This is a multi-part message in MIME format. ---------------0204120957420 Content-Type: text/plain; name="02-185.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="02-185.comments" 2000 Mathematics Subject Classification: 47B44 47E05 47A20 47A55 ---------------0204120957420 Content-Type: text/plain; name="02-185.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="02-185.keywords" open quantum system, dissipative Schroedinger operator, delta perturbation, dilation, characteristic function, generalized eigenfunctions, carrier and current densities, density matrix ---------------0204120957420 Content-Type: application/postscript; name="737.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="737.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.78 Copyright 1998 Radical Eye Software (www.radicaleye.com) %%Title: rec-gen-preprint2.dvi %%Pages: 29 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments 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(generalized)i(eigenfunctions)f(of)g(the)f(dilation)i(is)f(giv)n(en.)34 b(The)24 b(quan)n(tities)h(of)316 1982 y(carrier)d(and)f(curren)n(t)f (densities)i(are)f(rigorously)i(de\034ned.)32 b(F)-6 b(urthermore)19 b(w)n(e)j(will)g(sho)n(w)g(that)e(the)h(curren)n(t)316 2073 y(is)26 b(not)f(constan)n(t)g(and)f(that)h(the)f(v)l(ariation)i (of)g(the)e(curren)n(t)h(dep)r(end)e(essen)n(tially)k(on)d(the)h(c)n (hosen)g(densit)n(y)316 2165 y(matrix)g(and)h(the)g(imaginary)g(parts)h (of)g(the)e(delta)i(p)r(oten)n(tials,)g(i.e.)37 b FE(Im)o(\()p FD(\024)2497 2173 y FB(j)2530 2165 y FE(\))p FL(.)f(This)27 b(corresp)r(ondence)f(can)316 2256 y(b)r(e)g(used)f(to)h(mo)r(del)f(a)h (recom)n(bination-generation)g(rate)g(in)g(the)f(op)r(en)h(quan)n(tum)d (system.)109 2453 y FA(2000)31 b(Mathematics)g(Subje)l(ct)e (Classi\034c)l(ation:)39 b FO(47B44,)26 b(47E05,)g(47A20,)g(47A55)109 2599 y FA(Keywor)l(ds:)44 b FO(op)r(en)31 b(quan)n(tum)g(system,)g (dissipativ)n(e)f(Sc)n(hr\366)r(dinger)f(op)r(erator,)h(delta)h(p)r (erturbation,)f(recom)n(bi-)109 2699 y(nation,)24 b(generation,)e (dilation,)i(c)n(haracteristic)e(function,)i(generalized)e (eigenfunctions,)i(carrier)e(and)h(curren)n(t)109 2799 y(densities,)k(densit)n(y)h(matrix)109 3120 y Fz(Con)l(ten)l(ts)109 3350 y Fy(1)76 b(In)m(tro)s(duction)2845 b(2)109 3580 y(2)76 b(Dissipativ)m(e)31 b(Sc)m(hr\366)s(dinger)h(op)s(erators)1957 b(3)109 3810 y(3)76 b(The)32 b(c)m(haracteristic)i(function)2244 b(5)109 4040 y(4)76 b(Dilations)2995 b(7)109 4270 y(5)76 b(Eigenfunction)31 b(expansion)2304 b(12)109 4500 y(6)76 b(Carrier)33 b(densit)m(y)2689 b(14)109 4730 y(7)76 b(Curren)m(t)33 b(densit)m(y)2661 b(17)109 4959 y(8)76 b(Curren)m(t)33 b(v)-5 b(ariations)2554 b(21)109 5189 y(9)76 b(Remarks)2954 b(24)p eop %%Page: 2 4 2 3 bop 109 254 a FO(2)2519 b Fx(M.)31 b(Bar)n(o,)f(H.)h(Neidhardt)109 503 y Fz(1)131 b(In)l(tro)t(duction)109 732 y FO(T)-7 b(o)24 b(em)n(b)r(ed)i(a)f(quan)n(tum)g(mec)n(hanically)f(describ)r(ed) g(structure)h(in)n(to)g(a)g(macroscopic)e(\035o)n(w,)i(one)f(has)h(to)g (replace)109 831 y(the)36 b(selfadjoin)n(t)h(b)r(oundary)e(conditions)h (b)n(y)g(non-selfadjoin)n(t)g(ones)f([KNR01a)o(,)h(F)-7 b(re90)o(,)37 b(BA98)o(,)f(BADM97].)109 931 y(This)28 b(leads)g(to)g(so-called)f(op)r(en)i(quan)n(tum)f(systems.)39 b(One-dimensional)27 b(Sc)n(hr\366)r(dinger-t)n(yp)r(e)g(op)r(erators)g (with)109 1031 y(dissipativ)n(e)j(b)r(oundary)g(conditions)g(ha)n(v)n (e)g(b)r(een)h(in)n(tensely)g(studied)g(in)g([KNR01b)o(].)47 b(W)-7 b(e)32 b(extend)f(the)g(mo)r(del)109 1130 y(used)f(there)g(b)n (y)g(adding)g(dissipativ)n(e)f(delta)i(p)r(erturbations,)f(i.e.)45 b(w)n(e)30 b(consider)f(an)h(op)r(erator)f(formally)g(giv)n(en)109 1230 y(b)n(y)1471 1368 y Fw(H)h Fv(=)23 b Fu(\000)1733 1312 y Fv(1)p 1733 1349 42 4 v 1733 1425 a(2)1818 1312 y Fw(d)p 1794 1349 91 4 v 1794 1425 a(dx)1920 1312 y Fv(1)p 1904 1349 73 4 v 1904 1425 a Fw(m)2021 1312 y(d)p 1997 1349 91 4 v 1997 1425 a(dx)2116 1368 y Fv(+)2212 1347 y(~)2199 1368 y Fw(V)c(;)1192 b FO(\(1.1\))109 1537 y(where)1357 1677 y Fv(~)1344 1698 y Fw(V)19 b Fv(\()p Fu(\001)p Fv(\))24 b(=)e Fw(V)d Fv(\()p Fu(\001)p Fv(\))g Fu(\000)1896 1594 y Ft(N)1865 1619 y Fs(X)1868 1796 y Ft(j)s Fr(=1)1999 1698 y Fw(\024)2047 1710 y Ft(j)2082 1698 y Fw(\016)s Fv(\()p Fu(\001)g(\000)f Fw(x)2326 1710 y Ft(j)2361 1698 y Fv(\))p Fw(;)1065 b FO(\(1.2\))109 1933 y Fw(x)156 1945 y Ft(j)220 1933 y Fu(2)29 b Fv(\()p Fw(a;)14 b(b)p Fv(\))p FO(,)32 b Fw(x)587 1945 y Ft(j)651 1933 y Fu(6)p Fv(=)c Fw(x)791 1945 y Ft(i)850 1933 y FO(for)j Fw(i)d Fu(6)p Fv(=)g Fw(j)5 b FO(,)32 b(and)f Fw(\024)1438 1945 y Ft(j)1502 1933 y Fu(2)e Fq(C)1640 1945 y Fr(+)1729 1933 y Fv(:=)g Fu(f)p Fw(z)i Fu(2)e Fq(C)g Fu(j)14 b Fv(Im)6 b(\()p Fw(z)t Fv(\))29 b Fw(>)f Fv(0)p Fu(g)p FO(.)46 b(The)31 b(p)r(oten)n(tial)g Fw(V)50 b FO(is)31 b(assumed)109 2032 y(to)37 b(b)r(e)h(a)g(real-v)-5 b(alued)36 b Fp(L)900 2002 y Fr(2)937 2032 y Fv(\([)p Fw(a;)14 b(b)p Fv(]\))p FO(-function)38 b(and)g(the)g(e\033ectiv)n(e)f (mass)g Fw(m)j(>)g Fv(0)d FO(satis\034es)g Fw(m;)3141 2000 y Fr(1)p 3129 2014 59 4 v 3129 2061 a Ft(m)3237 2032 y Fu(2)k Fp(L)3378 2002 y Fo(1)3448 2032 y Fv(\()p Fw(a;)14 b(b)p Fv(\))p FO(.)109 2132 y(P)n(oin)n(t)32 b(in)n(teractions)g(of)h(this)g(kind)g(are)f(extensiv)n(ely)g(studied)h (in)g(the)g(literature)f(for)h(self-adjoin)n(t)f(b)r(oundary)109 2232 y(conditions)27 b(and)i(real)e(coupling)h(constan)n(ts)f Fw(\024)1580 2244 y Ft(j)1615 2232 y FO(,)h(cf.)40 b([A)n(GHH88)o(,)29 b(AK00)o(])f(and)g(references)f(therein.)39 b(Since)29 b(the)109 2331 y(b)r(oundary)36 b(conditions)g(are)f(not)i(self-adjoin) n(t)f(and)h(the)g(coupling)f(constan)n(t)g(are)f(complex)i(the)g (expression)109 2431 y(\(1.1\)-\(1.2\))32 b(generates)h(a)g(maximal)g (dissipativ)n(e)g(op)r(erator,)h(see)f(b)r(elo)n(w.)55 b(Suc)n(h)34 b(op)r(erators)e(naturally)h(arises)109 2530 y(if)38 b(one)g(is)g(in)n(terested)g(in)h(mathematical)e(mo)r (dels)h(for)g(semiconductor)f(devices)h(with)h(recom)n(bination)d(and) 109 2630 y(generation)25 b(pro)r(cesses)g(of)i(carriers)d(whic)n(h)j (are)f(em)n(b)r(edded)h(in)g(a)f(macrostructure.)35 b(This)26 b(case)g(is)h(not)g(treated)109 2730 y(in)32 b([A)n(GHH88])h(and)f ([AK00)o(].)52 b(T)-7 b(o)32 b(analyze)f(suc)n(h)h(op)r(erators)e(it)j (is)g(fruitful)g(to)g(use)f(the)h(dilation)f(theory)g(for)109 2829 y(maximal)26 b(dissipativ)n(e)g(op)r(erators)f([FN70,)i(D80)o(].) 37 b(T)-7 b(o)27 b(Sc)n(hr\366)r(dinger)e(op)r(erators)g(this)j (approac)n(h)d(w)n(as)h(sp)r(eci\034ed)109 2929 y(in)e([P77)o(,)g(P76,) g(P84)o(,)g(P96,)g(P99)o(],)h(see)f(also)e([A87,)i(A88)o(,)g(A89)o(,)g (A90,)g(A93)o(,)g(A97)o(].)36 b(In)24 b([KNR01c)o(])g(this)g(approac)n (h)109 3029 y(w)n(as)g(used)h(to)h(de\034ne)f(quan)n(tities)g(suc)n(h)g (as)g(steady)g(states,)h(carrier)d(and)i(curren)n(t)g(densit)n(y)g(for) g(devices)g(without)109 3128 y(recom)n(bination)c(and)h(generation.)34 b(In)23 b(the)g(presen)n(t)f(article)g(w)n(e)g(mo)r(di\034ed)h(this)g (approac)n(h)e(to)h(\034t)h(our)f(situation,)109 3228 y(i.e.)37 b(where)27 b(recom)n(bination)f(and)h(generation)f(pro)r (cesses)g(are)h(included.)109 3375 y(In)h(an)h(one-dimensional)e (drift-di\033usion)i(mo)r(del)g(\(cf.)40 b([S84)o(,)29 b(M86)o(,)g(MRS90)o(,)g(J01)o(]\))g(the)g(macroscopic)d(\035o)n(w)i(is) 109 3474 y(mo)r(deled)f(b)n(y)h(the)g(con)n(tin)n(uit)n(y)f(equations) 1225 3617 y Ft(@)p 1212 3631 65 4 v 1212 3679 a(@)t(t)1286 3650 y Fw(n)p Fv(\()p Fw(x;)14 b(t)p Fv(\))20 b Fu(\000)1645 3617 y Ft(@)p 1627 3631 77 4 v 1627 3679 a(@)t(x)1713 3650 y Fw(J)1759 3662 y Ft(n)1804 3650 y Fv(\()p Fw(x;)14 b(t)p Fv(\))84 b(=)f Fu(\000)p Fw(R)q Fv(\()p Fw(n;)14 b(p)p Fv(\))p Fw(;)1225 3768 y Ft(@)p 1212 3782 65 4 v 1212 3829 a(@)t(t)1286 3800 y Fw(p)p Fv(\()p Fw(x;)g(t)p Fv(\))19 b(+)1637 3768 y Ft(@)p 1618 3782 77 4 v 1618 3829 a(@)t(x)1705 3800 y Fw(J)1751 3812 y Ft(p)1790 3800 y Fv(\()p Fw(x;)14 b(t)p Fv(\))98 b(=)83 b Fu(\000)p Fw(R)q Fv(\()p Fw(n;)14 b(p)p Fv(\))p Fw(;)3481 3724 y FO(\(1.3\))109 3979 y(where)32 b Fw(n;)14 b(p)32 b FO(denote)h(the)g(electron)f(and)h(hole)f(densities,)j Fw(J)2016 3991 y Ft(n)2061 3979 y Fw(;)14 b(J)2144 3991 y Ft(p)2215 3979 y FO(the)33 b(corresp)r(onding)e(curren)n(ts)h(and)g Fw(R)q Fv(\()p Fw(n;)14 b(p)p Fv(\))109 4079 y FO(the)28 b(recom)n(bination-generation)c(rate.)38 b(The)28 b(stationary)e(con)n (tin)n(uit)n(y)h(equation)g(is)h(giv)n(en)f(b)n(y)h(equation)f (\(1.3\),)109 4178 y(where)372 4146 y Ft(@)p 359 4160 65 4 v 359 4207 a(@)t(t)433 4178 y Fw(n)p Fv(\()p Fw(x;)14 b(t)p Fv(\))25 b(=)797 4146 y Ft(@)p 784 4160 V 784 4207 a(@)t(t)858 4178 y Fw(p)p Fv(\()p Fw(x;)14 b(t)p Fv(\))25 b(=)e(0)28 b FO(and)f Fw(J)1468 4190 y Ft(k)1509 4178 y Fv(\()p Fw(x;)14 b(t)p Fv(\))25 b(=)e Fw(J)1846 4190 y Ft(k)1887 4178 y Fv(\()p Fw(x)p Fv(\))p FO(,)30 b Fw(k)c Fv(=)e Fw(n;)14 b(p)p FO(.)37 b(Replacing)28 b(carrier)e(densites)i Fw(n;)14 b(p)27 b FO(and)109 4278 y(curren)n(t)h(densities)h Fw(J)784 4290 y Ft(n)830 4278 y Fw(;)14 b(J)913 4290 y Ft(p)980 4278 y FO(b)n(y)29 b(quan)n(tum)g(mec)n(hanical)g (expressions)e(lik)n(e)i([KNR01c)o(])g(w)n(e)g(are)g(able)g(to)g (include)109 4378 y(recom)n(bination-generation)21 b(e\033ects)k(in)n (to)f(the)h(op)r(en)g(quan)n(tum)f(system)h(determined)g(b)n(y)f (\(1.1\))g(and)h(\(1.2\))f(and)109 4477 y(to)37 b(preserv)n(e)e(the)j (form)f(of)g(the)h(con)n(tin)n(uit)n(y)e(equations)h(\(1.3\))g(for)f (the)i(quan)n(tum)f(mec)n(hanically)f(describ)r(ed)109 4577 y(structure.)76 b(Ho)n(w)n(ev)n(er,)43 b(this)e(implies)g(that)h (one)e(has)h(to)g(c)n(hosen)f(the)h(imaginary)f(parts)g(of)h(the)h (coupling)109 4676 y(constan)n(ts)33 b Fw(\024)531 4688 y Ft(j)600 4676 y FO(in)h(an)g(appropriate)e(manner.)56 b(W)-7 b(e)35 b(outline)f(ho)n(w)g(the)g(imaginary)e(parts)i(has)f(to)h (b)r(e)h(c)n(hosen.)109 4776 y(In)29 b(forthcoming)f(pap)r(ers)g(w)n(e) h(ha)n(v)n(e)f(the)h(aim)g(to)g(consider)f(a)h(dissipativ)n(e)f(Sc)n (hr\366)r(dinger-P)n(oisson)e([BKNR02a)n(])109 4876 y(system)h(with)h (recom)n(bination)e(and)h(generation)f(e\033ects.)109 5023 y(The)32 b(pap)r(er)g(is)g(organized)f(as)g(follo)n(ws:)45 b(In)33 b(section)f(2)g(w)n(e)g(rigorously)d(de\034ne)k(the)g (dissipativ)n(e)e(Sc)n(hr\366)r(dinger-)109 5122 y(t)n(yp)r(e)23 b(op)r(erator)f(and)h(sho)n(w)g(that)h(the)f(op)r(erator)f Fw(H)30 b FO(is)24 b(maximal)f(dissipativ)n(e.)34 b(W)-7 b(e)24 b(use)f(the)h(w)n(ell-kno)n(w)e(dilation)109 5222 y(theory)f(as)h(a)g(main)g(to)r(ol)g(for)g(our)g(in)n(v)n(estigations.) 33 b(Therefore,)22 b(the)h(c)n(haracteristic)d(function)j(and)f(the)h (minimal)109 5321 y(dilation)28 b(corresp)r(onding)f(to)i(the)g (maximal)f(dissipativ)n(e)g(op)r(erator)f(are)h(the)i(main)e(ob)5 b(jectiv)n(e)28 b(in)i(the)f(sections)109 5421 y(3)k(and)g(4.)54 b(Section)33 b(5)g(is)g(dev)n(oted)g(to)g(the)h(generalized)e (eigenfunctions)h(of)g(the)h(dilation)f(op)r(erator.)52 b(In)34 b(the)109 5521 y(sections)26 b(6)h(and)f(7)h(w)n(e)g(giv)n(e)f (a)g(de\034nition)i(of)f(the)g(densit)n(y)g(matrix)f(and)h(de\034ne)g (the)h(quan)n(tities)e(of)h(carrier)e(and)109 5620 y(curren)n(t)32 b(densities)h(in)g(terms)g(of)g(the)h(generalized)e(eigenfunctions)g (of)i(the)f(dilation)g(\(cf.)54 b([KNR01c)o(]\).)g(The)109 5720 y(motiv)-5 b(ation)35 b(for)h(this)g(de\034nition)h(is,)h(that)e (the)h(selfadjoin)n(t)f(dilation)f(is)h(regarded)e(as)i(the)g (Hamiltonian)g(of)109 5820 y(a)d(larger)f(closed)h(system)h(whic)n(h)g (con)n(tains)f(the)h(op)r(en)g(system)g(giv)n(en)f(b)n(y)h(the)g (dissipativ)n(e)f(op)r(erator)f Fw(H)7 b FO(.)56 b(In)p eop %%Page: 3 5 3 4 bop 109 254 a FH(Dissipativ)n(e)27 b(Sc)n(hr\366)r(dinger-t)n(yp)r (e)e(op)r(erator)2125 b FO(3)109 503 y(section)21 b(8)h(w)n(e)g(sho)n (w)f(that,)j(dep)r(ending)e(on)g(the)h(densit)n(y)f(matrix,)h(loss)e (and/or)f(gain)i(e\033ect)g(of)h(the)f(op)r(en)g(system)109 602 y(can)k(b)r(e)i(ac)n(hiev)n(ed.)35 b(W)-7 b(e)28 b(close)e(with)i(some)e(remarks)g(and)h(a)g(discussion)f(on)h(ho)n(w)f (the)i(imaginary)d(parts)i(of)g(the)109 702 y(delta)k(p)r(oten)n(tials) f(ha)n(v)n(e)g(to)h(b)r(e)h(c)n(hosen)e(in)h(order)f(to)h(include)g (recom)n(bination-generation)d(pro)r(cesses)h(within)109 801 y(the)f(op)r(en)f(quan)n(tum)h(mo)r(del.)109 1123 y Fz(2)131 b(Dissipativ)l(e)45 b(Sc)l(hr\366)t(dinger)g(op)t(erators) 109 1352 y FO(Let)32 b Fw(x)309 1364 y Ft(j)345 1352 y FO(,)h Fw(j)j Fv(=)31 b(1)p Fw(;)14 b Fu(\001)g(\001)g(\001)27 b Fw(;)14 b(N)9 b FO(,)33 b(b)r(e)f(n)n(um)n(b)r(ers)g(con)n(tained)g (in)g(the)h(b)r(ounded)g(in)n(terv)-5 b(al)31 b Fv(\012)g(:=)g(\()p Fw(a;)14 b(b)p Fv(\))31 b Fu(\032)f Fq(R)p FO(,)40 b(suc)n(h)32 b(that)109 1452 y Fw(a)23 b(<)g(x)311 1464 y Fr(1)373 1452 y Fw(<)g(x)508 1464 y Fr(2)569 1452 y Fw(<)g Fu(\001)14 b(\001)g(\001)24 b Fw(<)f(x)913 1464 y Ft(N)1000 1452 y Fw(<)g(b)p FO(.)38 b(F)-7 b(urthermore)26 b(let)j Fw(V)42 b Fu(2)24 b Fp(L)2001 1421 y Fr(2)2038 1452 y Fv(\()p Fw(a;)14 b(b)p Fv(\))28 b FO(b)r(e)g(real-v)-5 b(alued,)27 b(and)h Fw(m)23 b Fu(2)h Fp(L)3188 1421 y Fo(1)3258 1452 y Fv(\()p Fw(a;)14 b(b)p Fv(\))p FO(,)29 b(with)109 1551 y Fw(m)23 b(>)f Fv(0)27 b FO(and)546 1519 y Fr(1)p 533 1533 59 4 v 533 1580 a Ft(m)625 1551 y Fu(2)c Fp(L)748 1521 y Fo(1)818 1551 y Fv(\()p Fw(a;)14 b(b)p Fv(\))p FO(.)37 b(In)28 b(accordance)e(with)i([KNR01b)o(])f(w)n(e)h(de\034ne)f (the)h(sesquilinear)f(form)806 1802 y Fn(t)p Fv([)p Fw(u;)14 b(v)s Fv(])23 b(:=)1141 1689 y Fs(Z)1224 1710 y Ft(b)1187 1878 y(a)1268 1746 y Fv(1)p 1268 1783 42 4 v 1268 1859 a(2)1401 1746 y(1)p 1329 1783 185 4 v 1329 1859 a Fw(m)p Fv(\()p Fw(x)p Fv(\))1524 1802 y Fw(u)p Fv(\()p Fw(x)p Fv(\))p 1683 1730 156 4 v Fw(v)s Fv(\()p Fw(x)p Fv(\))d(+)e Fw(V)h Fv(\()p Fw(x)p Fv(\))p Fw(u)p Fv(\()p Fw(x)p Fv(\))p 2277 1730 V Fw(v)s Fv(\()p Fw(x)p Fv(\))s Fw(dx)1276 2065 y Fu(\000)f Fw(\024)1407 2077 y Ft(a)1447 2065 y Fw(u)p Fv(\()p Fw(a)p Fv(\))p 1603 1993 152 4 v Fw(v)s Fv(\()p Fw(a)p Fv(\))h Fu(\000)1887 1961 y Ft(N)1856 1986 y Fs(X)1859 2163 y Ft(j)s Fr(=1)1990 2065 y Fw(\024)2038 2077 y Ft(j)2073 2065 y Fw(u)p Fv(\()p Fw(x)2200 2077 y Ft(j)2235 2065 y Fv(\))p 2267 1993 191 4 v Fw(v)s Fv(\()p Fw(x)2389 2077 y Ft(j)2425 2065 y Fv(\))g Fu(\000)f Fw(\024)2607 2077 y Ft(b)2640 2065 y Fw(u)p Fv(\()p Fw(b)p Fv(\))p 2788 1993 144 4 v Fw(v)s Fv(\()p Fw(b)p Fv(\))p Fw(;)109 2339 y FO(for)31 b Fw(u;)14 b(v)33 b Fu(2)e Fp(D)p Fv(\()p Fn(t)p Fv(\))f(=)g Fp(W)840 2309 y Fr(1)p Ft(;)p Fr(2)931 2339 y Fv(\()p Fw(a;)14 b(b)p Fv(\))32 b FO(and)g Fw(\024)1358 2351 y Ft(a)1398 2339 y Fw(;)14 b(\024)1483 2351 y Fr(1)1520 2339 y Fw(;)g Fu(\001)g(\001)g(\001)27 b Fw(;)14 b(\024)1766 2351 y Ft(N)1829 2339 y Fw(;)g(\024)1914 2351 y Ft(b)1977 2339 y Fu(2)31 b Fq(C)2117 2351 y Fr(+)2208 2339 y Fv(:=)f Fu(f)p Fw(z)j Fu(2)e Fq(C)e Fu(j)14 b Fv(Im)6 b(\()p Fw(z)t Fv(\))30 b Fw(>)g Fv(0)p Fu(g)p FO(.)49 b(Mimic)n(king)32 b(the)109 2439 y(pro)r(of)24 b(of)i(Theorem)e(2.20)g(in)i([KNR01a)o(],) g(w)n(e)f(get)g(that)h(the)f(form)h Fn(t)f FO(is)g(closed)g(on)g Fu(H)f Fv(=)f Fp(L)2926 2409 y Fr(2)2963 2439 y Fv(\()p Fw(a;)14 b(b)p Fv(\))25 b FO(and)g(sectorial.)109 2539 y(Th)n(us)32 b(w)n(e)h(get)g(the)g(existence)g(of)f(a)h(maximal)f (sectorial)g(op)r(erator)f Fw(H)7 b FO(,)34 b(suc)n(h)f(that)g Fv(\()p Fw(H)7 b(f)t(;)14 b(v)s Fv(\))32 b(=)g Fn(t)p Fv([)p Fw(f)t(;)14 b(v)s Fv(])33 b FO(for)f(all)109 2638 y Fw(f)f Fu(2)24 b Fp(D)p Fv(\()p Fw(H)7 b Fv(\))28 b FO(and)f Fw(v)f Fu(2)e Fp(D)p Fv(\()p Fn(t)p Fv(\))p FO(.)37 b(It)28 b(can)f(b)r(e)h(sho)n(wn)f(that)h(the)g(op)r(erator)d Fw(H)35 b FO(is)27 b(giv)n(en)g(b)n(y)266 2993 y Fp(D)p Fv(\()p Fw(H)7 b Fv(\))23 b(=)577 2749 y Fs(8)577 2823 y(>)577 2848 y(>)577 2873 y(>)577 2898 y(<)577 3047 y(>)577 3072 y(>)577 3097 y(>)577 3122 y(:)650 2993 y Fw(f)32 b Fu(2)23 b Fp(W)880 2959 y Fr(1)p Ft(;)p Fr(2)971 2993 y Fv(\()p Fw(a;)14 b(b)p Fv(\))1180 2749 y Fs(\014)1180 2798 y(\014)1180 2848 y(\014)1180 2898 y(\014)1180 2948 y(\014)1180 2998 y(\014)1180 3047 y(\014)1180 3097 y(\014)1180 3147 y(\014)1286 2799 y Fr(1)p 1273 2813 59 4 v 1273 2860 a Ft(m)1342 2831 y Fw(f)1392 2801 y Fo(0)1438 2831 y Fu(2)23 b Fp(W)1595 2801 y Fr(1)p Ft(;)p Fr(2)1686 2831 y Fv(\()p Fw(a;)14 b(x)1846 2843 y Fr(1)1883 2831 y Fv(\))19 b Fu(\010)2017 2769 y Fs(L)2109 2790 y Ft(N)6 b Fo(\000)p Fr(1)2109 2856 y Ft(j)s Fr(=1)2271 2831 y Fp(W)2350 2801 y Fr(1)p Ft(;)p Fr(2)2441 2831 y Fv(\()p Fw(x)2520 2843 y Ft(j)2555 2831 y Fw(;)14 b(x)2639 2843 y Ft(j)s Fr(+1)2759 2831 y Fv(\))19 b Fu(\010)f Fp(W)2972 2801 y Fr(1)p Ft(;)p Fr(2)3062 2831 y Fv(\()p Fw(x)3141 2843 y Ft(N)3205 2831 y Fw(;)c(b)p Fv(\))p Fw(;)1346 2905 y Fr(1)p 1273 2919 180 4 v 1273 2966 a(2)p Ft(m)p Fr(\()p Ft(a)p Fr(\))1463 2938 y Fw(f)1513 2908 y Fo(0)1535 2938 y Fv(\()p Fw(a)p Fv(\))24 b(=)f Fu(\000)p Fw(\024)1868 2950 y Ft(a)1907 2938 y Fw(f)9 b Fv(\()p Fw(a)p Fv(\))p Fw(;)2196 2905 y Fr(1)p 2126 2919 174 4 v 2126 2966 a(2)p Ft(m)p Fr(\()p Ft(b)p Fr(\))2309 2938 y Fw(f)2359 2908 y Fo(0)2382 2938 y Fv(\()p Fw(b)p Fv(\))23 b(=)g Fw(\024)2641 2950 y Ft(b)2674 2938 y Fw(f)9 b Fv(\()p Fw(b)p Fv(\))1405 3018 y Fr(1)p 1273 3032 297 4 v 1273 3080 a(2)p Ft(m)p Fr(\()p Ft(x)1429 3088 y Fm(j)1459 3080 y Fr(+0\))1579 3051 y Fw(f)1629 3021 y Fo(0)1652 3051 y Fv(\()p Fw(x)1731 3063 y Ft(j)1785 3051 y Fv(+)18 b(0\))h Fu(\000)2186 3018 y Fr(1)p 2054 3032 298 4 v 2054 3080 a(2)p Ft(m)p Fr(\()p Ft(x)2210 3088 y Fm(j)2240 3080 y Fo(\000)p Fr(0\))2361 3051 y Fw(f)2411 3021 y Fo(0)2434 3051 y Fv(\()p Fw(x)2513 3063 y Ft(j)2567 3051 y Fu(\000)f Fv(0\))23 b(=)g Fu(\000)p Fw(\024)2948 3063 y Ft(j)2982 3051 y Fw(f)9 b Fv(\()p Fw(x)3111 3063 y Ft(j)3146 3051 y Fv(\))p Fw(;)1263 3166 y Fu(8)p Fw(j)27 b Fv(=)c(1)p Fw(;)14 b Fu(\001)g(\001)g(\001)26 b Fw(;)14 b(N)3384 2749 y Fs(9)3384 2823 y(>)3384 2848 y(>)3384 2873 y(>)3384 2898 y(=)3384 3047 y(>)3384 3072 y(>)3384 3097 y(>)3384 3122 y(;)3472 2993 y Fw(;)109 3344 y FO(and)1268 3443 y Fv(\()p Fw(H)7 b(f)i Fv(\)\()p Fw(x)p Fv(\))24 b(=)f(\()p Fw(l)r Fv(\()p Fw(f)9 b Fv(\)\)\()p Fw(x)p Fv(\))p Fw(;)98 b(f)32 b Fu(2)23 b Fp(D)p Fv(\()p Fw(H)7 b Fv(\))p Fw(;)109 3591 y FO(where)1066 3714 y Fv(\()p Fw(l)r Fv(\()p Fw(f)i Fv(\)\)\()p Fw(x)p Fv(\))24 b(:=)e Fu(\000)1591 3657 y Fv(1)p 1591 3694 42 4 v 1591 3771 a(2)1676 3657 y Fw(d)p 1652 3694 91 4 v 1652 3771 a(dx)1835 3657 y Fv(1)p 1763 3694 185 4 v 1763 3771 a Fw(m)p Fv(\()p Fw(x)p Fv(\))1992 3657 y Fw(d)p 1968 3694 91 4 v 1968 3771 a(dx)2068 3714 y(f)9 b Fv(\()p Fw(x)p Fv(\))20 b(+)e Fw(V)g Fv(\()p Fw(x)p Fv(\))p Fw(f)9 b Fv(\()p Fw(x)p Fv(\))p Fw(:)109 3958 y FO(Since)29 b Fw(\024)375 3970 y Ft(a)415 3958 y Fw(;)14 b(\024)500 3970 y Fr(1)537 3958 y Fw(;)g Fu(\001)g(\001)g(\001)27 b Fw(;)14 b(\024)783 3970 y Ft(N)846 3958 y Fw(;)g(\024)931 3970 y Ft(b)990 3958 y Fu(2)26 b Fq(C)1125 3970 y Fr(+)1186 3958 y FO(,)j(the)h(op)r(erator)d Fw(H)36 b FO(is)29 b(dissipativ)n(e,)g(i.e.)41 b Fv(Im\()p Fw(H)7 b(f)t(;)14 b(f)9 b Fv(\))26 b Fu(\024)f Fv(0)p FO(,)k Fw(f)34 b Fu(2)26 b Fp(D)p Fv(\()p Fw(H)7 b Fv(\))p FO(.)42 b(This)109 4057 y(can)27 b(b)r(e)h(seen)f(b)n(y)664 4315 y Fv(Im\()p Fw(H)7 b(f)t(;)14 b(f)9 b Fv(\))23 b(=)f(Im)i Fn(t)p Fv([)p Fw(f)t(;)14 b(f)9 b Fv(])22 b(=)h Fu(\000)1659 4259 y Fw(\013)1712 4229 y Fr(2)1712 4279 y Ft(a)p 1658 4296 94 4 v 1684 4372 a Fv(2)1761 4315 y Fu(j)p Fw(f)9 b Fv(\()p Fw(a)p Fv(\))p Fu(j)1965 4281 y Fr(2)2021 4315 y Fu(\000)2114 4259 y Fw(\013)2167 4229 y Fr(2)2167 4282 y Ft(b)p 2114 4296 91 4 v 2139 4372 a Fv(2)2215 4315 y Fu(j)p Fw(f)g Fv(\()p Fw(b)p Fv(\))p Fu(j)2411 4281 y Fr(2)2466 4315 y Fu(\000)2580 4211 y Ft(N)2549 4236 y Fs(X)2552 4413 y Ft(j)s Fr(=1)2693 4250 y Fw(\013)2746 4219 y Fr(2)2746 4271 y Ft(j)p 2693 4296 V 2718 4372 a Fv(2)2794 4315 y Fu(j)p Fw(f)g Fv(\()p Fw(x)2946 4327 y Ft(j)2981 4315 y Fv(\))p Fu(j)3036 4281 y Fr(2)3074 4315 y Fw(;)109 4616 y FO(where)36 b Fw(\024)406 4628 y Ft(a)486 4616 y Fv(=)j Fw(q)627 4628 y Ft(a)692 4616 y Fv(+)791 4574 y Ft(i\013)857 4549 y Fl(2)857 4590 y Fm(a)p 791 4597 103 4 v 826 4645 a Fr(2)904 4616 y Fw(;)14 b Fu(\001)g(\001)g(\001)28 b Fw(;)14 b(\024)1151 4628 y Ft(b)1223 4616 y Fv(=)39 b Fw(q)1364 4628 y Ft(b)1422 4616 y Fv(+)1522 4574 y Ft(i\013)1588 4549 y Fl(2)1588 4590 y Fm(b)p 1522 4597 100 4 v 1555 4645 a Fr(2)1631 4616 y FO(,)h(for)d(some)f Fw(q)2085 4628 y Ft(a)2126 4616 y Fw(;)14 b Fu(\001)g(\001)g(\001)27 b Fw(;)14 b(q)2361 4628 y Ft(b)2394 4616 y Fw(;)g(\013)2484 4628 y Ft(a)2524 4616 y Fw(;)g Fu(\001)g(\001)g(\001)28 b Fw(;)14 b(\013)2776 4628 y Ft(b)2849 4616 y Fu(2)39 b Fq(R)p FO(,)46 b Fw(\013)3119 4628 y Ft(a)3159 4616 y Fw(;)14 b Fu(\001)g(\001)g(\001)28 b Fw(;)14 b(\013)3411 4628 y Ft(b)3483 4616 y Fw(>)39 b Fv(0)p FO(.)109 4716 y(A)34 b(dissipativ)n(e)f(op)r(erator)f(is)i (called)f(maximal)g(dissipativ)n(e,)i(if)f(it)h(do)r(es)e(not)h(admit)g (an)n(y)f(prop)r(er)g(dissipativ)n(e)109 4815 y(extension.)43 b(Since)30 b Fw(H)37 b FO(is)30 b(maximal)f(sectorial)f(it)j(is)e(also) g(maximal)h(dissipativ)n(e.)43 b(F)-7 b(urthermore,)29 b Fw(H)37 b FO(is)29 b(purely)109 4915 y(maximal)j(dissipativ)n(e)f (\(cf.)53 b([KNR01a)n(]\),)34 b(i.e.)52 b(it)33 b(has)f(no)g (selfadjoin)n(t)h(part)f([FN70)o(].)52 b(This)33 b(implies)f(that)h Fw(H)109 5014 y FO(has)27 b(no)g(real)g(eigen)n(v)-5 b(alues.)109 5161 y(Let)29 b(us)f(in)n(tro)r(duce)h(some)f(notions.)40 b(F)-7 b(or)28 b(simplicit)n(y)h(w)n(e)g(will)g(o)r(ccasional)e(write)h Fw(a)d Fv(=)g Fw(x)2900 5173 y Fr(0)2938 5161 y FO(,)k Fw(\024)3038 5173 y Ft(a)3103 5161 y Fv(=)c Fw(\024)3241 5173 y Fr(0)3278 5161 y FO(,)k Fw(\013)3383 5173 y Ft(a)3449 5161 y Fv(=)c Fw(\013)3592 5173 y Fr(0)3629 5161 y FO(,)109 5261 y Fw(b)d Fv(=)h Fw(x)302 5273 y Ft(N)6 b Fr(+1)449 5261 y FO(,)28 b Fw(\024)548 5273 y Ft(b)604 5261 y Fv(=)23 b Fw(\024)740 5273 y Ft(N)6 b Fr(+1)887 5261 y FO(,)28 b(and)f Fw(\013)1152 5273 y Ft(b)1209 5261 y Fv(=)22 b Fw(\013)1349 5273 y Ft(N)6 b Fr(+1)1496 5261 y FO(.)37 b(W)-7 b(e)28 b(set)395 5479 y Fv(\001\()p Fw(g)s(;)14 b(f)9 b Fv(\)\()p Fw(x)p Fv(\))24 b(:=)1078 5423 y(1)p 914 5460 370 4 v 914 5536 a(2)p Fw(m)p Fv(\()p Fw(x)18 b Fv(+)g(0\))1293 5479 y Fw(g)1336 5445 y Fo(0)1359 5479 y Fv(\()p Fw(x)h Fv(+)f(0\))g Fu(\000)1889 5423 y Fv(1)p 1725 5460 V 1725 5536 a(2)p Fw(m)p Fv(\()p Fw(x)h Fu(\000)f Fv(0\))2105 5479 y Fw(f)2155 5445 y Fo(0)2178 5479 y Fv(\()p Fw(x)h Fu(\000)f Fv(0\))p Fw(;)97 b Fv(\001\()p Fw(g)s Fv(\)\()p Fw(x)p Fv(\))24 b(:=)f(\001\()p Fw(g)s(;)14 b(g)s Fv(\)\()p Fw(x)p Fv(\))p Fw(:)109 5720 y FO(F)-7 b(or)26 b(a)h(matrix)f Fw(A)e Fv(=)e(\()p Fw(a)844 5732 y Ft(ij)903 5720 y Fv(\))p FO(,)28 b(with)f Fw(a)1218 5732 y Ft(ij)1300 5720 y Fu(2)c Fq(C)15 b FO(,)p 1488 5653 63 4 v 33 w Fw(A)28 b FO(will)f(denote)g(the)h(matrix)e(in)i(whic) n(h)f(ev)n(ery)e(elemen)n(t)j(is)f(complex)109 5820 y(conjugated,)g (i.e.)p 697 5753 V 37 w Fw(A)c Fv(=)g(\()p 902 5774 44 4 v Fw(a)946 5832 y Ft(ij)1004 5820 y Fv(\))p FO(.)p eop %%Page: 4 6 4 5 bop 109 254 a FO(4)2519 b Fx(M.)31 b(Bar)n(o,)f(H.)h(Neidhardt)109 503 y FO(In)c(order)e(to)i(get)f(an)h(explicit)g(description)f(of)h (the)g(resolv)n(en)n(t)e(of)i Fw(H)33 b FO(w)n(e)27 b(in)n(tro)r(duce)f (the)h(elemen)n(tary)f(solutions)109 602 y Fw(v)149 614 y Ft(a)189 602 y Fv(\()p Fw(x;)14 b(z)t Fv(\))28 b FO(and)f Fw(v)609 614 y Ft(b)643 602 y Fv(\()p Fw(x;)14 b(z)t Fv(\))28 b FO(de\034ned)g(b)n(y)724 795 y Fw(l)r Fv(\()p Fw(v)823 807 y Ft(a)863 795 y Fv(\()p Fu(\001)p Fw(;)14 b(z)t Fv(\)\)\()p Fw(x)p Fv(\))20 b Fu(\000)e Fw(z)t(v)1359 807 y Ft(a)1399 795 y Fv(\()p Fw(x;)c(z)t Fv(\))23 b(=)g(0)p Fw(;)36 b(v)1842 807 y Ft(a)1882 795 y Fv(\()p Fw(a;)14 b(z)t Fv(\))23 b(=)g(1)p Fw(;)2366 762 y Fr(1)p 2292 776 180 4 v 2292 823 a(2)p Ft(m)p Fr(\()p Ft(a)p Fr(\))2482 795 y Fw(v)2525 764 y Fo(0)2522 815 y Ft(a)2562 795 y Fv(\()p Fw(a;)14 b(z)t Fv(\))23 b(=)g Fu(\000)p Fw(\024)2974 807 y Ft(a)3013 795 y Fw(;)724 908 y Fv(\001\()p Fw(v)865 920 y Ft(a)906 908 y Fv(\()p Fu(\001)p Fw(;)14 b(z)t Fv(\)\)\()p Fw(x)1184 920 y Ft(j)1220 908 y Fv(\))23 b(=)g Fu(\000)p Fw(\024)1476 920 y Ft(j)1510 908 y Fw(v)1550 920 y Ft(a)1591 908 y Fv(\()p Fw(x)1670 920 y Ft(j)1705 908 y Fw(;)14 b(z)t Fv(\))p Fw(;)120 b Fu(8)p Fw(j)27 b Fv(=)c(1)p Fw(;)14 b Fu(\001)g(\001)g(\001)27 b Fw(;)14 b(N)t(;)724 1107 y(l)r Fv(\()p Fw(v)823 1119 y Ft(b)857 1107 y Fv(\()p Fu(\001)p Fw(;)g(z)t Fv(\)\)\()p Fw(x)p Fv(\))19 b Fu(\000)f Fw(z)t(v)1352 1119 y Ft(b)1385 1107 y Fv(\()p Fw(x;)c(z)t Fv(\))23 b(=)g(0)p Fw(;)36 b(v)1828 1119 y Ft(b)1862 1107 y Fv(\()p Fw(b;)14 b(z)t Fv(\))22 b(=)h(1)p Fw(;)2333 1074 y Fr(1)p 2263 1088 174 4 v 2263 1136 a(2)p Ft(m)p Fr(\()p Ft(b)p Fr(\))2446 1107 y Fw(v)2489 1077 y Fo(0)2486 1131 y Ft(b)2520 1107 y Fv(\()p Fw(b;)14 b(z)t Fv(\))22 b(=)h Fw(\024)2858 1119 y Ft(b)2891 1107 y Fw(;)724 1220 y Fv(\001\()p Fw(v)865 1232 y Ft(b)899 1220 y Fv(\()p Fu(\001)p Fw(;)14 b(z)t Fv(\)\)\()p Fw(x)1177 1232 y Ft(j)1213 1220 y Fv(\))23 b(=)g Fu(\000)p Fw(\024)1469 1232 y Ft(j)1504 1220 y Fw(v)1544 1232 y Ft(b)1577 1220 y Fv(\()p Fw(x)1656 1232 y Ft(j)1692 1220 y Fw(;)14 b(z)t Fv(\))p Fw(;)119 b Fu(8)p Fw(j)27 b Fv(=)c(1)p Fw(;)14 b Fu(\001)g(\001)g(\001)27 b Fw(;)14 b(N)t(:)109 1414 y FO(The)27 b(existence)g(of)h(these)g(solutions)e(can)i(b)r(e)g(pro)n (v)n(en)d(as)i(in)h([KNR01b)o(].)109 1561 y(The)f(W)-7 b(ronskian)27 b(of)g Fw(v)827 1573 y Ft(a)868 1561 y Fv(\()p Fw(x;)14 b(z)t Fv(\))27 b FO(and)h Fw(v)1288 1573 y Ft(b)1321 1561 y Fv(\()p Fw(x;)14 b(z)t Fv(\))28 b FO(is)g(de\034ned)g(b)n(y)573 1792 y Fw(W)12 b Fv(\()p Fw(v)735 1804 y Ft(a)776 1792 y Fv(\()p Fu(\001)p Fw(;)i(z)t Fv(\))p Fw(;)g(v)1020 1804 y Ft(b)1053 1792 y Fv(\()p Fu(\001)p Fw(;)g(z)t Fv(\)\)\()p Fw(x)p Fv(\))24 b(:=)f Fw(v)1538 1804 y Ft(a)1578 1792 y Fv(\()p Fw(x;)14 b(z)t Fv(\))1872 1736 y(1)p 1779 1773 227 4 v 1779 1849 a(2)p Fw(m)p Fv(\()p Fw(x)p Fv(\))2016 1792 y Fw(v)2059 1758 y Fo(0)2056 1813 y Ft(b)2089 1792 y Fv(\()p Fw(x;)g(z)t Fv(\))19 b Fu(\000)f Fw(v)2422 1804 y Ft(b)2455 1792 y Fv(\()p Fw(x;)c(z)t Fv(\))2749 1736 y(1)p 2656 1773 V 2656 1849 a(2)p Fw(m)p Fv(\()p Fw(x)p Fv(\))2893 1792 y Fw(v)2936 1758 y Fo(0)2933 1813 y Ft(a)2973 1792 y Fv(\()p Fw(x;)g(z)t Fv(\))p Fw(:)109 2034 y FO(W)-7 b(e)28 b(note)h(that)f Fw(W)12 b Fv(\()p Fw(v)780 2046 y Ft(a)821 2034 y Fv(\()p Fu(\001)p Fw(;)i(z)t Fv(\))p Fw(;)g(v)1065 2046 y Ft(b)1098 2034 y Fv(\()p Fu(\001)p Fw(;)g(z)t Fv(\)\)\()p Fw(x)p Fv(\))30 b FO(is)e(constan)n(t)g(for)g Fw(x)c Fu(2)h Fv(\()p Fw(x)2216 2046 y Ft(i)2245 2034 y Fw(;)14 b(x)2329 2046 y Ft(i)p Fr(+1)2441 2034 y Fv(\))p FO(.)39 b(F)-7 b(urthermore,)28 b(one)g(easily)f(c)n(hec)n(ks)109 2134 y(that)852 2248 y Fw(W)12 b Fv(\()p Fw(v)1014 2260 y Ft(a)1054 2248 y Fv(\()p Fu(\001)p Fw(;)i(z)t Fv(\))p Fw(;)g(v)1298 2260 y Ft(b)1332 2248 y Fv(\()p Fu(\001)p Fw(;)g(z)t Fv(\)\)\()p Fw(x)1610 2260 y Ft(i)1657 2248 y Fu(\000)k Fv(0\))k(=)h Fw(W)12 b Fv(\()p Fw(v)2086 2260 y Ft(a)2127 2248 y Fv(\()p Fu(\001)p Fw(;)i(z)t Fv(\))p Fw(;)g(v)2371 2260 y Ft(b)2404 2248 y Fv(\()p Fu(\001)p Fw(;)g(z)t Fv(\)\)\()p Fw(x)2682 2260 y Ft(i)2729 2248 y Fv(+)k(0\))p Fw(:)109 2413 y FO(Th)n(us)27 b(w)n(e)g(write)g Fw(W)12 b Fv(\()p Fw(z)t Fv(\))23 b(:=)g Fw(W)12 b Fv(\()p Fw(v)1146 2425 y Ft(a)1187 2413 y Fv(\()p Fu(\001)p Fw(;)i(z)t Fv(\))p Fw(;)g(v)1431 2425 y Ft(b)1464 2413 y Fv(\()p Fu(\001)p Fw(;)g(z)t Fv(\)\)\()p Fw(x)p Fv(\))p FO(.)109 2560 y(The)27 b(functions)h(de\034ned)g(b)n(y)913 2753 y Fw(v)953 2765 y Fo(\003)p Ft(a)1028 2753 y Fv(\()p Fw(x;)14 b(z)t Fv(\))23 b(:=)p 1353 2681 272 4 v 23 w Fw(v)1393 2765 y Ft(a)1433 2753 y Fv(\()p Fw(x;)p 1549 2707 43 4 v 14 w(z)t Fv(\))p Fw(;)180 b FO(and)166 b Fw(v)2167 2765 y Fo(\003)p Ft(b)2235 2753 y Fv(\()p Fw(x;)14 b(z)t Fv(\))23 b(:=)p 2559 2681 265 4 v 22 w Fw(v)2599 2765 y Ft(b)2633 2753 y Fv(\()p Fw(x;)p 2749 2707 43 4 v 14 w(z)t Fv(\))p Fw(;)109 2946 y FO(are)j(solutions)h(of)668 3117 y Fw(l)r Fv(\()p Fw(v)767 3129 y Fo(\003)p Ft(a)841 3117 y Fv(\()p Fu(\001)p Fw(;)14 b(z)t Fv(\)\)\()p Fw(x)p Fv(\))19 b Fu(\000)f Fw(z)t(v)1336 3129 y Fo(\003)p Ft(a)1410 3117 y Fv(\()p Fw(x;)c(z)t Fv(\))23 b(=)g(0)p Fw(;)37 b(v)1854 3129 y Fo(\003)p Ft(a)1928 3117 y Fv(\()p Fw(a;)14 b(z)t Fv(\))23 b(=)f(1)p Fw(;)2411 3084 y Fr(1)p 2338 3098 180 4 v 2338 3146 a(2)p Ft(m)p Fr(\()p Ft(a)p Fr(\))2528 3117 y Fw(v)2571 3087 y Fo(0)2568 3138 y(\003)p Ft(a)2642 3117 y Fv(\()p Fw(a;)14 b(z)t Fv(\))23 b(=)f Fu(\000)p 3005 3071 88 4 v Fw(\024)3053 3129 y Ft(a)668 3230 y Fv(\001\()p Fw(v)809 3242 y Fo(\003)p Ft(a)884 3230 y Fv(\()p Fu(\001)p Fw(;)14 b(z)t Fv(\)\)\()p Fw(x)1162 3242 y Ft(j)1197 3230 y Fv(\))24 b(=)e Fu(\000)p 1405 3184 83 4 v Fw(\024)1453 3242 y Ft(j)1488 3230 y Fw(v)1528 3242 y Fo(\003)p Ft(a)1602 3230 y Fv(\()p Fw(x)1681 3242 y Ft(j)1717 3230 y Fw(;)14 b(z)t Fv(\))p Fw(;)119 b Fu(8)p Fw(j)28 b Fv(=)22 b(1)p Fw(;)14 b Fu(\001)g(\001)g(\001)27 b Fw(;)14 b(N)t(;)668 3430 y(l)r Fv(\()p Fw(v)767 3442 y Fo(\003)p Ft(b)834 3430 y Fv(\()p Fu(\001)p Fw(;)g(z)t Fv(\)\)\()p Fw(x)p Fv(\))20 b Fu(\000)e Fw(z)t(v)1330 3442 y Fo(\003)p Ft(b)1397 3430 y Fv(\()p Fw(x;)c(z)t Fv(\))23 b(=)f(0)p Fw(;)37 b(v)1840 3442 y Fo(\003)p Ft(b)1907 3430 y Fv(\()p Fw(b;)14 b(z)t Fv(\))23 b(=)g(1)p Fw(;)2379 3397 y Fr(1)p 2309 3411 174 4 v 2309 3458 a(2)p Ft(m)p Fr(\()p Ft(b)p Fr(\))2492 3430 y Fw(v)2535 3400 y Fo(0)2532 3453 y(\003)p Ft(b)2600 3430 y Fv(\()p Fw(b;)14 b(z)t Fv(\))22 b(=)p 2890 3384 82 4 v 23 w Fw(\024)2938 3442 y Ft(b)668 3543 y Fv(\001\()p Fw(v)809 3555 y Fo(\003)p Ft(b)877 3543 y Fv(\()p Fu(\001)p Fw(;)14 b(z)t Fv(\)\)\()p Fw(x)1155 3555 y Ft(j)1191 3543 y Fv(\))23 b(=)g Fu(\000)p 1399 3497 83 4 v Fw(\024)1447 3555 y Ft(j)1481 3543 y Fw(v)1521 3555 y Fo(\003)p Ft(b)1589 3543 y Fv(\()p Fw(x)1668 3555 y Ft(j)1703 3543 y Fw(;)14 b(z)t Fv(\))p Fw(;)120 b Fu(8)p Fw(j)27 b Fv(=)c(1)p Fw(;)14 b Fu(\001)g(\001)g(\001)26 b Fw(;)14 b(N)t(:)109 3753 y FO(Ob)n(viously)-7 b(,)26 b(the)i(W)-7 b(ronskian)26 b Fw(W)1149 3765 y Fo(\003)1188 3753 y Fv(\()p Fw(z)t Fv(\))d(:=)f Fw(W)12 b Fv(\()p Fw(v)1590 3765 y Fo(\003)p Ft(a)1665 3753 y Fv(\()p Fu(\001)p Fw(;)i(z)t Fv(\))p Fw(;)g(v)1909 3765 y Fo(\003)p Ft(b)1976 3753 y Fv(\()p Fu(\001)p Fw(;)g(z)t Fv(\)\)\()p Fw(x)p Fv(\))29 b FO(satis\034es)e Fw(W)2699 3765 y Fo(\003)2737 3753 y Fv(\()p Fw(z)t Fv(\))c(=)p 2955 3681 197 4 v 23 w Fw(W)12 b Fv(\()p 3077 3707 43 4 v Fw(z)s Fv(\))q FO(.)109 3900 y(W)-7 b(e)28 b(are)f(no)n(w)g(able)h(to)g(write)g(the)g(resolv)n (en)n(ts)e(of)i Fw(H)35 b FO(and)28 b Fw(H)2017 3870 y Fo(\003)2083 3900 y FO(as)f(in)n(tegral)g(op)r(erators,)f(where)i (the)g(k)n(ernels)f(are)109 3999 y(giv)n(en)f(in)i(terms)f(of)h(the)g (elemen)n(tary)e(solutions)h(\(see)h(also)e([K80)o(]\).)109 4245 y Fy(Theorem)k(2.1)40 b FA(L)l(et)30 b Fw(V)43 b Fu(2)25 b Fp(L)1037 4214 y Fr(2)1074 4245 y Fv(\()p Fw(a;)14 b(b)p Fv(\))30 b FA(b)l(e)g(r)l(e)l(al)h(value)l(d)g(and)g Fw(\024)2009 4257 y Ft(a)2049 4245 y Fw(;)14 b(\024)2134 4257 y Fr(1)2171 4245 y Fw(;)g Fu(\001)g(\001)g(\001)27 b Fw(;)14 b(\024)2417 4257 y Ft(N)2480 4245 y Fw(;)g(\024)2565 4257 y Ft(b)2622 4245 y Fu(2)24 b Fq(C)2755 4257 y Fr(+)2816 4245 y FA(.)40 b(Then)31 b(the)g(r)l(esolvent)f(of)109 4344 y(the)g(maximal)g(dissip)l(ative)i(op)l(er)l(ator)f Fw(H)36 b FA(is)30 b(given)g(by)471 4595 y Fv(\(\()p Fw(H)c Fu(\000)18 b Fw(z)t Fv(\))788 4560 y Fo(\000)p Fr(1)877 4595 y Fw(f)9 b Fv(\)\()p Fw(x)p Fv(\))24 b(=)e Fu(\000)1256 4538 y Fw(v)1296 4550 y Ft(b)1330 4538 y Fv(\()p Fw(x;)14 b(z)t Fv(\))p 1256 4575 265 4 v 1290 4652 a Fw(W)e Fv(\()p Fw(z)t Fv(\))1545 4482 y Fs(Z)1628 4502 y Ft(x)1591 4670 y(a)1683 4595 y Fw(v)1723 4607 y Ft(a)1764 4595 y Fv(\()p Fw(y)s(;)i(z)t Fv(\))p Fw(f)9 b Fv(\()p Fw(y)s Fv(\))14 b Fw(dy)20 b Fu(\000)2321 4538 y Fw(v)2361 4550 y Ft(a)2401 4538 y Fv(\()p Fw(x;)14 b(z)t Fv(\))p 2321 4575 272 4 v 2358 4652 a Fw(W)e Fv(\()p Fw(z)t Fv(\))2616 4482 y Fs(Z)2699 4502 y Ft(b)2663 4670 y(x)2747 4595 y Fw(v)2787 4607 y Ft(b)2820 4595 y Fv(\()p Fw(y)s(;)i(z)t Fv(\))p Fw(f)9 b Fv(\()p Fw(y)s Fv(\))14 b Fw(dy)s(;)191 b FO(\(2.1\))109 4836 y FA(for)30 b Fw(f)i Fu(2)23 b(H)31 b FA(and)f Fw(z)c Fu(2)e Fw(\045)p Fv(\()p Fw(H)7 b Fv(\))p FA(.)109 4983 y(The)30 b(r)l(esolvent)g(of)h(the)f (adjoint)h(op)l(er)l(ator)f Fw(H)1545 4953 y Fo(\003)1613 4983 y FA(admits)g(the)g(r)l(epr)l(esentation)299 5233 y Fv(\(\()p Fw(H)439 5199 y Fo(\003)496 5233 y Fu(\000)18 b Fw(z)t Fv(\))654 5199 y Fo(\000)p Fr(1)743 5233 y Fw(f)9 b Fv(\)\()p Fw(x)p Fv(\))24 b(=)e Fu(\000)1122 5177 y Fw(v)1162 5189 y Fo(\003)p Ft(b)1229 5177 y Fv(\()p Fw(x;)14 b(z)t Fv(\))p 1122 5214 299 4 v 1159 5290 a Fw(W)1237 5302 y Fo(\003)1276 5290 y Fv(\()p Fw(z)t Fv(\))1444 5120 y Fs(Z)1527 5141 y Ft(x)1490 5309 y(a)1583 5233 y Fw(v)1623 5245 y Fo(\003)p Ft(a)1697 5233 y Fv(\()p Fw(y)s(;)g(z)t Fv(\))p Fw(f)9 b Fv(\()p Fw(y)s Fv(\))14 b Fw(dy)21 b Fu(\000)2255 5177 y Fw(v)2295 5189 y Fo(\003)p Ft(a)2369 5177 y Fv(\()p Fw(x;)14 b(z)t Fv(\))p 2255 5214 306 4 v 2296 5290 a Fw(W)2374 5302 y Fo(\003)2412 5290 y Fv(\()p Fw(z)t Fv(\))2584 5120 y Fs(Z)2667 5141 y Ft(b)2630 5309 y(x)2714 5233 y Fw(v)2754 5245 y Fo(\003)p Ft(b)2822 5233 y Fv(\()p Fw(y)s(;)g(z)t Fv(\))p Fw(f)9 b Fv(\()p Fw(y)s Fv(\))14 b Fw(dy)s(;)189 b FO(\(2.2\))109 5475 y FA(for)30 b Fw(f)i Fu(2)23 b(H)31 b FA(and)f Fw(z)c Fu(2)e Fw(\045)p Fv(\()p Fw(H)949 5445 y Fo(\003)987 5475 y Fv(\))p FA(.)109 5720 y FO(W)-7 b(e)28 b(omit)f(the)h(pro)r(of,) f(but)h(note)g(that)f Fw(W)12 b Fv(\()p Fw(z)t Fv(\))23 b(=)g(0)f Fu(,)h Fw(z)k Fu(2)c Fw(\033)s Fv(\()p Fw(H)7 b Fv(\))29 b FO(and)e Fw(W)2424 5732 y Fo(\003)2462 5720 y Fv(\()p Fw(z)t Fv(\))c(=)g(0)g Fu(,)g Fw(z)j Fu(2)d Fw(\033)s Fv(\()p Fw(H)3152 5690 y Fo(\003)3191 5720 y Fv(\))p FO(,)28 b(where)f Fw(\033)s Fv(\()p Fu(\001)p Fv(\))109 5820 y FO(denotes)g(the)h(sp)r(ectrum)f(of)h(the)g(corresp)r (onding)d(op)r(erator.)p eop %%Page: 5 7 5 6 bop 109 254 a FH(Dissipativ)n(e)27 b(Sc)n(hr\366)r(dinger-t)n(yp)r (e)e(op)r(erator)2125 b FO(5)109 503 y Fz(3)131 b(The)44 b(c)l(haracteristic)j(function)109 734 y FO(W)-7 b(e)28 b(de\034ne)f(the)h(unclosed)f(op)r(erator)f Fw(\013)e Fv(:)f Fu(H)h(!)f Fq(C)1683 704 y Ft(N)6 b Fr(+2)1864 734 y FO(b)n(y)1117 1145 y Fw(\013f)32 b Fv(=)1331 853 y Fs(0)1331 999 y(B)1331 1049 y(B)1331 1099 y(B)1331 1149 y(B)1331 1199 y(B)1331 1252 y(@)1529 917 y Fw(\013)1582 929 y Ft(b)1616 917 y Fw(f)9 b Fv(\()p Fw(b)p Fv(\))1445 1017 y Fu(\000)p Fw(\013)1563 1029 y Ft(N)1625 1017 y Fw(f)g Fv(\()p Fw(x)1754 1029 y Ft(N)1818 1017 y Fv(\))1636 1105 y FO(.)1636 1138 y(.)1636 1172 y(.)1471 1271 y Fu(\000)p Fw(\013)1589 1283 y Fr(1)1625 1271 y Fw(f)g Fv(\()p Fw(x)1754 1283 y Fr(1)1792 1271 y Fv(\))1489 1371 y Fu(\000)p Fw(\013)1607 1383 y Ft(a)1647 1371 y Fw(f)g Fv(\()p Fw(a)p Fv(\))1892 853 y Fs(1)1892 999 y(C)1892 1049 y(C)1892 1099 y(C)1892 1149 y(C)1892 1199 y(C)1892 1252 y(A)1978 1145 y Fw(;)97 b Fp(D)p Fv(\()p Fw(\013)p Fv(\))24 b(=)f Fp(C)p Fv(\()p Fw(a;)14 b(b)p Fv(\))p Fw(:)109 1615 y FO(Let)27 b(us)h(in)n(tro)r (duce)f(the)h(op)r(erator)e(v)-5 b(alued)27 b(function)h Fw(T)12 b Fv(\()p Fw(z)t Fv(\))22 b(:)h Fu(H)h(!)f Fq(C)2286 1585 y Ft(N)6 b Fr(+2)2467 1615 y FO(giv)n(en)27 b(b)n(y)1458 1803 y Fw(T)12 b Fv(\()p Fw(z)t Fv(\))p Fw(f)30 b Fv(:=)23 b Fw(\013)p Fv(\()p Fw(H)j Fu(\000)18 b Fw(z)t Fv(\))2146 1769 y Fo(\000)p Fr(1)2235 1803 y Fw(f)t(;)1178 b FO(\(3.1\))109 1991 y(for)27 b Fw(f)k Fu(2)24 b(H)k FO(and)g Fw(z)e Fu(2)d Fw(\045)p Fv(\()p Fw(H)7 b Fv(\))p FO(.)38 b(Using)27 b(the)h(expression)e(\(2.1\))h(w)n(e)g(get)357 2449 y Fw(T)12 b Fv(\()p Fw(z)t Fv(\))p Fw(f)30 b Fv(=)772 2392 y(1)p 694 2429 197 4 v 694 2506 a Fw(W)12 b Fv(\()p Fw(z)t Fv(\))914 2132 y Fs(0)914 2278 y(B)914 2328 y(B)914 2378 y(B)914 2428 y(B)914 2478 y(B)914 2527 y(B)914 2581 y(@)1744 2196 y Fu(\000)p Fw(\013)1862 2208 y Ft(b)1909 2129 y Fs(R)1964 2150 y Ft(b)1948 2225 y(a)2011 2196 y Fw(v)2051 2208 y Ft(a)2092 2196 y Fv(\()p Fw(y)s(;)i(z)t Fv(\))p Fw(f)9 b Fv(\()p Fw(y)s Fv(\))14 b Fw(dy)1029 2313 y(\013)1082 2325 y Ft(N)1145 2313 y Fw(v)1185 2325 y Ft(b)1218 2313 y Fv(\()p Fw(x)1297 2325 y Ft(N)1361 2313 y Fw(;)g(z)t Fv(\))1487 2246 y Fs(R)1542 2266 y Ft(x)1580 2274 y Fm(N)1525 2342 y Ft(a)1651 2313 y Fw(v)1691 2325 y Ft(a)1731 2313 y Fv(\()p Fw(y)s(;)g(z)t Fv(\))p Fw(f)9 b Fv(\()p Fw(y)s Fv(\))14 b Fw(dy)20 b Fv(+)e Fw(\013)2331 2325 y Ft(N)2395 2313 y Fw(v)2435 2325 y Ft(a)2475 2313 y Fv(\()p Fw(x)2554 2325 y Ft(N)2618 2313 y Fw(;)c(z)t Fv(\))2744 2246 y Fs(R)2798 2266 y Ft(b)2782 2342 y(x)2820 2350 y Fm(N)2891 2313 y Fw(v)2931 2325 y Ft(b)2965 2313 y Fv(\()p Fw(y)s(;)g(z)t Fv(\))g Fw(dy)2129 2409 y FO(.)2129 2442 y(.)2129 2475 y(.)1101 2592 y Fw(\013)1154 2604 y Fr(1)1191 2592 y Fw(v)1231 2604 y Ft(b)1265 2592 y Fv(\()p Fw(x)1344 2604 y Fr(1)1382 2592 y Fw(;)g(z)t Fv(\))1508 2525 y Fs(R)1563 2546 y Ft(x)1601 2554 y Fl(1)1546 2622 y Ft(a)1651 2592 y Fw(v)1691 2604 y Ft(a)1731 2592 y Fv(\()p Fw(y)s(;)g(z)t Fv(\))p Fw(f)9 b Fv(\()p Fw(y)s Fv(\))14 b Fw(dy)20 b Fv(+)e Fw(\013)2331 2604 y Fr(1)2369 2592 y Fw(v)2409 2604 y Ft(a)2449 2592 y Fv(\()p Fw(x)2528 2604 y Fr(1)2566 2592 y Fw(;)c(z)t Fv(\))2692 2525 y Fs(R)2747 2546 y Ft(b)2731 2622 y(x)2769 2630 y Fl(1)2819 2592 y Fw(v)2859 2604 y Ft(b)2892 2592 y Fv(\()p Fw(y)s(;)g(z)t Fv(\))g Fw(dy)1776 2717 y(\013)1829 2729 y Ft(a)1883 2650 y Fs(R)1939 2670 y Ft(b)1922 2746 y(a)1986 2717 y Fw(v)2026 2729 y Ft(b)2059 2717 y Fv(\()p Fw(y)s(;)g(z)t Fv(\))p Fw(f)9 b Fv(\()p Fw(y)s Fv(\))14 b Fw(dy)3294 2132 y Fs(1)3294 2278 y(C)3294 2328 y(C)3294 2378 y(C)3294 2428 y(C)3294 2478 y(C)3294 2527 y(C)3294 2581 y(A)3381 2449 y Fw(:)109 2906 y FO(The)27 b(adjoin)n(t)h(op)r(erator)d Fw(T)12 b Fv(\()p Fw(z)t Fv(\))1063 2876 y Fo(\003)1128 2906 y FO(is)28 b(giv)n(en)e(b)n(y)429 3133 y Fv(\()p Fw(T)12 b Fv(\()p Fw(z)t Fv(\))629 3099 y Fo(\003)666 3133 y Fw(\030)t Fv(\)\()p Fw(x)p Fv(\))25 b(=)1039 3077 y(1)p 948 3114 224 4 v 948 3190 a Fw(W)1026 3202 y Fo(\003)1065 3190 y Fv(\()p 1097 3144 43 4 v Fw(z)s Fv(\))1182 3133 y Fu(f\000)p Fw(\013)1342 3145 y Ft(b)1374 3133 y Fw(v)1414 3145 y Fo(\003)p Ft(a)1488 3133 y Fv(\()p Fw(x;)p 1604 3087 V 14 w(z)t Fv(\))p Fw(\030)1719 3099 y Ft(b)957 3318 y Fv(+)18 b Fw(\013)1093 3330 y Ft(N)1156 3318 y Fv(\()p Fw(v)1228 3330 y Fo(\003)p Ft(b)1296 3318 y Fv(\()p Fw(x)1375 3330 y Ft(N)1438 3318 y Fw(;)p 1475 3272 V 14 w(z)t Fv(\))p Fw(v)1590 3330 y Fo(\003)p Ft(a)1664 3318 y Fv(\()p Fw(x;)p 1780 3272 V 14 w(z)t Fv(\))p Fw(\037)1907 3333 y Fr([)p Ft(a;x)2020 3341 y Fl(1)2052 3333 y Fr(\))2082 3318 y Fv(\()p Fw(x)p Fv(\))h(+)g Fw(v)2336 3330 y Fo(\003)p Ft(a)2410 3318 y Fv(\()p Fw(x)2489 3330 y Ft(N)2553 3318 y Fw(;)p 2590 3272 V 14 w(z)s Fv(\))p Fw(v)2704 3330 y Fo(\003)p Ft(b)2772 3318 y Fv(\()p Fw(x;)p 2888 3272 V 14 w(z)t Fv(\))p Fw(\037)3015 3333 y Fr([)p Ft(x)3072 3341 y Fm(N)3125 3333 y Ft(;b)p Fr(])3196 3318 y Fv(\))p Fw(\030)3268 3284 y Ft(N)975 3440 y FO(.)975 3473 y(.)975 3506 y(.)957 3641 y Fv(+)f Fw(a)1084 3653 y Fr(1)1121 3641 y Fv(\()p Fw(v)1193 3653 y Fo(\003)p Ft(b)1261 3641 y Fv(\()p Fw(x)1340 3653 y Fr(1)1378 3641 y 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Ft(b)1738 4162 y Fw(\030)1778 4132 y Ft(N)1778 4250 y FO(.)1778 4283 y(.)1778 4317 y(.)1751 4416 y Fw(\030)1791 4386 y Fr(1)1750 4516 y Fw(\030)1790 4486 y Ft(a)1883 3998 y Fs(1)1883 4144 y(C)1883 4194 y(C)1883 4244 y(C)1883 4293 y(C)1883 4343 y(C)1883 4396 y(A)1979 4289 y Fu(2)23 b Fq(C)2111 4255 y Ft(N)6 b Fr(+2)2264 4289 y Fw(:)1194 b FO(\(3.2\))109 4729 y(Similarly)-7 b(,)27 b(w)n(e)g(de\034ne)h Fw(T)889 4741 y Fo(\003)926 4729 y Fv(\()p Fw(z)t Fv(\))23 b(:)g Fu(H)h(!)g Fq(C)1356 4699 y Ft(N)6 b Fr(+2)1537 4729 y FO(b)n(y)1425 4918 y Fw(T)1474 4930 y Fo(\003)1512 4918 y Fv(\()p Fw(z)t Fv(\))p Fw(f)32 b Fv(:=)22 b Fw(\013)p Fv(\()p Fw(H)1963 4883 y Fo(\003)2020 4918 y Fu(\000)c Fw(z)t Fv(\))2178 4883 y Fo(\000)p Fr(1)2267 4918 y Fw(f)t(;)109 5106 y FO(for)27 b Fw(f)k Fu(2)24 b(H)k FO(and)g Fw(z)e Fu(2)d Fw(\045)p Fv(\()p Fw(H)7 b Fv(\))p FO(.)38 b(Using)27 b(equation)g(\(2.2\))g(one)g(gets)275 5563 y Fw(T)324 5575 y Fo(\003)362 5563 y Fv(\()p Fw(z)t Fv(\))p Fw(f)32 b Fv(=)717 5507 y(1)p 639 5544 197 4 v 639 5620 a Fw(W)12 b Fv(\()p Fw(z)t Fv(\))860 5247 y Fs(0)860 5393 y(B)860 5443 y(B)860 5493 y(B)860 5542 y(B)860 5592 y(B)860 5642 y(B)860 5695 y(@)1740 5311 y Fu(\000)p Fw(\013)1858 5323 y Ft(b)1905 5244 y Fs(R)1960 5264 y Ft(b)1944 5340 y(a)2008 5311 y Fw(v)2048 5323 y Fo(\003)p Ft(a)2122 5311 y Fv(\()p Fw(y)s(;)i(z)t Fv(\))p Fw(f)9 b Fv(\()p Fw(y)s Fv(\))14 b Fw(dy)974 5427 y(\013)1027 5439 y Ft(N)1090 5427 y Fw(v)1130 5439 y Fo(\003)p Ft(b)1198 5427 y Fv(\()p Fw(x)1277 5439 y Ft(N)1340 5427 y Fw(;)g(z)t Fv(\))1466 5360 y Fs(R)1521 5381 y Ft(x)1559 5389 y Fm(N)1505 5457 y Ft(a)1630 5427 y Fw(v)1670 5439 y Fo(\003)p Ft(a)1744 5427 y Fv(\()p Fw(y)s(;)g(z)t Fv(\))p Fw(f)9 b Fv(\()p Fw(y)s Fv(\))14 b Fw(dy)21 b Fv(+)d Fw(\013)2345 5439 y Ft(N)2408 5427 y Fw(v)2448 5439 y Fo(\003)p Ft(a)2522 5427 y Fv(\()p Fw(x)2601 5439 y Ft(N)2665 5427 y Fw(;)c(z)t Fv(\))2791 5360 y Fs(R)2846 5381 y Ft(b)2829 5457 y(x)2867 5465 y Fm(N)2938 5427 y Fw(v)2978 5439 y Fo(\003)p Ft(b)3046 5427 y Fv(\()p Fw(y)s(;)g(z)t Fv(\))g Fw(dy)2143 5524 y FO(.)2143 5557 y(.)2143 5590 y(.)1046 5707 y Fw(\013)1099 5719 y Fr(1)1137 5707 y Fw(v)1177 5719 y Fo(\003)p Ft(b)1244 5707 y Fv(\()p Fw(x)1323 5719 y Fr(1)1361 5707 y Fw(;)g(z)t Fv(\))1487 5640 y Fs(R)1542 5660 y Ft(x)1580 5668 y Fl(1)1526 5736 y Ft(a)1630 5707 y Fw(v)1670 5719 y Fo(\003)p Ft(a)1744 5707 y Fv(\()p Fw(y)s(;)g(z)t Fv(\))p Fw(f)9 b Fv(\()p Fw(y)s Fv(\))14 b Fw(dy)21 b Fv(+)d Fw(\013)2345 5719 y Fr(1)2382 5707 y Fw(v)2422 5719 y Fo(\003)p Ft(a)2497 5707 y Fv(\()p Fw(x)2576 5719 y Fr(1)2614 5707 y Fw(;)c(z)t Fv(\))2740 5640 y Fs(R)2794 5660 y Ft(b)2778 5736 y(x)2816 5744 y Fl(1)2866 5707 y Fw(v)2906 5719 y Fo(\003)p Ft(b)2974 5707 y Fv(\()p Fw(y)s(;)g(z)t Fv(\))g Fw(dy)1772 5831 y(\013)1825 5843 y Ft(a)1880 5764 y Fs(R)1935 5785 y Ft(b)1919 5861 y(a)1982 5831 y Fw(v)2022 5843 y Fo(\003)p Ft(b)2090 5831 y Fv(\()p Fw(y)s(;)g(z)t Fv(\))p Fw(f)9 b Fv(\()p Fw(y)s Fv(\))14 b Fw(dy)3376 5247 y Fs(1)3376 5393 y(C)3376 5443 y(C)3376 5493 y(C)3376 5542 y(C)3376 5592 y(C)3376 5642 y(C)3376 5695 y(A)3462 5563 y Fw(:)p eop %%Page: 6 8 6 7 bop 109 254 a FO(6)2519 b Fx(M.)31 b(Bar)n(o,)f(H.)h(Neidhardt)109 503 y FO(The)c(adjoin)n(t)h(op)r(erator)d(has)i(the)h(represen)n (tation)473 722 y Fv(\()p Fw(T)554 734 y Fo(\003)592 722 y Fv(\()p Fw(z)t Fv(\))699 688 y Fo(\003)737 722 y Fw(\030)t Fv(\)\()p Fw(x)p Fv(\))d(=)1097 666 y(1)p 1019 703 197 4 v 1019 779 a Fw(W)12 b Fv(\()p 1141 733 43 4 v Fw(z)t Fv(\))1226 722 y Fu(f\000)p Fw(\013)1386 734 y Ft(b)1418 722 y Fw(v)1458 734 y Ft(a)1499 722 y Fv(\()p Fw(x;)p 1615 676 V 14 w(z)t Fv(\))p Fw(\030)1730 688 y Ft(b)1028 907 y Fv(+)18 b Fw(\013)1164 919 y Ft(N)1227 907 y Fv(\()p Fw(v)1299 919 y Ft(b)1333 907 y Fv(\()p Fw(x)1412 919 y Ft(N)1475 907 y Fw(;)p 1512 861 V 14 w(z)s Fv(\))p Fw(v)1626 919 y Ft(a)1667 907 y Fv(\()p Fw(x;)p 1783 861 V 14 w(z)t Fv(\))p Fw(\037)1910 922 y Fr([)p Ft(a;x)2023 930 y Fm(N)2076 922 y Fr(\))2106 907 y Fv(\()p Fw(x)p Fv(\))h(+)f Fw(v)2359 919 y Ft(a)2400 907 y Fv(\()p Fw(x)2479 919 y Ft(N)2542 907 y Fw(;)p 2579 861 V 14 w(z)t Fv(\))p Fw(v)2694 919 y Ft(b)2727 907 y Fv(\()p Fw(x;)p 2843 861 V 14 w(z)t Fv(\))p Fw(\037)2970 922 y Fr([)p Ft(x)3027 930 y Fm(N)3080 922 y Ft(;b)p Fr(])3152 907 y Fv(\))p Fw(\030)3224 872 y Ft(N)1046 1028 y FO(.)1046 1062 y(.)1046 1095 y(.)1028 1230 y Fv(+)g Fw(\013)1164 1242 y Fr(1)1201 1230 y Fv(\()p Fw(v)1273 1242 y Ft(b)1307 1230 y Fv(\()p Fw(x)1386 1242 y Fr(1)1424 1230 y Fw(;)p 1461 1184 V 14 w(z)s Fv(\))p Fw(v)1575 1242 y Ft(a)1616 1230 y Fv(\()p Fw(x;)p 1732 1184 V 14 w(z)t Fv(\))p Fw(\037)1859 1245 y Fr([)p Ft(a;x)1972 1253 y Fl(1)2003 1245 y Fr(\))2033 1230 y Fv(\()p Fw(x)p Fv(\))i(+)e Fw(v)2287 1242 y Ft(a)2327 1230 y Fv(\()p Fw(x)2406 1242 y Fr(1)2444 1230 y Fw(;)p 2481 1184 V 14 w(z)s Fv(\))p Fw(v)2595 1242 y Ft(b)2629 1230 y Fv(\()p Fw(x;)p 2745 1184 V 14 w(z)t Fv(\))p Fw(\037)2872 1245 y Fr([)p Ft(x)2929 1253 y Fl(1)2961 1245 y Ft(;b)p Fr(])3033 1230 y Fv(\))p Fw(\030)3105 1195 y Fr(1)1028 1354 y Fv(+)g Fw(\013)1164 1366 y Ft(a)1204 1354 y Fw(v)1244 1366 y Ft(b)1277 1354 y Fv(\()p Fw(x;)p 1393 1308 V 14 w(z)t Fv(\))p Fw(\030)1508 1320 y Ft(a)1549 1354 y Fu(g)p Fw(;)109 1539 y FO(for)27 b Fw(x)c Fu(2)h Fv([)p Fw(a;)14 b(b)p Fv(])p FO(.)109 1686 y(Let)27 b(us)h(collect)f(some)g(prop)r(erties)g (of)g(the)h(ab)r(o)n(v)n(e)e(op)r(erators.)109 1901 y Fy(Lemma)j(3.1)41 b FA(L)l(et)29 b Fw(V)42 b Fu(2)23 b Fp(L)967 1871 y Fr(2)1004 1901 y Fv(\()p Fw(a;)14 b(b)p Fv(\))30 b FA(b)l(e)g(r)l(e)l(al-value)l(d,)h(and)f Fw(\024)1962 1913 y Ft(a)2002 1901 y Fw(;)14 b(\024)2087 1913 y Fr(1)2124 1901 y Fw(;)g Fu(\001)g(\001)g(\001)27 b Fw(;)14 b(\024)2370 1913 y Ft(N)2433 1901 y Fw(;)g(\024)2518 1913 y Ft(b)2574 1901 y Fu(2)24 b Fq(C)2707 1913 y Fr(+)2768 1901 y FA(.)38 b(Then)31 b(we)f(have)823 2086 y Fv(\()p Fw(H)931 2052 y Fo(\003)988 2086 y Fu(\000)18 b Fw(z)t Fv(\))1146 2052 y Fo(\000)p Fr(1)1253 2086 y Fu(\000)g Fv(\()p Fw(H)26 b Fu(\000)18 b Fw(z)t Fv(\))1621 2052 y Fo(\000)p Fr(1)1732 2086 y Fv(=)23 b Fu(\000)p Fw(iT)1963 2098 y Fo(\003)2000 2086 y Fv(\()p 2032 2040 V Fw(z)s Fv(\))2106 2052 y Fo(\003)2145 2086 y Fw(T)2194 2098 y Fo(\003)2232 2086 y Fv(\()p Fw(z)t Fv(\))f(=)h Fu(\000)p Fw(iT)12 b Fv(\()p 2636 2040 V Fw(z)r Fv(\))2709 2052 y Fo(\003)2747 2086 y Fw(T)g Fv(\()p Fw(z)t Fv(\))p Fw(;)543 b FO(\(3.3\))109 2271 y FA(for)30 b Fw(z)d Fu(2)c Fw(\045)p Fv(\()p Fw(H)7 b Fv(\))19 b Fu(\\)f Fw(\045)p Fv(\()p Fw(H)811 2241 y Fo(\003)850 2271 y Fv(\))p FA(.)109 2486 y Fy(Pro)s(of.)36 b FO(A)28 b(straigh)n(tforw)n(ard)c(calculation)j(sho)n(ws)f(that)1122 2748 y Fv(\()p Fw(H)1230 2713 y Fo(\003)1268 2748 y Fw(f)t(;)14 b(g)s Fv(\))k Fu(\000)g Fv(\()p Fw(f)t(;)c(H)1716 2713 y Fo(\003)1754 2748 y Fw(g)s Fv(\))23 b(=)g Fw(i)1983 2644 y Ft(N)6 b Fr(+1)1994 2669 y Fs(X)1996 2846 y Ft(j)s Fr(=0)2139 2748 y Fw(\013)2192 2713 y Fr(2)2192 2768 y Ft(i)2230 2748 y Fw(f)j Fv(\()p Fw(x)2359 2760 y Ft(j)2394 2748 y Fv(\))p 2426 2675 190 4 v Fw(g)s Fv(\()p Fw(x)2548 2760 y Ft(j)2584 2748 y Fv(\))p Fw(;)109 3026 y FO(for)28 b Fw(f)t(;)14 b(g)27 b Fu(2)d Fp(D)p Fv(\()p Fw(H)633 2996 y Fo(\003)672 3026 y Fv(\))p FO(.)39 b(Setting)29 b Fw(f)k Fv(=)24 b(\()p Fw(H)1324 2996 y Fo(\003)1382 3026 y Fu(\000)18 b Fw(z)t Fv(\))1540 2996 y Fo(\000)p Fr(1)1629 3026 y Fw(h)28 b FO(and)h Fw(g)e Fv(=)d(\()p Fw(H)2132 2996 y Fo(\003)2189 3026 y Fu(\000)p 2273 2980 43 4 v 19 w Fw(z)s Fv(\))2347 2996 y Fo(\000)p Fr(1)2436 3026 y Fw(k)32 b FO(with)d Fw(h;)14 b(k)27 b Fu(2)e(H)q FO(,)j Fw(z)t(;)p 3138 2980 V 14 w(z)f Fu(2)e Fw(\045)p Fv(\()p Fw(H)3435 2996 y Fo(\003)3473 3026 y Fv(\))p FO(,)k(w)n(e)109 3126 y(obtain)733 3229 y Fv(\(\()p Fw(H)873 3194 y Fo(\003)930 3229 y Fu(\000)18 b Fw(z)t Fv(\))1088 3194 y Fo(\000)p Fr(1)1177 3229 y Fw(h;)c(k)s Fv(\))k Fu(\000)g Fv(\()p Fw(h;)c Fv(\()p Fw(H)1666 3194 y Fo(\003)1723 3229 y Fu(\000)p 1806 3183 V 18 w Fw(z)s Fv(\))1880 3194 y Fo(\000)p Fr(1)1970 3229 y Fw(k)s Fv(\))23 b(=)g Fu(\000)p Fw(i)p Fu(h)p Fw(T)2334 3241 y Fo(\003)2371 3229 y Fv(\()p Fw(z)t Fv(\))p Fw(h;)14 b(T)2612 3241 y Fo(\003)2649 3229 y Fv(\()p 2681 3183 V Fw(z)s Fv(\))p Fw(k)s Fu(i)2833 3245 y Fk(C)2870 3228 y Fm(N)5 b Fl(+2)3004 3229 y Fw(;)109 3381 y FO(and)25 b(the)h(\034rst)f(equalit)n(y)g(in)g(\(3.3\))g(is)h (pro)n(v)n(en.)34 b(The)26 b(second)e(relation)h(can)g(b)r(e)h(pro)n(v) n(en)e(in)h(the)h(same)f(fashion.)35 b Fj(\003)109 3528 y FO(The)c(c)n(haracteristic)f(function)i Fv(\002)1192 3540 y Ft(H)1255 3528 y Fv(\()p Fu(\001)p Fv(\))g FO(is)g(a)f(crucial)g (elemen)n(t)h(in)g(the)g(study)g(of)f(completely)h(non-selfadjoin)n(t) 109 3628 y(op)r(erators.)67 b(It)39 b(is)f(a)h(purely)f(con)n(tractiv)n (e)e(v)-5 b(alued)39 b(and)f(analytic)g(function)h(on)p 2754 3561 117 4 v 38 w Fq(C)2808 3640 y Fo(\000)2870 3628 y FO(,)i(where)d Fq(C)3239 3640 y Fo(\000)3343 3628 y Fv(:=)j Fu(f)p Fw(z)j Fu(2)109 3727 y Fq(C)29 b Fu(j)14 b Fv(Im)6 b(\()p Fw(z)t Fv(\))24 b Fw(<)g Fv(0)p Fu(g)p FO(,)k(cf.)39 b([FN70].)g(The)29 b(c)n(haracteristic)d(function)j Fv(\002)2180 3739 y Ft(H)2242 3727 y Fv(\()p Fu(\001)p Fv(\))g FO(of)g(the)g(maximal)e(dissipativ)n(e)h(op)r(erator)109 3827 y Fw(H)34 b FO(is)28 b(a)f Fv(\()p Fw(N)g Fv(+)18 b(2\))g Fu(\002)g Fv(\()p Fw(N)28 b Fv(+)18 b(2\))27 b FO(matrix-v)-5 b(alued)27 b(function)h(satisfying)f(the)h(relation) 918 4012 y Fv(\002)983 4024 y Ft(H)1045 4012 y Fv(\()p Fw(z)t Fv(\))p Fw(T)12 b Fv(\()p Fw(z)t Fv(\))p Fw(f)30 b Fv(=)23 b Fw(T)1528 4024 y Fo(\003)1566 4012 y Fv(\()p Fw(z)t Fv(\))p Fw(f)t(;)96 b(z)27 b Fu(2)c Fw(\045)p Fv(\()p Fw(H)7 b Fv(\))19 b Fu(\\)g Fw(\045)p Fv(\()p Fw(H)2408 3977 y Fo(\003)2446 4012 y Fv(\))p Fw(;)97 b(f)32 b Fu(2)23 b(H)q Fw(:)638 b FO(\(3.4\))109 4244 y(Let)27 b(us)h(compute)g(the)f(c)n(haracteristic)f(function)i Fv(\002)1744 4256 y Ft(H)1807 4244 y Fv(\()p Fu(\001)p Fv(\))g FO(of)f Fw(H)7 b FO(:)109 4459 y Fy(Theorem)30 b(3.2)40 b FA(L)l(et)21 b Fw(V)42 b Fu(2)24 b Fp(L)1026 4429 y Fr(2)1063 4459 y Fv(\()p Fw(a;)14 b(b)p Fv(\))21 b FA(b)l(e)h(r)l(e)l(al-value)l(d)g(and)f Fw(\024)1961 4471 y Ft(a)2001 4459 y Fw(;)14 b(\024)2086 4471 y Fr(1)2123 4459 y Fw(;)g Fu(\001)g(\001)g(\001)28 b Fw(;)14 b(\024)2370 4471 y Ft(N)2432 4459 y Fw(;)g(\024)2517 4471 y Ft(b)2574 4459 y Fu(2)23 b Fq(C)2706 4471 y Fo(\000)2768 4459 y FA(.)36 b(Then)22 b(the)f(char)l(acteristic)109 4559 y(function)29 b(of)i Fw(H)36 b FA(is)30 b(given)h(by)179 4779 y Fv(\002)244 4791 y Ft(H)306 4779 y Fv(\()p Fw(z)t Fv(\))23 b(=)g Fw(I)560 4796 y Fk(C)596 4779 y Fm(N)5 b Fl(+2)32 b Fu(\000)939 4723 y Fw(i)p 842 4760 224 4 v 842 4836 a(W)920 4848 y Fo(\003)959 4836 y Fv(\()p Fw(z)t Fv(\))1075 4779 y Fu(\002)179 4906 y Fi(0)179 5023 y(B)179 5062 y(B)179 5102 y(B)179 5142 y(B)179 5182 y(B)179 5225 y(@)338 4946 y Fh(\000)p Fg(\013)438 4922 y Ff(2)438 4969 y FB(b)473 4946 y Fg(v)507 4954 y Fe(\003)p FB(a)577 4946 y Fd(\()p Fg(b;)12 b(z)s Fd(\))315 b Fg(\013)1089 4958 y FB(b)1121 4946 y Fg(\013)1166 4957 y FB(N)1224 4946 y Fg(v)1258 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5025 y Fd(\()p Fg(x)2392 5034 y Ff(1)2426 5025 y Fg(;)12 b(z)s Fd(\))p Fg(v)2555 5037 y Fe(\003)p FB(b)2619 5025 y Fd(\()p Fg(x)2686 5036 y FB(N)2743 5025 y Fg(;)g(z)s Fd(\))83 b Fh(\000)p Fg(\013)3021 5036 y FB(N)3079 5025 y Fg(\013)3124 5033 y FB(a)3162 5025 y Fg(v)3196 5037 y Fe(\003)p FB(b)3259 5025 y Fd(\()p Fg(x)3326 5036 y FB(N)3384 5025 y Fg(;)12 b(z)s Fd(\))522 5106 y FI(.)522 5139 y(.)522 5172 y(.)1284 5106 y(.)1284 5139 y(.)1284 5172 y(.)2399 5106 y(.)2399 5139 y(.)2399 5172 y(.)3189 5106 y(.)3189 5139 y(.)3189 5172 y(.)305 5251 y Fg(\013)350 5260 y Ff(1)385 5251 y Fg(\013)430 5263 y FB(b)462 5251 y Fg(v)496 5259 y Fe(\003)p FB(a)566 5251 y Fd(\()p Fg(x)633 5260 y Ff(1)667 5251 y Fg(;)g(z)s Fd(\))106 b Fh(\000)p Fg(\013)968 5260 y Ff(1)1003 5251 y Fg(\013)1048 5262 y FB(N)1106 5251 y Fg(v)1140 5259 y Fe(\003)p FB(a)1210 5251 y Fd(\()p Fg(x)1277 5260 y Ff(1)1311 5251 y Fg(;)12 b(z)s Fd(\))p Fg(v)1440 5263 y Fe(\003)p FB(b)1503 5251 y Fd(\()p Fg(x)1570 5262 y FB(N)1628 5251 y 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b(z)s Fd(\))336 b Fh(\000)p Fg(\013)3102 5307 y Ff(2)3102 5347 y FB(a)3140 5331 y Fg(v)3174 5343 y Fe(\003)p FB(b)3238 5331 y Fd(\()p Fg(a;)12 b(z)s Fd(\))3520 4906 y Fi(1)3520 5023 y(C)3520 5062 y(C)3520 5102 y(C)3520 5142 y(C)3520 5182 y(C)3520 5225 y(A)3481 5425 y FO(\(3.5\))109 5604 y FA(for)30 b Fw(z)d Fu(2)c Fw(\045)p Fv(\()p Fw(H)7 b Fv(\))19 b Fu(\\)f Fw(\045)p Fv(\()p Fw(H)811 5574 y Fo(\003)850 5604 y Fv(\))p FA(.)109 5820 y Fy(Pro)s(of.)36 b FO(It)28 b(remains)e(to)i(v)n(erify)f(that)h Fv(\002)1381 5832 y Ft(H)1443 5820 y Fv(\()p Fu(\001)p Fv(\))g FO(giv)n(en)f(b)n(y)i (\(3.5\))e(satis\034es)g(\(3.4\))o(.)p eop %%Page: 7 9 7 8 bop 109 254 a FH(Dissipativ)n(e)27 b(Sc)n(hr\366)r(dinger-t)n(yp)r (e)e(op)r(erator)2125 b FO(7)109 503 y(One)27 b(easily)g(c)n(hec)n(ks,) f(that)i(equation)f(\(3.5\))g(can)g(b)r(e)h(written)g(as)1399 682 y Fv(\002)1464 694 y Ft(H)1526 682 y Fv(\()p Fw(z)t Fv(\))23 b(=)g Fw(I)1780 698 y Fk(C)1816 682 y Fm(N)5 b Fl(+2)32 b Fu(\000)18 b Fw(i\013T)12 b Fv(\()p 2227 636 43 4 v Fw(z)s Fv(\))2301 648 y Fo(\003)2339 682 y Fw(:)1119 b FO(\(3.6\))109 861 y(Using)27 b(equation)g(\(3.3\))g(w)n(e) g(get)812 1040 y Fv(\002)877 1052 y Ft(H)939 1040 y Fv(\()p Fw(z)t Fv(\))p Fw(T)12 b Fv(\()p Fw(z)t Fv(\))22 b(=)h Fw(T)12 b Fv(\()p Fw(z)t Fv(\))17 b Fu(\000)h Fw(i\013T)12 b Fv(\()p 1767 995 V Fw(z)s Fv(\))1841 1006 y Fo(\003)1879 1040 y Fw(T)g Fv(\()p Fw(z)t Fv(\))22 b(=)h Fw(\013)p Fv(\()p Fw(H)2318 1006 y Fo(\003)2375 1040 y Fu(\000)18 b Fw(z)t Fv(\))2533 1006 y Fo(\000)p Fr(1)2645 1040 y Fv(=)k Fw(T)2781 1052 y Fo(\003)2819 1040 y Fv(\()p Fw(z)t Fv(\))p Fw(:)109 1220 y FO(Th)n(us)27 b(equation)g(\(3.4\))g(is)h(v)n (eri\034ed.)2360 b Fj(\003)109 1366 y FO(Since)379 1546 y Fw(T)12 b Fv(\()p Fw(z)t Fv(\))547 1511 y Fo(\003)584 1546 y Fw(T)g Fv(\()p Fw(z)t Fv(\))17 b Fu(\000)h Fw(T)901 1558 y Fo(\003)939 1546 y Fv(\()p Fw(z)t Fv(\))1046 1511 y Fo(\003)1084 1546 y Fw(T)1133 1558 y Fo(\003)1170 1546 y Fv(\()p Fw(z)t Fv(\))23 b(=)g Fu(\000)p Fv(2Im)o(\()p Fw(z)t Fv(\))p Fw(T)12 b Fv(\()p Fw(z)t Fv(\))1868 1511 y Fo(\003)1905 1546 y Fw(T)g Fv(\()p 1998 1500 V Fw(z)s Fv(\))p Fw(T)g Fv(\()p 2165 1500 V Fw(z)r Fv(\))2238 1511 y Fo(\003)2277 1546 y Fw(T)g Fv(\()p Fw(z)t Fv(\))p Fw(;)123 b FO(for)27 b Fw(z)g Fu(2)c Fw(\045)p Fv(\()p Fw(H)7 b Fv(\))19 b Fu(\\)g Fw(\045)p Fv(\()p Fw(H)3289 1511 y Fo(\003)3327 1546 y Fv(\))p Fw(;)109 1725 y FO(w)n(e)27 b(ha)n(v)n(e)f(for)h Fw(z)g Fu(2)c Fq(C)747 1737 y Fo(\000)827 1725 y Fu(\\)c Fw(\045)p Fv(\()p Fw(H)7 b Fv(\))28 b FO(that)952 1904 y Fu(k)p Fv(\002)1059 1916 y Ft(H)1121 1904 y Fv(\()p Fw(z)t Fv(\))p Fw(T)12 b Fv(\()p Fw(z)t Fv(\))p Fw(f)d Fu(k)1488 1870 y Fr(2)1546 1904 y Fv(=)22 b Fu(k)p Fw(T)1724 1916 y Fo(\003)1762 1904 y Fv(\()p Fw(z)t Fv(\))p Fw(f)9 b Fu(k)1961 1870 y Fr(2)2020 1904 y Fu(\024)22 b(k)p Fw(T)12 b Fv(\()p Fw(z)t Fv(\))p Fw(f)d Fu(k)2409 1870 y Fr(2)2444 1904 y Fw(;)97 b(f)32 b Fu(2)23 b(H)q Fw(:)109 2083 y FO(Th)n(us)j Fv(\002)383 2095 y Ft(H)445 2083 y Fv(\()p Fw(z)t Fv(\))h FO(is)f(a)g(con)n(traction)f (for)h Fw(z)h Fu(2)c Fq(C)1488 2095 y Fo(\000)1566 2083 y Fu(\\)17 b Fw(\045)p Fv(\()p Fw(H)7 b Fv(\))p FO(.)37 b(Since)26 b(the)h(sp)r(ectrum)g(of)f Fw(H)34 b FO(consists)25 b(of)i(only)f(isolated)109 2183 y(eigen)n(v)-5 b(alues)26 b(in)i Fq(C)689 2195 y Fo(\000)751 2183 y FO(,)h(w)n(e)e(get)h(that)g (the)g(c)n(haracteristic)e(function)i Fv(\002)2288 2195 y Ft(H)2351 2183 y Fv(\()p Fw(z)t Fv(\))g FO(admits)g(a)f(unique)h(con) n(tin)n(uation)f(to)109 2283 y(all)g Fw(z)f Fu(2)p 368 2216 117 4 v 24 w Fq(C)422 2295 y Fo(\000)484 2283 y FO(,)h(cf.)38 b(\(3.6\))o(.)109 2604 y Fz(4)131 b(Dilations)109 2833 y FO(Since)27 b Fw(H)33 b FO(is)27 b(a)g(maximal)f(dissipativ)n(e) g(op)r(erator,)f(there)i(exists)g(a)f(larger)f(Hilb)r(ert)i(space)g Fu(K)h FO(con)n(taining)e Fu(H)q FO(,)h(i.e.)109 2933 y Fu(H)d(\022)e(K)q FO(,)28 b(and)g(a)f(selfadjoin)n(t)g(op)r(erator)f Fw(K)33 b FO(on)28 b Fu(K)h FO(suc)n(h)e(that)1191 3112 y Fw(P)1256 3078 y Fo(K)1244 3132 y(H)1310 3112 y Fv(\()p Fw(K)e Fu(\000)18 b Fw(z)t Fv(\))1596 3076 y Fo(\000)p Fr(1)1596 3140 y Fo(jH)1707 3112 y Fv(=)23 b(\()p Fw(H)i Fu(\000)18 b Fw(z)t Fv(\))2079 3078 y Fo(\000)p Fr(1)2168 3112 y Fw(;)97 b(z)26 b Fu(2)e Fq(C)2486 3124 y Fr(+)2547 3112 y Fw(;)911 b FO(\(4.1\))109 3305 y(\(see)35 b([FN70)o(]\).)61 b(The)35 b(op)r(erator)f Fw(K)41 b FO(is)35 b(called)g(a)g(dilation)g (of)g Fw(H)7 b FO(.)60 b Fw(K)41 b FO(is)35 b(said)g(to)g(b)r(e)h(a)f (minimal)g(selfadjoin)n(t)109 3404 y(dilation,)27 b(if)1425 3504 y Fv(clospan)1460 3578 y Ft(z)r Fo(2)p Fk(C)t Fo(n)p Fk(R)1693 3504 y Fv(\()p Fw(K)d Fu(\000)18 b Fw(z)t Fv(\))1978 3470 y Fo(\000)p Fr(1)2067 3504 y Fu(H)24 b Fv(=)e Fu(K)q Fw(:)1146 b FO(\(4.2\))109 3714 y(All)32 b(minimal)h(selfadjoin)n(t)e (dilation)h(of)g(a)g(maximal)f(dissipativ)n(e)h(op)r(erator)e(are)h (isomorphic.)49 b(In)32 b(particular,)109 3813 y(all)27 b(minimal)h(selfadjoin)n(t)f(dilation)g(are)g(unitarily)g(equiv)-5 b(alen)n(t.)109 3960 y(The)25 b(next)g(step)h(in)f(our)f(in)n(v)n (estigations)g(is)h(to)g(obtain)g(an)g(explicit)g(description)g(of)g (the)g(selfadjoin)n(t)g(dilation)g(of)109 4060 y Fw(H)7 b FO(.)36 b(Let)28 b(us)g(in)n(tro)r(duce)f(the)h(Hilb)r(ert)g(space)f Fu(K)i FO(de\034ned)f(b)n(y)1513 4239 y Fu(K)d Fv(:=)d Fu(D)1775 4251 y Fo(\000)1850 4239 y Fu(\010)c(H)h(\010)f(D)2169 4251 y Fr(+)2224 4239 y Fw(;)109 4418 y FO(with)28 b Fu(D)362 4430 y Fo(\006)441 4418 y Fv(:=)23 b Fp(L)597 4388 y Fr(2)634 4418 y Fv(\()p Fq(R)720 4430 y Fo(\006)782 4418 y Fw(;)14 b Fq(C)873 4388 y Ft(N)6 b Fr(+2)1026 4418 y Fv(\))p FO(.)37 b(In)n(tro)r(ducing)27 b(the)h(domain)2015 4397 y Fv(^)2006 4418 y(\012)1329 5784 y Fq(R)1383 5796 y Fo(\000)1329 5535 y Fq(R)1383 5547 y Fo(\000)2325 5784 y Fq(R)2379 5796 y Fr(+)2325 5535 y Fq(R)2379 5547 y Fr(+)1926 5716 y Fv([)p Fw(a;)14 b(x)2077 5728 y Fr(1)2115 5716 y Fv(])p 1013 5823 1661 7 v 1013 5574 V 1840 5820 7 250 v 1832 5463 a FO(.)1832 5496 y(.)1832 5529 y(.)1329 5368 y Fq(R)1383 5380 y Fo(\000)1329 5119 y Fq(R)1383 5131 y Fo(\000)2325 5368 y Fq(R)2379 5380 y Fr(+)2325 5119 y Fq(R)2379 5131 y Fr(+)1926 5301 y Fv([)p Fw(x)1996 5313 y Ft(i)2024 5301 y Fw(;)g(x)2108 5313 y Ft(i)p Fr(+1)2220 5301 y Fv(])p 1013 5408 1661 7 v 1013 5159 V 1840 5404 7 250 v 1832 5047 a FO(.)1832 5081 y(.)1832 5114 y(.)1329 4953 y Fq(R)1383 4965 y Fo(\000)1329 4704 y Fq(R)1383 4716 y Fo(\000)2325 4953 y Fq(R)2379 4965 y Fr(+)2325 4704 y Fq(R)2379 4716 y Fr(+)1926 4886 y Fv([)p Fw(x)1996 4898 y Ft(N)2060 4886 y Fw(;)g(b)p Fv(])p 1013 4993 1661 7 v 1013 4744 V 1840 4989 7 250 v eop %%Page: 8 10 8 9 bop 109 254 a FO(8)2519 b Fx(M.)31 b(Bar)n(o,)f(H.)h(Neidhardt)109 503 y FO(w)n(e)c(get)g Fu(K)e Fv(=)d Fp(L)589 472 y Fr(2)627 503 y Fv(\()668 482 y(^)659 503 y(\012)p Fw(;)14 b(dx)p Fv(\))p FO(.)38 b(F)-7 b(or)23 b Fw(~)-38 b(g)25 b Fu(2)f(K)29 b FO(w)n(e)e(write)1559 681 y Fw(~)-38 b(g)25 b Fv(:=)e Fw(g)1779 693 y Fo(\000)1853 681 y Fu(\010)18 b Fw(g)j Fu(\010)d Fw(g)2120 693 y Fr(+)2175 681 y Fw(;)109 860 y FO(where)1053 1167 y Fw(g)1093 1179 y Fo(\000)1172 1167 y Fv(=)1260 875 y Fs(0)1260 1021 y(B)1260 1071 y(B)1260 1121 y(B)1260 1171 y(B)1260 1220 y(B)1260 1274 y(@)1379 939 y Fw(g)1422 909 y Ft(b)1419 960 y Fo(\000)1474 939 y Fv(\()p Fw(x)p Fv(\))1374 1039 y Fw(g)1417 1009 y Ft(N)1414 1059 y Fo(\000)1479 1039 y Fv(\()p Fw(x)p Fv(\))1471 1127 y FO(.)1471 1161 y(.)1471 1194 y(.)1379 1293 y Fw(g)1422 1263 y Fr(1)1419 1314 y Fo(\000)1474 1293 y Fv(\()p Fw(x)p Fv(\))1379 1393 y Fw(g)1422 1363 y Ft(a)1419 1414 y Fo(\000)1474 1393 y Fv(\()p Fw(x)p Fv(\))1633 875 y Fs(1)1633 1021 y(C)1633 1071 y(C)1633 1121 y(C)1633 1171 y(C)1633 1220 y(C)1633 1274 y(A)1802 1167 y FO(and)83 b Fw(g)2059 1179 y Fr(+)2137 1167 y Fv(=)2225 875 y Fs(0)2225 1021 y(B)2225 1071 y(B)2225 1121 y(B)2225 1171 y(B)2225 1220 y(B)2225 1274 y(@)2344 939 y Fw(g)2387 909 y Ft(b)2384 960 y Fr(+)2439 939 y Fv(\()p Fw(x)p Fv(\))2339 1039 y Fw(g)2382 1009 y Ft(N)2379 1059 y Fr(+)2444 1039 y Fv(\()p Fw(x)p Fv(\))2436 1127 y FO(.)2436 1161 y(.)2436 1194 y(.)2344 1293 y Fw(g)2387 1263 y Fr(1)2384 1314 y(+)2439 1293 y Fv(\()p Fw(x)p Fv(\))2344 1393 y Fw(g)2387 1363 y Ft(a)2384 1414 y Fr(+)2439 1393 y Fv(\()p Fw(x)p Fv(\))2598 875 y Fs(1)2598 1021 y(C)2598 1071 y(C)2598 1121 y(C)2598 1171 y(C)2598 1220 y(C)2598 1274 y(A)2684 1167 y Fw(;)109 1541 y FO(for)30 b Fw(x)f Fu(2)f Fq(R)452 1553 y Fo(\000)545 1541 y FO(and)j Fw(x)d Fu(2)h Fq(R)923 1553 y Fr(+)984 1541 y FO(,)j(resp)r(ectiv)n (ely)-7 b(.)45 b(F)-7 b(urthermore)30 b(w)n(e)g(will)h(need)g(the)g Fv(\()p Fw(N)e Fv(+)20 b(2\))h Fu(\002)f Fv(\()p Fw(N)29 b Fv(+)20 b(2\))p FO(-matrices)109 1641 y Fw(K)186 1610 y Ft(b)180 1661 y Fo(\006)235 1641 y FO(,)28 b Fw(K)363 1610 y Ft(a)357 1661 y Fo(\006)412 1641 y FO(,)g(and)g Fw(K)702 1601 y Ft(j)696 1661 y Fo(\006)751 1641 y FO(,)g Fw(j)g Fv(=)23 b(1)p Fw(;)14 b Fu(\001)g(\001)g(\001)26 b Fw(;)14 b(N)9 b FO(,)28 b(de\034ned)g(b)n(y)499 2003 y Fw(K)576 1969 y Ft(b)570 2024 y Fo(\000)648 2003 y Fv(:=)791 1947 y(1)p 769 1984 87 4 v 769 2060 a Fw(\013)822 2072 y Ft(b)879 1762 y Fs(0)879 1908 y(B)879 1958 y(B)879 2008 y(B)879 2061 y(@)993 1825 y Fv(1)83 b(0)f Fu(\001)14 b(\001)g(\001)97 b Fv(0)83 b Fu(\000)p Fw(\024)1674 1837 y Ft(b)993 1925 y Fv(0)207 b Fu(\001)14 b(\001)g(\001)274 b Fv(0)1002 2014 y FO(.)1002 2047 y(.)1002 2080 y(.)1622 2014 y(.)1622 2047 y(.)1622 2080 y(.)993 2180 y Fv(0)207 b Fu(\001)14 b(\001)g(\001)274 b Fv(0)1748 1762 y Fs(1)1748 1908 y(C)1748 1958 y(C)1748 2008 y(C)1748 2061 y(A)1834 2003 y Fw(;)97 b(K)2031 1969 y Ft(a)2025 2024 y Fo(\000)2104 2003 y Fv(:=)2250 1947 y(1)p 2224 1984 94 4 v 2224 2060 a Fw(\013)2277 2072 y Ft(a)2342 1762 y Fs(0)2342 1908 y(B)2342 1958 y(B)2342 2008 y(B)2342 2061 y(@)2456 1825 y Fv(0)207 b Fu(\001)14 b(\001)g(\001)244 b Fv(0)2465 1914 y FO(.)2465 1947 y(.)2465 1980 y(.)3055 1914 y(.)3055 1947 y(.)3055 1980 y(.)2456 2080 y Fv(0)207 b Fu(\001)14 b(\001)g(\001)244 b Fv(0)2456 2180 y(1)82 b(0)h Fu(\001)14 b(\001)g(\001)96 b Fv(0)83 b Fw(\024)3071 2192 y Ft(a)3153 1762 y Fs(1)3153 1908 y(C)3153 1958 y(C)3153 2008 y(C)3153 2061 y(A)3239 2003 y Fw(;)918 2622 y(K)995 2582 y Ft(j)989 2643 y Fo(\000)1068 2622 y Fv(:=)1212 2566 y(1)p 1188 2603 89 4 v 1188 2679 a Fw(\013)1241 2691 y Ft(j)1300 2306 y Fs(0)1300 2452 y(B)1300 2502 y(B)1300 2552 y(B)1300 2601 y(B)1300 2651 y(B)1300 2701 y(B)1300 2754 y(@)1414 2367 y Fv(0)207 b Fu(\001)14 b(\001)g(\001)242 b Fv(0)1424 2455 y FO(.)1424 2488 y(.)1424 2522 y(.)2011 2455 y(.)2011 2488 y(.)2011 2522 y(.)1414 2621 y Fv(1)83 b(0)f Fu(\001)14 b(\001)g(\001)97 b Fv(0)83 b Fw(\024)2030 2633 y Ft(j)1424 2710 y FO(.)1424 2743 y(.)1424 2776 y(.)2011 2710 y(.)2011 2743 y(.)2011 2776 y(.)1414 2876 y Fv(0)207 b Fu(\001)14 b(\001)g(\001)242 b Fv(0)2106 2306 y Fs(1)2106 2452 y(C)2106 2502 y(C)2106 2552 y(C)2106 2601 y(C)2106 2651 y(C)2106 2701 y(C)2106 2754 y(A)2202 2622 y Fu( )23 b Fv(\()p Fw(j)g Fv(+)18 b(1\))p FO(-th)28 b(ro)n(w)o Fw(;)109 3057 y FO(and)f Fw(K)347 3026 y Ft(b)341 3077 y Fr(+)419 3057 y Fv(:=)p 529 2978 127 4 v 22 w Fw(K)606 3028 y Ft(b)600 3077 y Fo(\000)656 3057 y FO(,)h Fw(K)784 3026 y Ft(a)778 3077 y Fr(+)855 3057 y Fv(:=)p 966 2990 V 23 w Fw(K)1043 3028 y Ft(a)1037 3077 y Fo(\000)1093 3057 y FO(,)f Fw(K)1220 3017 y Ft(j)1214 3077 y Fr(+)1292 3057 y Fv(:=)p 1403 2968 V 23 w Fw(K)1480 3017 y Ft(j)1474 3077 y Fo(\000)1529 3057 y FO(,)h(as)f(w)n(ell)g(as)1253 3469 y Fw(E)i Fv(=)1430 3178 y Fs(0)1430 3324 y(B)1430 3374 y(B)1430 3423 y(B)1430 3473 y(B)1430 3523 y(B)1430 3576 y(@)1577 3241 y Fv(0)114 b Fu(\001)14 b(\001)g(\001)97 b(\001)14 b(\001)g(\001)131 b Fv(0)118 b(1)1577 3341 y(0)c Fu(\001)14 b(\001)g(\001)97 b(\001)14 b(\001)g(\001)131 b Fv(0)118 b(0)1586 3429 y FO(.)1586 3463 y(.)1586 3496 y(.)2324 3429 y(.)2324 3463 y(.)2324 3496 y(.)1577 3596 y Fv(0)149 b(0)117 b Fu(\001)14 b(\001)g(\001)97 b(\001)14 b(\001)g(\001)97 b Fv(0)1544 3695 y Fu(\000)p Fv(1)117 b(0)g Fu(\001)14 b(\001)g(\001)97 b(\001)14 b(\001)g(\001)97 b Fv(0)2398 3178 y Fs(1)2398 3324 y(C)2398 3374 y(C)2398 3423 y(C)2398 3473 y(C)2398 3523 y(C)2398 3576 y(A)2484 3469 y Fw(:)109 3870 y FO(W)-7 b(e)28 b(set)469 4262 y Fw(g)509 4274 y Ft(a)572 4262 y Fv(=)660 3970 y Fs(0)660 4116 y(B)660 4166 y(B)660 4216 y(B)660 4266 y(B)660 4316 y(B)660 4369 y(@)857 3995 y Fr(1)p 784 4009 180 4 v 784 4056 a(2)p Ft(m)p Fr(\()p Ft(a)p Fr(\))974 4027 y Fw(g)1017 3997 y Fo(0)1040 4027 y Fv(\()p Fw(a)p Fv(\))940 4140 y(0)949 4229 y FO(.)949 4262 y(.)949 4295 y(.)940 4395 y Fv(0)885 4494 y Fw(g)s Fv(\()p Fw(a)p Fv(\))1190 3970 y Fs(1)1190 4116 y(C)1190 4166 y(C)1190 4216 y(C)1190 4266 y(C)1190 4316 y(C)1190 4369 y(A)1276 4262 y Fw(;)97 b(g)1436 4274 y Ft(b)1492 4262 y Fv(=)1580 3970 y Fs(0)1580 4116 y(B)1580 4166 y(B)1580 4216 y(B)1580 4266 y(B)1580 4316 y(B)1580 4369 y(@)1774 3995 y Fr(1)p 1704 4009 174 4 v 1704 4056 a(2)p Ft(m)p Fr(\()p Ft(b)p Fr(\))1887 4027 y Fw(g)1930 3997 y Fo(0)1953 4027 y Fv(\()p Fw(b)p Fv(\))1853 4140 y(0)1862 4229 y FO(.)1862 4262 y(.)1862 4295 y(.)1853 4395 y Fv(0)1802 4494 y Fw(g)s Fv(\()p Fw(b)p Fv(\))2094 3970 y Fs(1)2094 4116 y(C)2094 4166 y(C)2094 4216 y(C)2094 4266 y(C)2094 4316 y(C)2094 4369 y(A)2181 4262 y Fw(;)g FO(and)82 b Fw(g)2557 4274 y Ft(j)2615 4262 y Fv(=)2703 3970 y Fs(0)2703 4116 y(B)2703 4166 y(B)2703 4216 y(B)2703 4266 y(B)2703 4316 y(B)2703 4369 y(@)2817 4034 y Fv(\001\()p Fw(g)s Fv(\)\()p Fw(x)3072 4046 y Ft(j)3108 4034 y Fv(\))2958 4133 y(0)2967 4222 y FO(.)2967 4255 y(.)2967 4288 y(.)2958 4388 y Fv(0)2884 4488 y Fw(g)s Fv(\()p Fw(x)3006 4500 y Ft(j)3041 4488 y Fv(\))3182 3970 y Fs(1)3182 4116 y(C)3182 4166 y(C)3182 4216 y(C)3182 4266 y(C)3182 4316 y(C)3182 4369 y(A)3268 4262 y Fw(:)109 4727 y Fy(Theorem)30 b(4.1)40 b FA(L)l(et)c Fw(V)54 b Fu(2)36 b Fp(L)1065 4697 y Fr(2)1102 4727 y Fv(\()p Fw(a;)14 b(b)p Fv(\))p FA(,)39 b Fv(Im\()p Fw(V)19 b Fv(\))35 b(=)g(0)p FA(,)j Fw(\024)1865 4739 y Ft(a)1905 4727 y Fw(;)14 b(\024)1990 4739 y Fr(1)2027 4727 y Fw(;)g Fu(\001)g(\001)g(\001)28 b Fw(;)14 b(\024)2274 4739 y Ft(N)2336 4727 y Fw(;)g(\024)2421 4739 y Ft(b)2489 4727 y Fu(2)36 b Fq(C)2634 4739 y Fr(+)2695 4727 y FA(,)j(and)d Fw(x)2973 4739 y Fr(1)3011 4727 y Fw(;)14 b Fu(\001)g(\001)g(\001)28 b Fw(;)14 b(x)3257 4739 y Ft(N)3355 4727 y Fu(2)36 b Fv(\()p Fw(a;)14 b(b)p Fv(\))p FA(,)109 4826 y(such)29 b(that)h Fw(x)514 4838 y Fr(1)575 4826 y Fw(<)23 b(x)710 4838 y Fr(2)770 4826 y Fw(<)g Fu(\001)14 b(\001)g(\001)23 b Fw(<)f(x)1112 4838 y Ft(N)1176 4826 y FA(.)39 b(Then)30 b(the)g(op)l(er)l(ator)g Fw(K)36 b FA(de\034ne)l(d)30 b(by)462 5177 y Fp(D)p Fv(\()p Fw(K)6 b Fv(\))23 b(:=)797 4932 y Fs(8)797 5007 y(>)797 5032 y(>)797 5057 y(>)797 5082 y(<)797 5231 y(>)797 5256 y(>)797 5281 y(>)797 5306 y(:)867 5177 y Fw(~)-38 b(g)25 b Fu(2)f(K)1093 4932 y Fs(\014)1093 4982 y(\014)1093 5032 y(\014)1093 5082 y(\014)1093 5131 y(\014)1093 5181 y(\014)1093 5231 y(\014)1093 5281 y(\014)1093 5331 y(\014)1162 5000 y Fw(g)1202 5012 y Fo(\006)1281 5000 y Fu(2)f Fp(W)1438 4970 y Fr(1)p Ft(;)p Fr(2)1529 5000 y Fv(\()p Fq(R)1615 5012 y Fo(\006)1677 5000 y Fw(;)14 b Fq(C)1768 4970 y Ft(N)6 b Fr(+2)1921 5000 y Fv(\))p Fw(;)14 b(g)26 b Fu(2)d Fp(W)2213 4970 y Fr(1)p Ft(;)p Fr(2)2304 5000 y Fv(\()p Fw(a;)14 b(b)p Fv(\))p Fw(;)1185 5079 y Fr(1)p 1172 5093 59 4 v 1172 5140 a Ft(m)1241 5111 y Fw(g)1284 5081 y Fo(0)1330 5111 y Fu(2)23 b Fp(W)1487 5081 y Fr(1)p Ft(;)p Fr(2)1578 5111 y Fv(\()p Fw(a;)14 b(x)1738 5123 y Fr(1)1776 5111 y Fv(\))k Fu(\010)1909 5049 y Fs(L)2002 5069 y Ft(N)2002 5136 y(j)s Fr(=1)2135 5111 y Fp(W)2214 5081 y Fr(1)p Ft(;)p Fr(2)2304 5111 y Fv(\()p Fw(x)2383 5123 y Ft(j)2419 5111 y Fw(;)c(x)2503 5123 y Ft(j)s Fr(+1)2622 5111 y Fv(\))19 b Fu(\010)f Fp(W)2835 5081 y Fr(1)p Ft(;)p Fr(2)2926 5111 y Fv(\()p Fw(x)3005 5123 y Ft(N)3068 5111 y Fw(;)c(b)p Fv(\))1162 5229 y Fw(K)1239 5199 y Ft(a)1233 5249 y Fo(\000)1289 5229 y Fw(g)1329 5241 y Ft(a)1387 5229 y Fv(+)1470 5167 y Fs(P)1558 5187 y Ft(N)1558 5254 y(j)s Fr(=1)1690 5229 y Fw(K)1767 5189 y Ft(j)1761 5250 y Fo(\000)1817 5229 y Fw(g)1857 5241 y Ft(j)1910 5229 y Fv(+)k Fw(K)2070 5199 y Ft(b)2064 5249 y Fo(\000)2120 5229 y Fw(g)2160 5241 y Ft(b)2216 5229 y Fv(=)k Fw(g)2343 5241 y Fo(\000)2399 5229 y Fv(\(0\))p Fw(;)1162 5347 y(K)1239 5316 y Ft(a)1233 5367 y Fr(+)1288 5347 y Fw(g)1328 5359 y Ft(a)1386 5347 y Fv(+)1469 5284 y Fs(P)1557 5305 y Ft(N)1557 5371 y(j)s Fr(=1)1690 5347 y Fw(K)1767 5307 y Ft(j)1761 5367 y Fr(+)1815 5347 y Fw(g)1855 5359 y Ft(j)1908 5347 y Fv(+)c Fw(K)2068 5316 y Ft(b)2062 5367 y Fr(+)2117 5347 y Fw(g)2157 5359 y Ft(b)2213 5347 y Fv(=)23 b Fw(g)2341 5359 y Fr(+)2395 5347 y Fv(\(0\))3225 4932 y Fs(9)3225 5007 y(>)3225 5032 y(>)3225 5057 y(>)3225 5082 y(=)3225 5231 y(>)3225 5256 y(>)3225 5281 y(>)3225 5306 y(;)3481 5177 y FO(\(4.3\))109 5527 y FA(and)999 5650 y Fw(K)r(~)-38 b(g)25 b Fv(:=)e Fu(\000)p Fw(i)1379 5593 y(d)p 1356 5631 91 4 v 1356 5707 a(dx)1456 5650 y(g)1496 5662 y Fo(\000)1570 5650 y Fu(\010)18 b Fw(l)r Fv(\()p Fw(g)s Fv(\))g Fu(\010)g(\000)p Fw(i)2015 5593 y(d)p 1992 5631 V 1992 5707 a(dx)2092 5650 y(g)2132 5662 y Fr(+)2187 5650 y Fw(;)180 b(~)-39 b(g)26 b Fu(2)d Fp(D)p Fv(\()p Fw(K)6 b Fv(\))p Fw(;)109 5820 y FA(is)30 b(selfadjoint.)p eop %%Page: 9 11 9 10 bop 109 254 a FH(Dissipativ)n(e)27 b(Sc)n(hr\366)r(dinger-t)n(yp)r (e)e(op)r(erator)2125 b FO(9)109 503 y(The)27 b(pro)r(of)g(is)h(essen)n (tially)e(the)i(same)f(as)g(in)h([KNR01b)o(],)g(so)f(w)n(e)g(omit)g (it.)109 649 y(Figure)g(1)g(sho)n(ws)f(the)i(b)r(oundary)f(conditions)g (of)g(the)h(op)r(erator)e Fw(K)33 b FO(with)28 b(resp)r(ect)g(to)f(the) h(domain)3290 628 y Fv(^)3281 649 y(\012)p FO(.)109 796 y(T)-7 b(o)27 b(sho)n(w)g(that)g Fw(K)33 b FO(is)28 b(the)g(minimal)g (dilation)f(corresp)r(onding)f(to)h Fw(H)7 b FO(,)27 b(let)h(us)g(compute)f(the)h(resolv)n(en)n(t)e(of)i Fw(K)6 b FO(.)109 1068 y Fy(Theorem)30 b(4.2)40 b FA(Assume)d(that)g Fw(V)56 b Fu(2)37 b Fp(L)1413 1038 y Fr(2)1450 1068 y Fv(\()p Fw(a;)14 b(b)p Fv(\))38 b FA(is)f(r)l(e)l(al-value)l(d)h(and)g Fw(\024)2400 1080 y Ft(a)2440 1068 y Fw(;)14 b(\024)2525 1080 y Fr(1)2562 1068 y Fw(;)g Fu(\001)g(\001)g(\001)28 b Fw(;)14 b(\024)2809 1080 y Ft(N)2871 1068 y Fw(;)g(\024)2956 1080 y Ft(b)3026 1068 y Fu(2)37 b Fq(C)3172 1080 y Fr(+)3233 1068 y FA(.)62 b(Then)38 b(the)109 1168 y(r)l(esolvent)29 b(of)i Fw(K)k FA(admits)c(the)e(r)l(epr)l(esentation)282 1379 y Fv(\()p Fw(K)24 b Fu(\000)19 b Fw(z)t Fv(\))568 1344 y Fo(\000)p Fr(1)656 1379 y Fv(\()p Fw(f)729 1391 y Fo(\000)804 1379 y Fu(\010)f Fw(f)27 b Fu(\010)18 b Fw(f)1079 1391 y Fr(+)1133 1379 y Fv(\)\()p Fw(x)p Fv(\))2205 b FO(\(4.4\))592 1565 y Fv(=)o Fw(i)699 1452 y Fs(Z)782 1473 y Ft(x)745 1641 y Fo(\0001)881 1565 y Fw(e)920 1531 y Ft(i)p Fr(\()p Ft(x)p Fo(\000)p Ft(y)r Fr(\))p Ft(z)1158 1565 y Fw(f)1199 1577 y Fo(\000)1255 1565 y Fv(\()p Fw(y)s Fv(\))14 b Fw(dy)22 b Fu(\010)c Fv(\(\()p Fw(H)25 b Fu(\000)18 b Fw(z)t Fv(\))1882 1531 y Fo(\000)p Fr(1)1971 1565 y Fw(f)9 b Fv(\)\()p Fw(x)p Fv(\))19 b(+)f Fw(i)2309 1448 y Fs(\022)2370 1565 y Fw(T)2419 1577 y Fo(\003)2457 1565 y Fv(\()p 2489 1520 43 4 v Fw(z)s Fv(\))2563 1531 y Fo(\003)2615 1452 y Fs(Z)2698 1473 y Fr(0)2662 1641 y Fo(\0001)2798 1565 y Fw(e)2837 1531 y Fo(\000)p Ft(iy)r(z)2986 1565 y Fw(f)3027 1577 y Fo(\000)3082 1565 y Fv(\()p Fw(y)s Fv(\))c Fw(dy)3291 1448 y Fs(\023)3366 1565 y Fv(\()p Fw(x)p Fv(\))675 1811 y Fu(\010)k Fw(i)801 1698 y Fs(Z)883 1719 y Ft(x)846 1887 y Fr(0)939 1811 y Fw(e)978 1777 y Ft(i)p Fr(\()p Ft(x)p Fo(\000)p Ft(y)r Fr(\))p Ft(z)1216 1811 y Fw(f)1257 1823 y Fr(+)1312 1811 y Fv(\()p Fw(y)s Fv(\))c Fw(dy)22 b Fv(+)c Fw(ie)1691 1777 y Ft(iz)r(x)1789 1811 y Fw(T)12 b Fv(\()p Fw(z)t Fv(\))p Fw(f)26 b Fv(+)18 b Fw(i)p Fv(\002)2201 1823 y Ft(H)2263 1811 y Fv(\()p 2295 1765 V Fw(z)t Fv(\))2370 1777 y Fo(\003)2422 1698 y Fs(Z)2505 1719 y Fr(0)2468 1887 y Fo(\0001)2604 1811 y Fw(e)2643 1777 y Ft(i)p Fr(\()p Ft(x)p Fo(\000)p Ft(y)r Fr(\))p Ft(z)2882 1811 y Fw(f)2923 1823 y Fo(\000)2979 1811 y Fv(\()p Fw(y)s Fv(\))c Fw(dy)s(;)109 2075 y FA(for)30 b Fv(Im\()p Fw(z)t Fv(\))23 b Fw(>)g Fv(0)29 b FA(and)198 2286 y Fv(\()p Fw(K)24 b Fu(\000)18 b Fw(z)t Fv(\))483 2252 y Fo(\000)p Fr(1)572 2286 y Fv(\()p Fw(f)645 2298 y Fo(\000)719 2286 y Fu(\010)g Fw(f)27 b Fu(\010)18 b Fw(f)994 2298 y Fr(+)1049 2286 y Fv(\)\()p Fw(x)p Fv(\))2289 b FO(\(4.5\))507 2473 y Fv(=)18 b Fu(\000)g Fw(i)716 2360 y Fs(Z)799 2380 y Fr(0)762 2548 y Ft(x)850 2473 y Fw(e)889 2438 y Ft(i)p Fr(\()p Ft(x)p Fo(\000)p Ft(y)r Fr(\))p Ft(z)1127 2473 y Fw(f)1168 2485 y Fo(\000)1224 2473 y Fv(\()p Fw(y)s Fv(\))c Fw(dy)21 b Fu(\000)e Fw(ie)1603 2438 y Ft(iz)r(x)1701 2473 y Fw(T)1750 2485 y Fo(\003)1788 2473 y Fv(\()p Fw(z)t Fv(\))p Fw(f)27 b Fu(\000)18 b Fw(i)p Fv(\002)2140 2485 y Ft(H)2202 2473 y Fv(\()p Fw(z)t Fv(\))2323 2360 y Fs(Z)2406 2380 y Fo(1)2369 2548 y Fr(0)2490 2473 y Fw(e)2529 2438 y Ft(i)p Fr(\()p Ft(x)p Fo(\000)p Ft(y)r Fr(\))p Ft(z)2767 2473 y Fw(f)2808 2485 y Fr(+)2863 2473 y Fv(\()p Fw(y)s Fv(\))c Fw(dy)590 2702 y Fu(\010)k Fv(\(\()p Fw(H)813 2668 y Fo(\003)870 2702 y Fu(\000)g Fw(z)t Fv(\))1028 2668 y Fo(\000)p Fr(1)1117 2702 y Fw(f)9 b Fv(\)\()p Fw(x)p Fv(\))19 b Fu(\000)f Fw(i)1455 2585 y Fs(\022)1516 2702 y Fw(T)12 b Fv(\()p 1609 2656 V Fw(z)r Fv(\))1682 2668 y Fo(\003)1735 2589 y Fs(Z)1818 2609 y Fo(1)1781 2778 y Fr(0)1902 2702 y Fw(e)1941 2668 y Fo(\000)p Ft(iy)r(z)2090 2702 y Fw(f)2131 2714 y Fr(+)2186 2702 y Fv(\()p Fw(y)s Fv(\))i Fw(dy)2395 2585 y Fs(\023)2470 2702 y Fv(\()p Fw(x)p Fv(\))19 b Fu(\010)f(\000)p Fw(i)2791 2589 y Fs(Z)2873 2609 y Fo(1)2836 2778 y Ft(x)2957 2702 y Fw(e)2996 2668 y Ft(i)p Fr(\()p Ft(x)p Fo(\000)p Ft(y)r Fr(\))p Ft(z)3235 2702 y Fw(f)3276 2714 y Fr(+)3331 2702 y Fv(\()p Fw(y)s Fv(\))c Fw(dy)s(;)109 2981 y FA(for)30 b Fv(Im\()p Fw(z)t Fv(\))23 b Fw(<)g Fv(0)p FA(,)30 b(wher)l(e)900 2959 y Fw(~)889 2981 y(f)i Fv(=)22 b Fw(f)1090 2993 y Fo(\000)1164 2981 y Fu(\010)c Fw(f)27 b Fu(\010)18 b Fw(f)1439 2993 y Fr(+)1517 2981 y Fu(2)24 b(K)q FA(.)999 3395 y FO(Figure)j(1:)36 b(Boundary)26 b(conditions)h(of)h(the)g (dilation)f Fw(K)p 150 5457 1578 7 v 2060 5457 V 150 5125 V 2060 5125 V 176 5551 a Fu(\000)p Fw(i)298 5519 y Ft(d)p 279 5533 73 4 v 279 5580 a(dx)361 5551 y Fw(g)404 5521 y Ft(a)401 5572 y Fo(\000)3331 5551 y Fu(\000)p Fw(i)3453 5519 y Ft(d)p 3434 5533 V 3434 5580 a(dx)3516 5551 y Fw(g)3559 5521 y Ft(a)3556 5572 y Fr(+)561 5394 y Fw(\013)614 5406 y Ft(a)655 5394 y Fw(g)698 5364 y Ft(a)695 5414 y Fo(\000)750 5394 y Fv(\(0\))c(=)1050 5361 y Fr(1)p 977 5375 180 4 v 977 5422 a(2)p Ft(m)p Fr(\()p Ft(a)p Fr(\))1167 5394 y Fw(g)1210 5364 y Fo(0)1233 5394 y Fv(\()p Fw(a)p Fv(\))c(+)f Fw(\024)1491 5406 y Ft(a)1531 5394 y Fw(g)s Fv(\()p Fw(a)p Fv(\))2164 5361 y Fr(1)p 2091 5375 V 2091 5422 a(2)p Ft(m)p Fr(\()p Ft(a)p Fr(\))2281 5394 y Fw(g)2324 5364 y Fo(0)2347 5394 y Fv(\()p Fw(a)p Fv(\))h(+)p 2557 5348 88 4 v 18 w Fw(\024)2605 5406 y Ft(a)2645 5394 y Fw(g)s Fv(\()p Fw(a)p Fv(\))k(=)f Fw(\013)2959 5406 y Ft(a)3000 5394 y Fw(g)3043 5364 y Ft(a)3040 5414 y Fr(+)3094 5394 y Fv(\(0\))1711 5474 y FO(\))301 b(\()p 1890 5412 7 250 v 176 5219 a Fu(\000)p Fw(i)298 5187 y Ft(d)p 279 5201 73 4 v 279 5248 a(dx)361 5219 y Fw(g)404 5189 y Fr(1)401 5240 y Fo(\000)1935 5333 y Fw(l)r Fv(\()p Fw(g)s Fv(\))3331 5219 y Fu(\000)p Fw(i)3453 5187 y Ft(d)p 3434 5201 V 3434 5248 a(dx)3516 5219 y Fw(g)3559 5189 y Fr(1)3556 5240 y(+)568 5077 y Fw(\013)621 5089 y Fr(1)658 5077 y Fw(g)701 5047 y Fr(1)698 5098 y Fo(\000)754 5077 y Fv(\(0\))23 b(=)g(\001\()p Fw(g)s Fv(\)\()p Fw(x)1226 5089 y Fr(1)1264 5077 y Fv(\))c(+)f Fw(\024)1446 5089 y Fr(1)1483 5077 y Fw(g)s Fv(\()p Fw(x)1605 5089 y Fr(1)1643 5077 y Fv(\))437 b(\001\()p Fw(g)s Fv(\)\()p Fw(c)p Fv(\))20 b(+)p 2491 5032 86 4 v 18 w Fw(\024)2539 5089 y Fr(1)2576 5077 y Fw(g)s Fv(\()p Fw(x)2698 5089 y Fr(1)2735 5077 y Fv(\))k(=)e Fw(\013)2931 5089 y Fr(1)2969 5077 y Fw(g)3012 5047 y Fr(1)3009 5098 y(+)3063 5077 y Fv(\(0\))1711 5142 y FO(\))301 b(\()1882 4889 y(.)1882 4922 y(.)1882 4955 y(.)p 150 4710 1578 7 v 2060 4710 V 176 4801 a Fu(\000)p Fw(i)298 4769 y Ft(d)p 279 4783 73 4 v 279 4830 a(dx)361 4801 y Fw(g)404 4761 y Ft(j)401 4822 y Fo(\000)3331 4801 y Fu(\000)p Fw(i)3453 4769 y Ft(d)p 3434 4783 V 3434 4830 a(dx)3516 4801 y Fw(g)3559 4761 y Ft(j)3556 4822 y Fr(+)573 4662 y Fw(\013)626 4674 y Ft(j)661 4662 y Fw(g)704 4622 y Ft(j)701 4683 y Fo(\000)756 4662 y Fv(\(0\))24 b(=)e(\001\()p Fw(g)s Fv(\)\()p Fw(x)1228 4674 y Ft(j)1264 4662 y Fv(\))d(+)f Fw(\024)1446 4674 y Ft(j)1481 4662 y Fw(g)s Fv(\()p Fw(x)1603 4674 y Ft(j)1638 4662 y Fv(\))422 b(\001\()p Fw(g)s Fv(\)\()p Fw(x)2347 4674 y Ft(j)2383 4662 y Fv(\))19 b(+)p 2517 4616 83 4 v 18 w Fw(\024)2565 4674 y Ft(j)2600 4662 y Fw(g)s Fv(\()p Fw(x)2722 4674 y Ft(j)2757 4662 y Fv(\))24 b(=)e Fw(\013)2953 4674 y Ft(j)2988 4662 y Fw(g)3031 4622 y Ft(j)3028 4683 y Fr(+)3083 4662 y Fv(\(0\))1711 4727 y FO(\))301 b(\()p 1890 4665 7 250 v 1935 4586 a Fw(l)r Fv(\()p Fw(g)s Fv(\))p 150 4377 1578 7 v 2060 4377 V 143 4477 a Fu(\000)p Fw(i)265 4445 y Ft(d)p 246 4459 73 4 v 246 4506 a(dx)328 4477 y Fw(g)371 4438 y Ft(j)s Fr(+1)368 4498 y Fo(\000)3322 4477 y Fu(\000)p Fw(i)3444 4445 y Ft(d)p 3426 4459 V 3426 4506 a(dx)3508 4477 y Fw(g)3551 4438 y Ft(j)s Fr(+1)3548 4498 y(+)206 4330 y Fw(\013)259 4342 y Ft(j)s Fr(+1)378 4330 y Fw(g)421 4290 y Ft(j)s Fr(+1)418 4351 y Fo(\000)539 4330 y Fv(\(0\))23 b(=)g(\001\()p Fw(g)s Fv(\)\()p Fw(x)1011 4342 y Ft(j)s Fr(+1)1131 4330 y Fv(\))c(+)f Fw(\024)1313 4342 y Ft(j)s Fr(+1)1432 4330 y Fw(g)s Fv(\()p Fw(x)1554 4342 y Ft(j)s Fr(+1)1673 4330 y Fv(\))393 b(\001\()p Fw(g)s Fv(\)\()p Fw(x)2353 4342 y Ft(j)s Fr(+1)2474 4330 y Fv(\))18 b(+)p 2607 4284 167 4 v 18 w Fw(\024)2655 4342 y Ft(j)s Fr(+1)2774 4330 y Fw(g)s Fv(\()p Fw(x)2896 4342 y Ft(j)s Fr(+1)3016 4330 y Fv(\))23 b(=)g Fw(\013)3212 4342 y Ft(j)s Fr(+1)3331 4330 y Fw(g)3374 4290 y Ft(j)s Fr(+1)3371 4351 y(+)3492 4330 y Fv(\(0\))1711 4395 y FO(\))301 b(\()1882 4142 y(.)1882 4175 y(.)1882 4208 y(.)p 150 3962 1578 7 v 2060 3962 V 171 4057 a Fu(\000)p Fw(i)293 4024 y Ft(d)p 274 4038 73 4 v 274 4086 a(dx)356 4057 y Fw(g)399 4027 y Ft(N)396 4078 y Fo(\000)3326 4057 y Fu(\000)p Fw(i)3448 4024 y Ft(d)p 3429 4038 V 3429 4086 a(dx)3511 4057 y Fw(g)3554 4027 y Ft(N)3551 4078 y Fr(+)454 3915 y Fw(\013)507 3927 y Ft(N)570 3915 y Fw(g)613 3885 y Ft(N)610 3936 y Fo(\000)675 3915 y Fv(\(0\))23 b(=)g(\001\()p Fw(g)s Fv(\)\()p Fw(x)1147 3927 y Ft(N)1211 3915 y Fv(\))c(+)f Fw(\024)1393 3927 y Ft(N)1456 3915 y Fw(g)s Fv(\()p Fw(x)1578 3927 y Ft(N)1641 3915 y Fv(\))416 b(\001\()p Fw(g)s Fv(\)\()p Fw(x)2344 3927 y Ft(N)2408 3915 y Fv(\))19 b(+)p 2542 3869 111 4 v 18 w Fw(\024)2590 3927 y Ft(N)2653 3915 y Fw(g)s Fv(\()p Fw(x)2775 3927 y Ft(N)2838 3915 y Fv(\))k(=)g Fw(\013)3034 3927 y Ft(N)3097 3915 y Fw(g)3140 3885 y Ft(N)3137 3936 y Fr(+)3203 3915 y Fv(\(0\))1711 3980 y FO(\))301 b(\()p 1890 3918 7 250 v 1935 3839 a Fw(l)r Fv(\()p Fw(g)s Fv(\))p 150 3630 1578 7 v 2060 3630 V 176 3723 a Fu(\000)p Fw(i)298 3690 y Ft(d)p 279 3704 73 4 v 279 3751 a(dx)361 3723 y Fw(g)404 3693 y Ft(b)401 3743 y Fo(\000)3331 3723 y Fu(\000)p Fw(i)3453 3690 y Ft(d)p 3434 3704 V 3434 3751 a(dx)3516 3723 y Fw(g)3559 3693 y Ft(b)3556 3743 y Fr(+)605 3567 y Fw(\013)658 3579 y Ft(b)691 3567 y Fw(g)734 3537 y Ft(b)731 3588 y Fo(\000)787 3567 y Fv(\(0\))23 b(=)1084 3535 y Fr(1)p 1014 3549 174 4 v 1014 3596 a(2)p Ft(m)p Fr(\()p Ft(b)p Fr(\))1197 3567 y Fw(g)1240 3537 y Fo(0)1262 3567 y Fv(\()p Fw(b)p Fv(\))c Fu(\000)f Fw(\024)1512 3579 y Ft(b)1545 3567 y Fw(g)s Fv(\()p Fw(b)p Fv(\))2180 3535 y Fr(1)p 2110 3549 V 2110 3596 a(2)p Ft(m)p Fr(\()p Ft(b)p Fr(\))2293 3567 y Fw(g)2336 3537 y Fo(0)2358 3567 y Fv(\()p Fw(b)p Fv(\))h Fu(\000)p 2560 3521 82 4 v 18 w Fw(\024)2608 3579 y Ft(b)2641 3567 y Fw(g)s Fv(\()p Fw(b)p Fv(\))k(=)g Fw(\013)2948 3579 y Ft(b)2981 3567 y Fw(g)3024 3537 y Ft(b)3021 3588 y Fr(+)3076 3567 y Fv(\(0\))1711 3648 y FO(\))301 b(\()109 5820 y Fy(Pro)s(of.)53 b FO(W)-7 b(e)33 b(will)h(only)e(pro)n(v)n(e)g(equation)g(\(4.4\))h (since)g(the)h(equalit)n(y)e(of)40 b(\(4.5\))33 b(can)g(b)r(e)g(sho)n (wn)g(in)g(the)h(same)p eop %%Page: 10 12 10 11 bop 109 254 a FO(10)2477 b Fx(M.)31 b(Bar)n(o,)f(H.)h(Neidhardt) 109 503 y FO(fashion.)36 b(Let)28 b Fv(Im\()p Fw(z)t Fv(\))23 b Fw(>)f Fv(0)p FO(,)994 481 y Fw(~)983 503 y(f)31 b Fu(2)24 b(K)q FO(.)37 b(W)-7 b(e)28 b(set)483 735 y Fw(g)523 747 y Fo(\000)579 735 y Fv(\()p Fw(x)p Fv(\))c(:=)f Fw(i)868 622 y Fs(Z)950 643 y Ft(x)913 811 y Fo(\0001)1049 735 y Fw(e)1088 701 y Ft(i)p Fr(\()p Ft(x)p Fo(\000)p Ft(y)r Fr(\))p Ft(z)1327 735 y Fw(f)1368 747 y Fo(\000)1423 735 y Fv(\()p Fw(y)s Fv(\))14 b Fw(dy)s(;)536 981 y(g)s Fv(\()p Fw(x)p Fv(\))24 b(:=)f(\(\()p Fw(H)i Fu(\000)18 b Fw(z)t Fv(\))1141 946 y Fo(\000)p Fr(1)1230 981 y Fw(f)9 b Fv(\)\()p Fw(x)p Fv(\))19 b(+)f Fw(i)1568 864 y Fs(\022)1629 981 y Fw(T)1678 993 y Fo(\003)1716 981 y Fv(\()p 1748 935 43 4 v Fw(z)s Fv(\))1822 946 y Fo(\003)1874 868 y Fs(Z)1957 888 y Fr(0)1921 1056 y Fo(\0001)2057 981 y Fw(e)2096 946 y Fo(\000)p Ft(iy)r(z)2245 981 y Fw(f)2286 993 y Fo(\000)2341 981 y Fv(\()p Fw(y)s Fv(\))c Fw(dy)2550 864 y Fs(\023)2625 981 y Fv(\()p Fw(x)p Fv(\))484 1226 y Fw(g)524 1238 y Fr(+)579 1226 y Fv(\()p Fw(x)p Fv(\))24 b(:=)f Fw(i)868 1113 y Fs(Z)950 1134 y Ft(x)913 1302 y Fr(0)1006 1226 y Fw(e)1045 1192 y Ft(i)p Fr(\()p Ft(x)p Fo(\000)p Ft(y)r Fr(\))p Ft(z)1283 1226 y Fw(f)1324 1238 y Fr(+)1379 1226 y Fv(\()p Fw(y)s Fv(\))14 b Fw(dy)21 b Fv(+)d Fw(ie)1757 1192 y Ft(iz)r(x)1856 1226 y Fw(T)12 b Fv(\()p Fw(z)t Fv(\))p Fw(f)26 b Fv(+)18 b Fw(i)p Fv(\002)2268 1238 y Ft(H)2330 1226 y Fv(\()p 2362 1181 V Fw(z)t Fv(\))2437 1192 y Fo(\003)2489 1113 y Fs(Z)2572 1134 y Fr(0)2535 1302 y Fo(\0001)2671 1226 y Fw(e)2710 1192 y Ft(i)p Fr(\()p Ft(x)p Fo(\000)p Ft(y)r Fr(\))p Ft(z)2949 1226 y Fw(f)2990 1238 y Fo(\000)3045 1226 y Fv(\()p Fw(y)s Fv(\))c Fw(dy)s(:)204 b FO(\(4.6\))109 1460 y(One)27 b(easily)g(v)n(eri\034es)f(that)1198 1564 y Fv(\()p Fw(K)f Fu(\000)18 b Fw(z)t Fv(\)\()p Fw(g)1556 1576 y Fo(\000)1629 1564 y Fu(\010)g Fw(g)k Fu(\010)c Fw(g)1897 1576 y Fr(+)1951 1564 y Fv(\))24 b(=)e Fw(f)2135 1576 y Fo(\000)2209 1564 y Fu(\010)c Fw(f)27 b Fu(\010)18 b Fw(f)2484 1576 y Fr(+)2539 1564 y Fw(:)109 1718 y FO(Th)n(us)31 b(it)h(remains)e(to)h(sho)n(w)g(that)d Fw(~)-38 b(g)34 b FO(satis\034es)c(the)i(b)r(oundary)f(conditions)f(\(4.3\).)49 b(W)-7 b(e)31 b(set)h Fw(h)d Fv(=)g(\()p Fw(H)f Fu(\000)21 b Fw(z)t Fv(\))3514 1688 y Fo(\000)p Fr(1)3602 1718 y Fw(f)109 1818 y FO(and)27 b(get)338 2003 y Fw(g)378 2015 y Ft(a)440 2003 y Fv(=)c Fw(h)576 2015 y Ft(a)635 2003 y Fv(+)18 b Fw(G)783 2015 y Ft(a)823 2003 y Fw(g)863 2015 y Fo(\000)919 2003 y Fv(\(0\))p Fw(;)97 b(g)1185 2015 y Ft(j)1242 2003 y Fv(=)23 b Fw(h)1378 2015 y Ft(j)1431 2003 y Fv(+)18 b Fw(G)1579 2015 y Ft(j)1614 2003 y Fw(g)1654 2015 y Fo(\000)1710 2003 y Fv(\(0\))p Fw(;)97 b(g)1976 2015 y Ft(b)2032 2003 y Fv(=)23 b Fw(h)2168 2015 y Ft(b)2219 2003 y Fv(+)18 b Fw(G)2367 2015 y Ft(b)2401 2003 y Fw(g)2441 2015 y Fo(\000)2497 2003 y Fv(\(0\))p Fw(;)97 b Fu(8)p Fw(j)27 b Fv(=)22 b(1)p Fw(;)14 b Fu(\001)g(\001)g(\001)27 b Fw(;)14 b(N)t(;)229 b FO(\(4.7\))109 2188 y(where)464 2587 y Fw(G)529 2599 y Ft(a)592 2587 y Fv(:=)703 2296 y Fs(0)703 2442 y(B)703 2491 y(B)703 2541 y(B)703 2591 y(B)703 2641 y(B)703 2694 y(@)817 2353 y Fu(\000)p Fw(\024)930 2365 y Ft(a)970 2353 y Fv(\()p Fw(T)1051 2365 y Fo(\003)1089 2353 y Fv(\()p 1121 2307 V Fw(z)s Fv(\))1195 2323 y Fo(\003)1234 2353 y Fw(e)1273 2365 y Ft(b)1306 2353 y Fv(\)\()p Fw(a)p Fv(\))83 b Fu(\001)14 b(\001)g(\001)97 b(\000)p Fw(\024)1836 2365 y Ft(a)1876 2353 y Fv(\()p Fw(T)1957 2365 y Fo(\003)1995 2353 y Fv(\()p 2027 2307 V Fw(z)s Fv(\))2101 2323 y Fo(\003)2140 2353 y Fw(e)2179 2365 y Fr(1)2215 2353 y Fv(\)\()p Fw(a)p Fv(\))2523 2320 y Fr(1)p 2449 2334 180 4 v 2449 2381 a(2)p Ft(m)p Fr(\()p Ft(a)p Fr(\))2639 2353 y Fv(\()p Fw(T)2720 2365 y Fo(\003)2758 2353 y Fv(\()p 2790 2307 43 4 v Fw(z)t Fv(\))2865 2323 y Fo(\003)2903 2353 y Fw(e)2942 2365 y Ft(a)2982 2353 y Fv(\))3014 2323 y Fo(0)3037 2353 y Fv(\()p Fw(a)p Fv(\))1111 2466 y(0)376 b Fu(\001)14 b(\001)g(\001)393 b Fv(0)711 b(0)1120 2554 y FO(.)1120 2587 y(.)1120 2621 y(.)2028 2554 y(.)2028 2587 y(.)2028 2621 y(.)2781 2554 y(.)2781 2587 y(.)2781 2621 y(.)1111 2720 y Fv(0)376 b Fu(\001)14 b(\001)g(\001)393 b Fv(0)711 b(0)893 2820 y(\()p Fw(T)974 2832 y Fo(\003)1012 2820 y Fv(\()p 1044 2774 V Fw(z)t Fv(\))1119 2790 y Fo(\003)1157 2820 y Fw(e)1196 2832 y Ft(b)1229 2820 y Fv(\)\()p Fw(a)p Fv(\))160 b Fu(\001)14 b(\001)g(\001)173 b Fv(\()p Fw(T)1880 2832 y Fo(\003)1918 2820 y Fv(\()p 1950 2774 V Fw(z)t Fv(\))2025 2790 y Fo(\003)2063 2820 y Fw(e)2102 2832 y Fr(1)2139 2820 y Fv(\)\()p Fw(a)p Fv(\))272 b(\()p Fw(T)2632 2832 y Fo(\003)2670 2820 y Fv(\()p 2702 2774 V Fw(z)s Fv(\))2776 2790 y Fo(\003)2815 2820 y Fw(e)2854 2832 y Ft(a)2894 2820 y Fv(\)\()p Fw(a)p Fv(\))3187 2296 y Fs(1)3187 2442 y(C)3187 2491 y(C)3187 2541 y(C)3187 2591 y(C)3187 2641 y(C)3187 2694 y(A)3274 2587 y Fw(;)541 3252 y(G)606 3264 y Ft(b)663 3252 y Fv(:=)774 2961 y Fs(0)774 3107 y(B)774 3157 y(B)774 3207 y(B)774 3257 y(B)774 3306 y(B)774 3360 y(@)968 2986 y Fr(1)p 898 3000 174 4 v 898 3047 a(2)p Ft(m)p Fr(\()p Ft(b)p Fr(\))1081 3018 y Fv(\()p Fw(T)1162 3030 y Fo(\003)1200 3018 y Fv(\()p 1232 2973 43 4 v Fw(z)s Fv(\))1306 2988 y Fo(\003)1345 3018 y Fw(e)1384 3030 y Ft(b)1417 3018 y Fv(\))1449 2988 y Fo(0)1472 3018 y Fv(\()p Fw(b)p Fv(\))84 b Fw(\024)1704 3030 y Ft(b)1737 3018 y Fv(\()p Fw(T)1818 3030 y Fo(\003)1856 3018 y Fv(\()p 1888 2973 V Fw(z)s Fv(\))1962 2988 y Fo(\003)2001 3018 y Fw(e)2040 3030 y Ft(N)2102 3018 y Fv(\)\()p Fw(b)p Fv(\))g Fu(\001)14 b(\001)g(\001)97 b Fw(\024)2560 3030 y Ft(b)2593 3018 y Fv(\()p Fw(T)2674 3030 y Fo(\003)2712 3018 y Fv(\()p 2744 2973 V Fw(z)s Fv(\))2818 2988 y Fo(\003)2857 3018 y Fw(e)2896 3030 y Ft(a)2936 3018 y Fv(\)\()p Fw(b)p Fv(\))1210 3131 y(0)673 b(0)351 b Fu(\001)14 b(\001)g(\001)354 b Fv(0)1219 3220 y FO(.)1219 3253 y(.)1219 3286 y(.)1934 3220 y(.)1934 3253 y(.)1934 3286 y(.)2778 3220 y(.)2778 3253 y(.)2778 3286 y(.)1210 3386 y Fv(0)673 b(0)351 b Fu(\001)14 b(\001)g(\001)354 b Fv(0)996 3485 y(\()p Fw(T)1077 3497 y Fo(\003)1115 3485 y Fv(\()p 1147 3440 V Fw(z)t Fv(\))1222 3455 y Fo(\003)1260 3485 y Fw(e)1299 3497 y Ft(b)1332 3485 y Fv(\)\()p Fw(b)p Fv(\))232 b(\()p Fw(T)1777 3497 y Fo(\003)1815 3485 y Fv(\()p 1847 3440 V Fw(z)t Fv(\))1922 3455 y Fo(\003)1960 3485 y Fw(e)1999 3497 y Ft(N)2062 3485 y Fv(\)\()p Fw(b)p Fv(\))124 b Fu(\001)14 b(\001)g(\001)137 b Fv(\()p Fw(T)2633 3497 y Fo(\003)2671 3485 y Fv(\()p 2703 3440 V Fw(z)t Fv(\))2778 3455 y Fo(\003)2816 3485 y Fw(e)2855 3497 y Ft(a)2895 3485 y Fv(\)\()p Fw(b)p Fv(\))3110 2961 y Fs(1)3110 3107 y(C)3110 3157 y(C)3110 3207 y(C)3110 3257 y(C)3110 3306 y(C)3110 3360 y(A)3196 3252 y Fw(;)109 3635 y FO(and)389 4026 y Fw(G)454 4038 y Ft(j)512 4026 y Fv(:=)623 3735 y Fs(0)623 3881 y(B)623 3931 y(B)623 3981 y(B)623 4031 y(B)623 4080 y(B)623 4133 y(@)737 3799 y Fu(\000)p Fw(\024)850 3811 y Ft(j)885 3799 y Fv(\()p Fw(T)966 3811 y Fo(\003)1004 3799 y Fv(\()p 1036 3753 V Fw(z)s Fv(\))1110 3769 y Fo(\003)1149 3799 y Fw(e)1188 3811 y Ft(b)1221 3799 y Fv(\)\()p Fw(x)1332 3811 y Ft(j)1368 3799 y Fv(\))83 b Fu(\001)14 b(\001)g(\001)97 b Fv(\001\()p Fw(T)1827 3811 y Fo(\003)1865 3799 y Fv(\()p 1897 3753 V Fw(z)s Fv(\))1971 3769 y Fo(\003)2010 3799 y Fw(e)2049 3811 y Ft(j)2083 3799 y Fv(\)\()p Fw(x)2194 3811 y Ft(j)2230 3799 y Fv(\))84 b Fu(\001)14 b(\001)g(\001)96 b(\000)p Fw(\024)2652 3811 y Ft(j)2687 3799 y Fv(\()p Fw(T)2780 3769 y Fo(\003)2817 3799 y Fv(\()p 2849 3753 V Fw(z)t Fv(\))2924 3769 y Fo(\003)2962 3799 y Fw(e)3001 3811 y Ft(a)3041 3799 y Fv(\)\()p Fw(x)3152 3811 y Ft(j)3188 3799 y Fv(\))1048 3898 y(0)393 b Fu(\001)14 b(\001)g(\001)369 b Fv(0)355 b Fu(\001)14 b(\001)g(\001)416 b Fv(0)1057 3987 y FO(.)1057 4020 y(.)1057 4053 y(.)1958 3987 y(.)1958 4020 y(.)1958 4053 y(.)2868 3987 y(.)2868 4020 y(.)2868 4053 y(.)1048 4153 y Fv(0)393 b Fu(\001)14 b(\001)g(\001)369 b Fv(0)355 b Fu(\001)14 b(\001)g(\001)416 b Fv(0)811 4252 y(\()p Fw(T)892 4264 y Fo(\003)930 4252 y Fv(\()p 962 4207 V Fw(z)s Fv(\))1036 4222 y Fo(\003)1075 4252 y Fw(e)1114 4264 y Ft(b)1147 4252 y Fv(\)\()p Fw(x)1258 4264 y Ft(j)1294 4252 y Fv(\))157 b Fu(\001)14 b(\001)g(\001)131 b Fv(\()p Fw(T)1792 4264 y Fo(\003)1830 4252 y Fv(\()p 1862 4207 V Fw(z)t Fv(\))1937 4222 y Fo(\003)1975 4252 y Fw(e)2014 4264 y Ft(j)2049 4252 y Fv(\)\()p Fw(x)2160 4264 y Ft(j)2196 4252 y Fv(\))118 b Fu(\001)14 b(\001)g(\001)170 b Fv(\()p Fw(T)2706 4222 y Fo(\003)2744 4252 y Fv(\()p 2776 4207 V Fw(z)s Fv(\))2850 4222 y Fo(\003)2889 4252 y Fw(e)2928 4264 y Ft(a)2967 4252 y Fv(\)\()p Fw(x)3078 4264 y Ft(j)3115 4252 y Fv(\))3262 3735 y Fs(1)3262 3881 y(C)3262 3931 y(C)3262 3981 y(C)3262 4031 y(C)3262 4080 y(C)3262 4133 y(A)3348 4026 y Fw(;)109 4446 y FO(where)27 b Fw(e)388 4458 y Ft(a)427 4446 y Fw(;)14 b(e)503 4458 y Ft(j)538 4446 y Fw(;)g(e)614 4458 y Ft(b)670 4446 y Fu(2)23 b Fq(C)802 4416 y Ft(N)6 b Fr(+2)983 4446 y FO(are)27 b(giv)n(en)f(b)n(y)1045 4889 y Fw(e)1084 4901 y Ft(b)1140 4889 y Fv(=)1228 4648 y Fs(0)1228 4794 y(B)1228 4844 y(B)1228 4893 y(B)1228 4947 y(@)1342 4711 y Fv(1)1342 4811 y(0)1351 4899 y FO(.)1351 4933 y(.)1351 4966 y(.)1342 5066 y Fv(0)1425 4648 y Fs(1)1425 4794 y(C)1425 4844 y(C)1425 4893 y(C)1425 4947 y(A)1511 4889 y Fw(;)97 b(e)1670 4901 y Ft(a)1733 4889 y Fv(=)1821 4648 y Fs(0)1821 4794 y(B)1821 4844 y(B)1821 4893 y(B)1821 4947 y(@)1935 4711 y Fv(0)1944 4800 y FO(.)1944 4833 y(.)1944 4866 y(.)1935 4966 y Fv(0)1935 5066 y(1)2018 4648 y Fs(1)2018 4794 y(C)2018 4844 y(C)2018 4893 y(C)2018 4947 y(A)2105 4889 y Fw(;)f(e)2263 4901 y Ft(j)2321 4889 y Fv(=)2409 4573 y Fs(0)2409 4719 y(B)2409 4769 y(B)2409 4819 y(B)2409 4869 y(B)2409 4918 y(B)2409 4968 y(B)2409 5021 y(@)2523 4634 y Fv(0)2532 4722 y FO(.)2532 4756 y(.)2532 4789 y(.)2523 4888 y Fv(1)2532 4977 y FO(.)2532 5010 y(.)2532 5043 y(.)2523 5143 y Fv(0)2606 4573 y Fs(1)2606 4719 y(C)2606 4769 y(C)2606 4819 y(C)2606 4869 y(C)2606 4918 y(C)2606 4968 y(C)2606 5021 y(A)2692 4889 y Fw(:)109 5377 y FO(Since)27 b Fw(h)c Fu(2)h Fp(D)p Fv(\()p Fw(H)7 b Fv(\))p FO(,)1053 5480 y Fw(K)1130 5446 y Ft(a)1124 5501 y Fo(\000)1179 5480 y Fw(h)1227 5492 y Ft(a)1290 5480 y Fv(=)23 b Fw(K)1455 5441 y Ft(j)1449 5501 y Fo(\000)1505 5480 y Fw(h)1553 5492 y Ft(j)1611 5480 y Fv(=)f Fw(K)1775 5446 y Ft(b)1769 5501 y Fo(\000)1825 5480 y Fw(h)1873 5492 y Ft(b)1929 5480 y Fv(=)h(0)p Fw(;)96 b Fu(8)p Fw(j)27 b Fv(=)c(1)p Fw(;)14 b Fu(\001)g(\001)g(\001)27 b Fw(;)14 b(N)t(:)773 b FO(\(4.8\))109 5634 y(A)28 b(straigh)n(tforw)n(ard)c (calculation)j(sho)n(ws,)f(that)629 5820 y Fw(K)706 5785 y Ft(a)700 5840 y Fo(\000)756 5820 y Fw(G)821 5832 y Ft(a)861 5820 y Fw(\030)h Fv(=)c Fw(\030)1052 5785 y Ft(a)1092 5820 y Fw(e)1131 5832 y Ft(a)1171 5820 y Fw(;)97 b(K)1368 5785 y Ft(b)1362 5840 y Fo(\000)1418 5820 y Fw(G)1483 5832 y Ft(b)1516 5820 y Fw(\030)27 b Fv(=)c Fw(\030)1707 5785 y Ft(b)1740 5820 y Fw(e)1779 5832 y Ft(b)1812 5820 y Fw(;)97 b(K)2009 5780 y Ft(j)2003 5840 y Fo(\000)2059 5820 y Fw(G)2124 5832 y Ft(j)2159 5820 y Fw(\030)27 b Fv(=)c Fw(\030)2350 5785 y Ft(j)2385 5820 y Fw(e)2424 5832 y Ft(j)2459 5820 y Fw(;)120 b Fu(8)p Fw(j)27 b Fv(=)22 b(1)p Fw(;)14 b Fu(\001)g(\001)g(\001)27 b Fw(;)14 b(N)t(;)p eop %%Page: 11 13 11 12 bop 109 254 a FH(Dissipativ)n(e)27 b(Sc)n(hr\366)r(dinger-t)n(yp) r(e)e(op)r(erator)2084 b FO(11)109 503 y(where)27 b Fw(\030)g Fu(2)c Fq(C)544 472 y Ft(N)6 b Fr(+2)725 503 y FO(\(see)27 b(\(3.2\)\).)37 b(Therefore)27 b(w)n(e)g(get)g(b)n(y)h(\(4.7\))g(and)f (\(4.8\))1241 741 y Fw(K)1318 707 y Ft(a)1312 762 y Fo(\000)1367 741 y Fw(g)1407 753 y Ft(a)1466 741 y Fv(+)1579 638 y Ft(N)1549 663 y Fs(X)1551 839 y Ft(j)s Fr(=1)1682 741 y Fw(K)1759 701 y Ft(j)1753 762 y Fo(\000)1809 741 y Fw(g)1849 753 y Ft(j)1902 741 y Fv(+)18 b Fw(K)2062 707 y Ft(b)2056 762 y Fo(\000)2111 741 y Fw(g)2151 753 y Ft(b)2207 741 y Fv(=)23 b Fw(g)2335 753 y Fo(\000)2391 741 y Fv(\(0\))p Fw(:)109 987 y FO(Using)28 b(\(4.6\))f(w)n(e)h(get) 1303 1087 y Fw(g)1343 1099 y Fr(+)1397 1087 y Fv(\(0\))23 b(=)g Fw(iT)12 b Fv(\()p Fw(z)t Fv(\))p Fw(f)26 b Fv(+)18 b(\002)2026 1099 y Ft(H)2088 1087 y Fv(\()p 2120 1041 43 4 v Fw(z)t Fv(\))2195 1053 y Fo(\003)2233 1087 y Fw(g)2273 1099 y Fo(\000)2329 1087 y Fv(\(0\))p Fw(:)109 1221 y FO(Since)382 1379 y Fw(K)459 1345 y Ft(a)453 1399 y Fr(+)507 1379 y Fw(h)555 1391 y Ft(a)619 1379 y Fv(=)k Fu(\000)p Fw(i\013)853 1391 y Ft(a)893 1379 y Fw(h)p Fv(\()p Fw(a)p Fv(\))p Fw(e)1088 1391 y Ft(a)1128 1379 y Fw(;)97 b(K)1325 1345 y Ft(b)1319 1399 y Fr(+)1373 1379 y Fw(h)1421 1391 y Ft(b)1477 1379 y Fv(=)23 b Fw(i\013)1647 1391 y Ft(b)1680 1379 y Fw(h)p Fv(\()p Fw(b)p Fv(\))p Fw(e)1867 1391 y Ft(b)1900 1379 y Fw(;)97 b(K)2097 1339 y Ft(j)2091 1400 y Fr(+)2146 1379 y Fw(h)2194 1391 y Ft(j)2252 1379 y Fv(=)22 b Fu(\000)p Fw(i\013)2486 1391 y Ft(j)2521 1379 y Fw(h)p Fv(\()p Fw(x)2648 1391 y Ft(j)2683 1379 y Fv(\))p Fw(e)2754 1391 y Ft(j)2789 1379 y Fw(;)37 b Fu(8)p Fw(j)27 b Fv(=)c(1)p Fw(;)14 b Fu(\001)g(\001)g(\001)27 b Fw(;)14 b(N)t(;)109 1537 y FO(w)n(e)27 b(\034nd)h(b)n(y)f(equation)g(\(3.1\)) 1210 1717 y Fw(K)1287 1683 y Ft(a)1281 1738 y Fr(+)1336 1717 y Fw(h)1384 1729 y Ft(a)1442 1717 y Fv(+)1556 1614 y Ft(N)1526 1638 y Fs(X)1528 1815 y Ft(j)s Fr(=1)1659 1717 y Fw(K)1736 1677 y Ft(j)1730 1738 y Fr(+)1785 1717 y Fw(h)1833 1729 y Ft(j)1886 1717 y Fv(+)18 b Fw(K)2046 1683 y Ft(b)2040 1738 y Fr(+)2095 1717 y Fw(h)2143 1729 y Ft(b)2199 1717 y Fv(=)23 b Fw(iT)12 b Fv(\()p Fw(z)t Fv(\))p Fw(f)t(:)109 1935 y FO(Note)27 b(that)1449 2035 y Fv(\002)1514 2047 y Ft(H)1576 2035 y Fv(\()p 1608 1989 V Fw(z)t Fv(\))1683 2001 y Fo(\003)1744 2035 y Fv(=)c(1)18 b(+)g Fw(i\013T)2106 2047 y Fo(\003)2144 2035 y Fv(\()p 2176 1989 V Fw(z)s Fv(\))2250 2001 y Fo(\003)2289 2035 y Fw(:)1169 b FO(\(4.9\))109 2169 y(W)-7 b(e)28 b(ha)n(v)n(e)627 2483 y Fw(K)704 2449 y Ft(a)698 2504 y Fr(+)753 2483 y Fw(G)818 2495 y Ft(a)881 2483 y Fv(=)969 2241 y Fs(0)969 2388 y(B)969 2437 y(B)969 2487 y(B)969 2540 y(@)1394 2305 y Fv(0)393 b Fu(\001)14 b(\001)g(\001)450 b Fv(0)1403 2394 y FO(.)1403 2427 y(.)1403 2460 y(.)2386 2394 y(.)2386 2427 y(.)2386 2460 y(.)1394 2560 y Fv(0)393 b Fu(\001)14 b(\001)g(\001)450 b Fv(0)1083 2659 y Fu(\000)p Fw(i\013)1230 2671 y Ft(a)1270 2659 y Fv(\()p Fw(T)1351 2671 y Fo(\003)1389 2659 y Fv(\()p 1421 2614 V Fw(z)s Fv(\))1495 2629 y Fo(\003)1534 2659 y Fw(e)1573 2671 y Ft(b)1606 2659 y Fv(\)\()p Fw(a)p Fv(\))83 b Fu(\001)14 b(\001)g(\001)97 b Fv(1)18 b Fu(\000)g Fw(i\013)2248 2671 y Ft(a)2288 2659 y Fv(\()p Fw(T)2369 2671 y Fo(\003)2407 2659 y Fv(\()p 2439 2614 V Fw(z)s Fv(\))2513 2629 y Fo(\003)2552 2659 y Fw(e)2591 2671 y Ft(a)2631 2659 y Fv(\)\()p Fw(a)p Fv(\))2813 2241 y Fs(1)2813 2388 y(C)2813 2437 y(C)2813 2487 y(C)2813 2540 y(A)881 2970 y Fv(=)969 2728 y Fs(0)969 2875 y(B)969 2924 y(B)969 2974 y(B)969 3027 y(@)1431 2792 y Fv(0)430 b Fu(\001)14 b(\001)g(\001)519 b Fv(0)1440 2881 y FO(.)1440 2914 y(.)1440 2947 y(.)2528 2881 y(.)2528 2914 y(.)2528 2947 y(.)1431 3047 y Fv(0)430 b Fu(\001)14 b(\001)g(\001)519 b Fv(0)1083 3146 y Fw(i)p Fu(h)p Fw(\013T)1246 3158 y Fo(\003)1284 3146 y Fv(\()p 1316 3101 V Fw(z)t Fv(\))1391 3116 y Fo(\003)1429 3146 y Fw(e)1468 3158 y Ft(b)1501 3146 y Fw(;)14 b(e)1577 3158 y Ft(a)1617 3146 y Fu(i)1649 3163 y Fk(C)1685 3146 y Fm(N)5 b Fl(+2)97 b Fu(\001)14 b(\001)g(\001)96 b Fv(1)18 b(+)g Fw(i)p Fu(h)p Fw(\013T)2402 3158 y Fo(\003)2440 3146 y Fv(\()p 2472 3101 V Fw(z)t Fv(\))2547 3116 y Fo(\003)2585 3146 y Fw(e)2624 3158 y Ft(a)2664 3146 y Fw(;)c(e)2740 3158 y Ft(a)2780 3146 y Fu(i)2812 3163 y Fk(C)2848 3146 y Fm(N)5 b Fl(+2)3024 2728 y Fs(1)3024 2875 y(C)3024 2924 y(C)3024 2974 y(C)3024 3027 y(A)3110 2970 y Fw(;)109 3300 y FO(and)27 b(similar)637 3613 y Fw(K)714 3579 y Ft(b)708 3634 y Fr(+)763 3613 y Fw(G)828 3625 y Ft(b)885 3613 y Fv(=)972 3372 y Fs(0)972 3518 y(B)972 3568 y(B)972 3618 y(B)972 3671 y(@)1087 3435 y Fv(1)17 b(+)i Fw(i)p Fu(h)p Fw(\013T)1393 3447 y Fo(\003)1430 3435 y Fv(\()p 1462 3390 V Fw(z)t Fv(\))1537 3405 y Fo(\003)1575 3435 y Fw(e)1614 3447 y Ft(b)1647 3435 y Fw(;)14 b(e)1723 3447 y Ft(b)1756 3435 y Fu(i)1788 3452 y Fk(C)1825 3435 y Fm(N)5 b Fl(+2)96 b Fu(\001)14 b(\001)g(\001)97 b Fw(i)p Fu(h)p Fw(\013T)2399 3447 y Fo(\003)2437 3435 y Fv(\()p 2469 3390 V Fw(z)s Fv(\))2543 3405 y Fo(\003)2582 3435 y Fw(e)2621 3447 y Ft(a)2661 3435 y Fw(;)14 b(e)2737 3447 y Ft(b)2770 3435 y Fu(i)2802 3452 y Fk(C)2838 3435 y Fm(N)5 b Fl(+2)1502 3535 y Fv(0)498 b Fu(\001)14 b(\001)g(\001)444 b Fv(0)1511 3624 y FO(.)1511 3657 y(.)1511 3690 y(.)2592 3624 y(.)2592 3657 y(.)2592 3690 y(.)1502 3790 y Fv(0)498 b Fu(\001)14 b(\001)g(\001)444 b Fv(0)3014 3372 y Fs(1)3014 3518 y(C)3014 3568 y(C)3014 3618 y(C)3014 3671 y(A)3100 3613 y Fw(;)130 4310 y(K)207 4271 y Ft(j)201 4331 y Fr(+)256 4310 y Fw(G)321 4322 y Ft(j)379 4310 y Fv(=)467 3895 y Fs(0)467 4041 y(B)467 4090 y(B)467 4140 y(B)467 4190 y(B)467 4240 y(B)467 4290 y(B)467 4340 y(B)467 4389 y(B)467 4439 y(B)467 4489 y(B)467 4542 y(@)926 3955 y Fv(0)427 b Fu(\001)14 b(\001)g(\001)514 b Fv(0)500 b Fu(\001)14 b(\001)g(\001)445 b Fv(0)935 4044 y FO(.)935 4077 y(.)935 4110 y(.)2015 4044 y(.)2015 4077 y(.)2015 4110 y(.)3099 4044 y(.)3099 4077 y(.)3099 4110 y(.)926 4210 y Fv(0)427 b Fu(\001)14 b(\001)g(\001)514 b Fv(0)500 b Fu(\001)14 b(\001)g(\001)445 b Fv(0)581 4310 y Fw(i)p Fu(h)p Fw(\013T)744 4322 y Fo(\003)782 4310 y Fv(\()p 814 4264 V Fw(z)s Fv(\))888 4279 y Fo(\003)927 4310 y Fw(e)966 4322 y Ft(b)999 4310 y Fw(;)14 b(e)1075 4322 y Ft(j)1109 4310 y Fu(i)1141 4326 y Fk(C)1178 4309 y Fm(N)5 b Fl(+2)1395 4310 y Fu(\001)14 b(\001)g(\001)97 b Fv(1)18 b(+)g Fw(i)p Fu(h)p Fw(\013T)1895 4322 y Fo(\003)1933 4310 y Fv(\()p 1965 4264 V Fw(z)s Fv(\))2039 4279 y Fo(\003)2078 4310 y Fw(e)2117 4322 y Ft(j)2151 4310 y Fw(;)c(e)2227 4322 y Ft(j)2262 4310 y Fu(i)2294 4326 y Fk(C)2330 4309 y Fm(N)5 b Fl(+2)2548 4310 y Fu(\001)14 b(\001)g(\001)96 b Fw(i)p Fu(h)p Fw(\013T)2904 4322 y Fo(\003)2942 4310 y Fv(\()p 2974 4264 V Fw(z)t Fv(\))3049 4279 y Fo(\003)3087 4310 y Fw(e)3126 4322 y Ft(a)3166 4310 y Fw(;)14 b(e)3242 4322 y Ft(j)3277 4310 y Fu(i)3309 4326 y Fk(C)3345 4309 y Fm(N)5 b Fl(+2)926 4409 y Fv(0)427 b Fu(\001)14 b(\001)g(\001)514 b Fv(0)500 b Fu(\001)14 b(\001)g(\001)445 b Fv(0)935 4498 y FO(.)935 4531 y(.)935 4564 y(.)2015 4498 y(.)2015 4531 y(.)2015 4564 y(.)3099 4498 y(.)3099 4531 y(.)3099 4564 y(.)926 4664 y Fv(0)427 b Fu(\001)14 b(\001)g(\001)514 b Fv(0)500 b Fu(\001)14 b(\001)g(\001)445 b Fv(0)3521 3895 y Fs(1)3521 4041 y(C)3521 4090 y(C)3521 4140 y(C)3521 4190 y(C)3521 4240 y(C)3521 4290 y(C)3521 4340 y(C)3521 4389 y(C)3521 4439 y(C)3521 4489 y(C)3521 4542 y(A)3607 4310 y Fw(;)109 4799 y FO(for)27 b(all)g Fw(j)h Fv(=)23 b(1)p Fw(;)14 b Fu(\001)g(\001)g(\001)27 b Fw(;)14 b(N)9 b FO(.)36 b(Hence)28 b(w)n(e)f(get)g(b)n(y)i(\(4.9\))1180 5038 y Fw(K)1257 5003 y Ft(a)1251 5058 y Fr(+)1305 5038 y Fw(G)1370 5050 y Ft(a)1429 5038 y Fv(+)1543 4934 y Ft(N)1512 4959 y Fs(X)1515 5136 y Ft(j)s Fr(=1)1646 5038 y Fw(K)1717 5050 y Fr(+)1772 5038 y Fw(G)1837 5050 y Ft(j)1890 5038 y Fv(+)18 b Fw(K)2044 5050 y Fr(+)2099 5038 y Fw(G)2164 5050 y Ft(b)2221 5038 y Fv(=)k(\002)2373 5050 y Ft(H)2436 5038 y Fv(\()p 2468 4992 V Fw(z)s Fv(\))2542 5003 y Fo(\003)109 5279 y FO(and)27 b(\034nally)1254 5455 y Fw(K)1325 5467 y Fr(+)1380 5455 y Fw(g)1420 5467 y Ft(a)1478 5455 y Fv(+)1592 5351 y Ft(N)1561 5376 y Fs(X)1564 5553 y Ft(j)s Fr(=1)1695 5455 y Fw(K)1766 5467 y Fr(+)1820 5455 y Fw(g)1860 5467 y Ft(j)1913 5455 y Fv(+)18 b Fw(K)2073 5421 y Ft(b)2067 5475 y Fr(+)2122 5455 y Fw(g)2162 5467 y Ft(b)2218 5455 y Fv(=)23 b Fw(g)2346 5467 y Fr(+)2400 5455 y Fv(\(0\))109 5673 y FO(whic)n(h)k(completes)g (the)h(pro)r(of.)2502 b Fj(\003)109 5820 y FO(Using)27 b(the)h(expressions)e(\(4.4\))h(and)g(\(4.5\))h(for)f(the)h(resolv)n (en)n(t)d(of)j Fw(K)6 b FO(,)27 b(one)g(easily)g(v)n(eri\034es)f (\(4.1\))i(and)f(\(4.2\).)p eop %%Page: 12 14 12 13 bop 109 254 a FO(12)2477 b Fx(M.)31 b(Bar)n(o,)f(H.)h(Neidhardt) 109 503 y Fz(5)131 b(Eigenfunction)45 b(expansion)109 732 y FO(The)31 b(selfadjoin)n(t)g(op)r(erator)f Fw(K)37 b FO(is)31 b(absolutely)g(con)n(tin)n(uous)f(and)h Fw(\033)s Fv(\()p Fw(K)6 b Fv(\))30 b(=)f Fw(\033)2565 744 y Ft(ac)2636 732 y Fv(\()p Fw(K)6 b Fv(\))29 b(=)g Fq(R)p FO(,)39 b(i.e.)48 b(the)32 b(sp)r(ectrum)109 831 y(of)i Fw(K)40 b FO(coincides)34 b(with)h(the)f(real)g(line.)57 b(Its)35 b(m)n(ultiplicit)n(y)f(is)h Fw(N)d Fv(+)22 b(2)p FO(.)57 b(W)-7 b(e)35 b(are)e(no)n(w)h(going)f(to)h(compute)h(the)109 931 y(generalized)26 b(eigenfunction)1052 909 y Fw(~)1047 931 y(\036)q Fv(\()p Fu(\001)p Fw(;)14 b(\025)p Fv(\))p FO(,)28 b Fw(\025)c Fu(2)f Fq(R)p FO(,)34 b(of)28 b Fw(K)6 b FO(.)36 b(W)-7 b(e)28 b(set)1167 1091 y Fw(~)1163 1113 y(\036)p Fv(\()p Fw(x;)14 b(\025)p Fv(\))25 b(=)d Fw(\036)1569 1125 y Fo(\000)1626 1113 y Fv(\()p Fw(x;)14 b(\025)p Fv(\))20 b Fu(\010)e Fw(\036)p Fv(\()p Fw(x;)c(\025)p Fv(\))20 b Fu(\010)e Fw(\036)2322 1125 y Fr(+)2377 1113 y Fv(\()p Fw(x;)c(\025)p Fv(\))p Fw(;)109 1283 y FO(for)27 b Fw(x)c Fu(2)394 1263 y Fv(^)385 1283 y(\012)p FO(,)k(where)459 1677 y Fw(\036)508 1689 y Fo(\000)565 1677 y Fv(\()p Fw(x;)14 b(\025)p Fv(\))24 b(=)873 1386 y Fs(0)873 1532 y(B)873 1582 y(B)873 1631 y(B)873 1681 y(B)873 1731 y(B)873 1784 y(@)991 1450 y Fw(\036)1040 1420 y Ft(b)1040 1470 y Fo(\000)1096 1450 y Fv(\()p Fw(x;)14 b(\025)p Fv(\))987 1549 y Fw(\036)1036 1519 y Ft(N)1036 1570 y Fo(\000)1100 1549 y Fv(\()p Fw(x;)g(\025)p Fv(\))1131 1638 y FO(.)1131 1671 y(.)1131 1704 y(.)991 1804 y Fw(\036)1040 1774 y Fr(1)1040 1825 y Fo(\000)1096 1804 y Fv(\()p Fw(x;)g(\025)p Fv(\))991 1904 y Fw(\036)1040 1873 y Ft(a)1040 1924 y Fo(\000)1096 1904 y Fv(\()p Fw(x;)g(\025)p Fv(\))1338 1386 y Fs(1)1338 1532 y(C)1338 1582 y(C)1338 1631 y(C)1338 1681 y(C)1338 1731 y(C)1338 1784 y(A)1425 1677 y Fw(;)97 b(x)23 b Fu(2)h Fq(R)1748 1689 y Fo(\000)1810 1677 y Fw(;)97 b(\036)1979 1689 y Fr(+)2034 1677 y Fv(\()p Fw(x;)14 b(\025)p Fv(\))25 b(=)2342 1386 y Fs(0)2342 1532 y(B)2342 1582 y(B)2342 1631 y(B)2342 1681 y(B)2342 1731 y(B)2342 1784 y(@)2460 1450 y Fw(\036)2509 1420 y Ft(b)2509 1470 y Fr(+)2565 1450 y Fv(\()p Fw(x;)14 b(\025)p Fv(\))2456 1549 y Fw(\036)2505 1519 y Ft(N)2505 1570 y Fr(+)2569 1549 y Fv(\()p Fw(x;)g(\025)p Fv(\))2600 1638 y FO(.)2600 1671 y(.)2600 1704 y(.)2460 1804 y Fw(\036)2509 1774 y Fr(1)2509 1825 y(+)2565 1804 y Fv(\()p Fw(x;)g(\025)p Fv(\))2460 1904 y Fw(\036)2509 1873 y Ft(a)2509 1924 y Fr(+)2565 1904 y Fv(\()p Fw(x;)g(\025)p Fv(\))2808 1386 y Fs(1)2808 1532 y(C)2808 1582 y(C)2808 1631 y(C)2808 1681 y(C)2808 1731 y(C)2808 1784 y(A)2894 1677 y Fw(;)97 b(x)24 b Fu(2)f Fq(R)3217 1689 y Fr(+)3278 1677 y Fw(:)109 2118 y FO(F)-7 b(rom)811 2311 y Fv(\()p Fw(K)924 2289 y(~)920 2311 y(\036)p Fv(\)\()p Fw(x;)14 b(\025)p Fv(\))25 b(=)d Fu(\000)p Fw(i)1436 2255 y(d)p 1413 2292 91 4 v 1413 2368 a(dx)1513 2311 y(\036)1562 2323 y Fo(\000)1619 2311 y Fv(\()p Fw(x;)14 b(\025)p Fv(\))19 b Fu(\010)f Fw(l)r Fv(\()p Fw(\036)p Fv(\()p Fu(\001)p Fw(;)c(\025)p Fv(\)\)\()p Fw(x)p Fv(\))21 b Fu(\010)d(\000)p Fw(i)2571 2255 y(d)p 2548 2292 V 2548 2368 a(dx)2648 2311 y(\036)2697 2323 y Fr(+)2753 2311 y Fv(\()p Fw(x;)c(\025)p Fv(\))1222 2463 y(=)22 b Fw(\025)p Fv(\()p Fw(\036)1438 2475 y Fo(\000)1495 2463 y Fv(\()p Fw(x;)14 b(\025)p Fv(\))20 b Fu(\010)e Fw(\036)p Fv(\()p Fw(x;)c(\025)p Fv(\))21 b Fu(\010)d Fw(\036)2192 2475 y Fr(+)2247 2463 y Fv(\()p Fw(x;)c(\025)p Fv(\)\))p Fw(;)109 2645 y(x)23 b Fu(2)267 2624 y Fv(^)258 2645 y(\012)p FO(,)k(w)n(e)g(\034nd)h(the)g(equations)936 3039 y Fu(\000)p Fw(i)1063 2983 y(d)p 1040 3020 V 1040 3096 a(dx)1154 2748 y Fs(0)1154 2894 y(B)1154 2944 y(B)1154 2993 y(B)1154 3043 y(B)1154 3093 y(B)1154 3146 y(@)1271 2812 y Fw(\036)1320 2781 y Ft(b)1320 2832 y Fo(\000)1377 2812 y Fv(\()p Fw(x;)14 b(\025)p Fv(\))1268 2911 y Fw(\036)1317 2881 y Ft(N)1317 2932 y Fo(\000)1380 2911 y Fv(\()p Fw(x;)g(\025)p Fv(\))1411 3000 y FO(.)1411 3033 y(.)1411 3066 y(.)1271 3166 y Fw(\036)1320 3136 y Fr(1)1320 3186 y Fo(\000)1377 3166 y Fv(\()p Fw(x;)g(\025)p Fv(\))1271 3265 y Fw(\036)1320 3235 y Ft(a)1320 3286 y Fo(\000)1377 3265 y Fv(\()p Fw(x;)g(\025)p Fv(\))1619 2748 y Fs(1)1619 2894 y(C)1619 2944 y(C)1619 2993 y(C)1619 3043 y(C)1619 3093 y(C)1619 3146 y(A)1715 3039 y Fv(=)22 b Fw(\025)1864 2748 y Fs(0)1864 2894 y(B)1864 2944 y(B)1864 2993 y(B)1864 3043 y(B)1864 3093 y(B)1864 3146 y(@)1982 2812 y Fw(\036)2031 2781 y Ft(b)2031 2832 y Fo(\000)2088 2812 y Fv(\()p Fw(x;)14 b(\025)p Fv(\))1979 2911 y Fw(\036)2028 2881 y Ft(N)2028 2932 y Fo(\000)2091 2911 y Fv(\()p Fw(x;)g(\025)p Fv(\))2122 3000 y FO(.)2122 3033 y(.)2122 3066 y(.)1982 3166 y Fw(\036)2031 3136 y Fr(1)2031 3186 y Fo(\000)2088 3166 y Fv(\()p Fw(x;)g(\025)p Fv(\))1982 3265 y Fw(\036)2031 3235 y Ft(a)2031 3286 y Fo(\000)2088 3265 y Fv(\()p Fw(x;)g(\025)p Fv(\))2330 2748 y Fs(1)2330 2894 y(C)2330 2944 y(C)2330 2993 y(C)2330 3043 y(C)2330 3093 y(C)2330 3146 y(A)2417 3039 y Fw(;)97 b(x)23 b Fu(2)g Fq(R)2739 3051 y Fo(\000)2801 3039 y Fw(;)657 b FO(\(5.1\))1237 3437 y Fw(l)r Fv(\()p Fw(\036)p Fv(\()p Fu(\001)p Fw(;)14 b(\025)p Fv(\)\)\()p Fw(x)p Fv(\))26 b(=)d Fw(\025\036)p Fv(\()p Fw(x;)14 b(\025)p Fv(\))p Fw(;)99 b(x)23 b Fu(2)h Fv([)p Fw(a;)14 b(b)p Fv(])p Fw(;)957 b FO(\(5.2\))109 3580 y(and)937 3886 y Fu(\000)p Fw(i)1063 3830 y(d)p 1041 3867 V 1041 3943 a(dx)1154 3594 y Fs(0)1154 3741 y(B)1154 3790 y(B)1154 3840 y(B)1154 3890 y(B)1154 3940 y(B)1154 3993 y(@)1272 3658 y Fw(\036)1321 3628 y Ft(b)1321 3679 y Fr(+)1377 3658 y Fv(\()p Fw(x;)14 b(\025)p Fv(\))1268 3758 y Fw(\036)1317 3728 y Ft(N)1317 3779 y Fr(+)1381 3758 y Fv(\()p Fw(x;)g(\025)p Fv(\))1412 3847 y FO(.)1412 3880 y(.)1412 3913 y(.)1272 4013 y Fw(\036)1321 3983 y Fr(1)1321 4033 y(+)1377 4013 y Fv(\()p Fw(x;)g(\025)p Fv(\))1272 4112 y Fw(\036)1321 4082 y Ft(a)1321 4133 y Fr(+)1377 4112 y Fv(\()p Fw(x;)g(\025)p Fv(\))1620 3594 y Fs(1)1620 3741 y(C)1620 3790 y(C)1620 3840 y(C)1620 3890 y(C)1620 3940 y(C)1620 3993 y(A)1715 3886 y Fv(=)23 b Fw(\025)1865 3594 y Fs(0)1865 3741 y(B)1865 3790 y(B)1865 3840 y(B)1865 3890 y(B)1865 3940 y(B)1865 3993 y(@)1983 3658 y Fw(\036)2032 3628 y Ft(b)2032 3679 y Fo(\000)2088 3658 y Fv(\()p Fw(x;)14 b(\025)p Fv(\))1979 3758 y Fw(\036)2028 3728 y Ft(N)2028 3779 y Fo(\000)2092 3758 y Fv(\()p Fw(x;)g(\025)p Fv(\))2123 3847 y FO(.)2123 3880 y(.)2123 3913 y(.)1983 4013 y Fw(\036)2032 3983 y Fr(1)2032 4033 y Fo(\000)2088 4013 y Fv(\()p Fw(x;)g(\025)p Fv(\))1983 4112 y Fw(\036)2032 4082 y Ft(a)2032 4133 y Fo(\000)2088 4112 y Fv(\()p Fw(x;)g(\025)p Fv(\))2331 3594 y Fs(1)2331 3741 y(C)2331 3790 y(C)2331 3840 y(C)2331 3890 y(C)2331 3940 y(C)2331 3993 y(A)2417 3886 y Fw(;)97 b(x)24 b Fu(2)f Fq(R)2740 3898 y Fr(+)2801 3886 y Fw(:)657 b FO(\(5.3\))109 4256 y(The)27 b(equation)g(\(5.1\))g(and)h(\(5.3\))f (ha)n(v)n(e)g(the)g(solutions)1098 4427 y Fw(\036)1147 4439 y Fo(\000)1204 4427 y Fv(\()p Fw(x;)14 b(\025)p Fv(\))24 b(=)f Fw(C)1571 4439 y Fo(\000)1627 4427 y Fw(e)1666 4392 y Ft(ix\025)1771 4427 y Fw(;)97 b(C)1950 4439 y Fo(\000)2029 4427 y Fu(2)23 b Fq(C)2161 4392 y Ft(N)6 b Fr(+2)2314 4427 y Fw(;)37 b(x)24 b Fu(2)f Fq(R)2577 4439 y Fo(\000)2639 4427 y Fw(;)109 4598 y FO(and)1100 4697 y Fw(\036)1149 4709 y Fr(+)1205 4697 y Fv(\()p Fw(x;)14 b(\025)p Fv(\))24 b(=)f Fw(C)1572 4709 y Fr(+)1627 4697 y Fw(e)1666 4663 y Ft(ix\025)1771 4697 y Fw(;)97 b(C)1950 4709 y Fr(+)2028 4697 y Fu(2)23 b Fq(C)2160 4663 y Ft(N)6 b Fr(+2)2313 4697 y Fw(;)37 b(x)24 b Fu(2)f Fq(R)2576 4709 y Fr(+)2637 4697 y Fw(:)109 4840 y FO(The)36 b(solution)f(of)42 b(\(5.2\))35 b(is)h(giv)n(en)f(b)n(y)h(a)f(linear)g(com)n(bination)g (of)h(the)g(elemen)n(tary)f(solutions)g Fw(v)3245 4852 y Ft(a)3285 4840 y Fv(\()p Fw(x;)14 b(\025)p Fv(\))37 b FO(and)109 4939 y Fw(v)149 4951 y Ft(b)182 4939 y Fv(\()p Fw(x;)14 b(\025)p Fv(\))29 b FO(on)f(eac)n(h)e(in)n(terv)-5 b(al)27 b Fv([)p Fw(x)1079 4951 y Ft(j)1115 4939 y Fw(;)14 b(x)1199 4951 y Ft(j)s Fr(+1)1318 4939 y Fv(\))p FO(,)28 b(i.e.)929 5194 y Fw(\036)p Fv(\()p Fw(x;)14 b(\025)p Fv(\))25 b(=)1317 5091 y Ft(N)1287 5115 y Fs(X)1289 5292 y Ft(j)s Fr(=0)1420 5102 y Fs(\020)1470 5194 y Fw(c)1506 5160 y Ft(j)1506 5215 y(a)1546 5194 y Fw(v)1586 5206 y Ft(a)1627 5194 y Fv(\()p Fw(x;)14 b(\025)p Fv(\))19 b(+)f Fw(c)1961 5154 y Ft(j)1961 5219 y(b)1996 5194 y Fw(v)2036 5206 y Ft(b)2070 5194 y Fv(\()p Fw(x;)c(\025)p Fv(\))p Fw(;)2303 5102 y Fs(\021)2368 5194 y Fw(\037)2420 5209 y Fr([)p Ft(x)2477 5217 y Fm(j)2507 5209 y Ft(;x)2565 5217 y Fm(j)r Fl(+1)2666 5209 y Fr(\))2696 5194 y Fv(\()p Fw(x)p Fv(\))p Fw(:)651 b FO(\(5.4\))109 5496 y(The)27 b(eigenfunctions)h(ha)n(v)n(e)e(to)h(satisfy)h(the)f(b)r(oundary)g (conditions,)g(i.e.)1009 5744 y Fw(K)1086 5709 y Ft(a)1080 5764 y Fo(\000)1135 5744 y Fw(\036)1184 5756 y Ft(a)1225 5744 y Fv(\()p Fw(\025)p Fv(\))19 b(+)1470 5640 y Ft(N)1440 5665 y Fs(X)1442 5841 y Ft(j)s Fr(=1)1573 5744 y Fw(K)1650 5704 y Ft(j)1644 5764 y Fo(\000)1700 5744 y Fw(\036)1749 5756 y Ft(j)1784 5744 y Fv(\()p Fw(\025)p Fv(\))h(+)e Fw(K)2076 5709 y Ft(b)2070 5764 y Fo(\000)2125 5744 y Fw(\036)2174 5756 y Ft(b)2208 5744 y Fv(\()p Fw(\025)p Fv(\))24 b(=)f Fw(\036)2481 5756 y Fo(\000)2537 5744 y Fv(\(0)p Fw(;)14 b(\025)p Fv(\))p Fw(:)730 b FO(\(5.5\))p eop %%Page: 13 15 13 14 bop 109 254 a FH(Dissipativ)n(e)27 b(Sc)n(hr\366)r(dinger-t)n(yp) r(e)e(op)r(erator)2084 b FO(13)109 503 y(and)1011 663 y Fw(K)1088 628 y Ft(a)1082 683 y Fr(+)1136 663 y Fw(\036)1185 675 y Ft(a)1226 663 y Fv(\()p Fw(\025)p Fv(\))19 b(+)1471 559 y Ft(N)1440 584 y Fs(X)1443 761 y Ft(j)s Fr(=1)1574 663 y Fw(K)1651 623 y Ft(j)1645 683 y Fr(+)1700 663 y Fw(\036)1749 675 y Ft(j)1784 663 y Fv(\()p Fw(\025)p Fv(\))h(+)e Fw(K)2076 628 y Ft(b)2070 683 y Fr(+)2124 663 y Fw(\036)2173 675 y Ft(b)2207 663 y Fv(\()p Fw(\025)p Fv(\))24 b(=)f Fw(\036)2480 675 y Fr(+)2535 663 y Fv(\(0)p Fw(;)14 b(\025)p Fv(\))p Fw(:)732 b FO(\(5.6\))109 888 y(F)-7 b(urthermore)26 b(the)i(condition)651 1059 y Fv(\()p Fw(c)719 1025 y Ft(j)s Fr(+1)719 1080 y Ft(a)857 1059 y Fu(\000)18 b Fw(c)976 1025 y Ft(j)976 1080 y(a)1016 1059 y Fv(\))p Fw(v)1088 1071 y Ft(a)1128 1059 y Fv(\()p Fw(x)1207 1071 y Ft(j)1243 1059 y Fw(;)c(\025)p Fv(\))19 b(+)f(\()p Fw(c)1530 1019 y Ft(j)s Fr(+1)1530 1084 y Ft(b)1668 1059 y Fu(\000)g Fw(c)1787 1019 y Ft(j)1787 1084 y(b)1822 1059 y Fv(\))p Fw(v)1894 1071 y Ft(b)1928 1059 y Fv(\()p Fw(x)2007 1071 y Ft(j)2043 1059 y Fw(;)c(\025)p Fv(\))23 b(=)g(0)p Fw(;)96 b Fu(8)p Fw(j)27 b Fv(=)c(0)p Fw(;)14 b Fu(\001)g(\001)g(\001)27 b Fw(;)14 b(N)27 b Fv(+)18 b(1)p Fw(;)109 1230 y FO(has)27 b(to)g(b)r(e)h(satis\034ed.)36 b(A)28 b(straigh)n(tforw)n(ard)d(calculation)h(sho)n(ws)h(that)669 1444 y Fw(c)705 1410 y Ft(N)6 b Fr(+1)705 1465 y Ft(a)875 1444 y Fv(=)22 b Fu(\000)1092 1388 y Fw(\013)1145 1400 y Ft(b)p 1037 1425 197 4 v 1037 1501 a Fw(W)12 b Fv(\()p Fw(z)t Fv(\))1244 1444 y Fw(;)i(C)1346 1410 y Ft(b)1340 1465 y Fo(\000)1396 1444 y Fw(;)97 b(c)1552 1410 y Ft(j)1552 1465 y(a)1615 1444 y Fv(=)23 b Fw(c)1739 1410 y Ft(j)s Fr(+1)1739 1465 y Ft(a)1876 1444 y Fv(+)1969 1388 y Fