sp)n(e)n(ctr)n(al) 25 b(the)n(ory)f(of)h(self-adjoint)i(op)n(er)n(ators)c(in)189 1509 y(Hilb)n(ert)18 b(sp)n(ac)n(e)p Fz(,)e(D.)g(Reidel)f(Publishing)h (Co.,)g(Dordrec)o(h)o(t,)f(1987.)114 1625 y([4])23 b(Ben)29 b(Ab)q(dallah,)j(N.;)j(Degond,)e(P)l(.,)e(Mark)o(o)o(wic)o(h,)g(P)l (.A.:)46 b(On)30 b(a)f(one-dimensional)189 1685 y(Sc)o(hr\366)q (dinger-P)o(oisson)16 b(scattering)e(mo)q(del.)f Fc(Z.)i(A)o(ngew.)j (Math.)d(Phys.)f Fq(48)h Fz(\(1997\),)g(No.1,)189 1745 y(135-155.)114 1861 y([5])23 b(Ben)g(Ab)q(dallah,)h(N.:)34 b(A)23 b(h)o(ybrid)g(kinetic-quan)o(tum)d(mo)q(del)i(for)h(stationary)h (electron)189 1921 y(transp)q(ort.)18 b Fc(J.)f(Stat.)h(Phys.)d Fq(90)i Fz(\(1998\),)g(No.3-4,)f(627-662.)114 2037 y([6])23 b(Ben)15 b(Ab)q(dallah,)f(N.;)g(Ke\034,)h(J.:)20 b(Limite)12 b(semi-classique)h(du)i(probleme)e(de)h(Sc)o(hro)q(edinger)189 2097 y(a)o(v)o(ec)h(masse)h(v)m(ariable.)f Fc(C.)i(R.)g(A)n(c)n(ad.)f (Sci.,)i(Paris,)f(Ser.)g(I,)g(Math.)f Fq(331)p Fz(,)g(No.2,)f(165-170) 189 2158 y(\(2000\).)114 2273 y([7])23 b(Buslaev,)15 b(V.;)f(F)l(omin,)f(V.:)20 b(An)15 b(in)o(v)o(erse)e(scattering)i (problem)f(for)h(the)g(one-dimensional)189 2334 y(Sc)o(hr\366)q(dinger) 20 b(equation)g(on)g(the)g(en)o(tire)e(axis.)i Fc(V)l(estnik)i(L)n (eningr)n(ad.)e(Univ.)h Fq(17)p Fz(,)g(No.)e(1,)189 2394 y(56\02564)g(\(1962\).)114 2509 y([8])k(F)l(oias,)13 b(C.;)g(Sz.-Nagy)l(,)f(B.:)18 b Fc(Harmonic)13 b(analysis)h(of)f(op)n (er)n(ators)f(on)i(Hilb)n(ert)g(sp)n(ac)n(e)p Fz(,)e(North-)189 2570 y(Holland)k(Publishing)g(Compan)o(y)l(,)f(Amsterdam-London,)g (1970.)114 2685 y([9])23 b(Kaiser,)16 b(H.-Ch.;)g(Reh)o(b)q(erg,)g(J.:) 21 b(Ab)q(out)c(a)g(one-dimensional)e(stationary)j(Sc)o(hr\366)q (dinger-)189 2746 y(P)o(oisson)i(system)e(with)h(K)o(ohn-Sham)g(p)q (oten)o(tial.)f Fc(Z.)i(A)o(ngew.)i(Math.)d(Phys.)g Fq(50)p Fz(,)g(No.3,)189 2806 y(423-458)g(\(1999\).)p eop %%Page: 29 31 29 30 bop 89 23 a Fs(Density)18 b(and)g(current)h(of)f(a)g(dissip)l(a)l (tive)g(Schr)o(\366dinger)h(opera)l(tor)187 b Fz(29)89 147 y([10])24 b(Kaiser,)d(H.-Ch.;)g(Reh)o(b)q(erg,)f(J.:)29 b(Ab)q(out)21 b(a)f(stationary)h(Sc)o(hr\366)q(dinger-P)o(oisson)g (system)189 208 y(with)i(K)o(ohn-Sham)f(p)q(oten)o(tial)h(in)f(a)i(b)q (ounded)f(t)o(w)o(o-)g(or)g(three-dimensional)e(domain.)189 268 y Fc(Nonline)n(ar)d(A)o(nal.,)g(The)n(ory)f(Metho)n(ds)g(Appl.)f Fq(41A)p Fz(,)g(No.)g(1-2,)h(33-72)h(\(2000\).)89 369 y([11])24 b(Kaiser,)43 b(H-Ch.;)49 b(Neidhardt,)43 b(H.;)48 b(Reh)o(b)q(erg,)43 b(J.:)65 b Fc(Macr)n(osc)n(opic)36 b(curr)n(ent)i(in-)189 430 y(duc)n(e)n(d)h(b)n(oundary)g(c)n(onditions) g(for)g(Schr)n(\366)n(dinger-typ)n(e)g(p)n(er)n(ators)p Fz(.)e(WIAS-preprin)o(t)189 490 y Fq(No.)g(650)p Fz(,)g(Berlin)31 b(2001)j(\(see)e(also)i(Mathematical)c(Ph)o(ysics)i(Preprin)o(t)g(Arc)o (hiv)o(e)189 550 y(h)o(ttp://146.6.139.68/mp_arc/)18 b(No.)e(01-182\).)89 652 y([12])24 b(Kaiser,)38 b(H-Ch.;)k(Neidhardt,) 37 b(H.;)k(Reh)o(b)q(erg,)d(J.:)56 b Fc(On)35 b(1-dimensional)f(dissip) n(a-)189 712 y(tive)23 b(Schr)n(\366)n(dinger-typ)n(e)f(op)n(er)n (ators,)g(their)g(dilations)g(and)h(eigenfunction)i(exp)n(ansions)p Fz(.)189 772 y(WIAS-preprin)o(t)15 b Fq(No.)j(664)p Fz(,)d(Berlin)g (2001)i(\(see)f(also)g(Mathematical)e(Ph)o(ysics)i(Preprin)o(t)189 832 y(Arc)o(hiv)o(e)e(h)o(ttp://146.6.139.68/mp_arc/)k(No.)e(01-368\).) 89 934 y([13])24 b(Kaiser,)15 b(H-Ch.;)g(Neidhardt,)f(H.;)h(Reh)o(b)q (erg,)f(J.:)21 b(On)15 b(dissipativ)o(e)f(Sc)o(hr\366)q(dinger-P)o (oisson)189 994 y(systems.)h(WIAS-preprin)o(t)f Fq(No.)k(719)p Fz(,)e(Berlin)e(2002)k(\(see)d(also)i(Mathematical)c(Ph)o(ysics)189 1054 y(Preprin)o(t)j(Arc)o(hiv)o(e)e(h)o(ttp://146.6.139.68/mp_arc/)j (No.)f(02-93\).)89 1156 y([14])24 b(Kirkner,)17 b(D.;)h(Len)o(t,)f(C.:) 24 b(The)17 b(quan)o(tum)g(transmitting)f(b)q(oundary)j(metho)q(d.)d Fc(J.)i(Appl.)189 1216 y(Phys.)e Fq(67)g Fz(\(1990\),)i(6353-6359.)89 1318 y([15])24 b(Landau,)k(L.A.,)d(Lifsc)o(hitz,)f(E.M.:)37 b Fc(Quantenme)n(chanik)p Fz(,)28 b(Ak)m(ademie-V)l(erl)o(ag,)c (Berlin,)189 1378 y(1971.)89 1480 y([16])g(Lax,)13 b(P)l(.D.;)f (Phillips,)f(R.S.:)18 b Fc(Sc)n(attering)d(the)n(ory)p Fz(,)c(A)o(cademic)d(Press,)13 b(New)e(Y)l(ork-London,)189 1540 y(1967.)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF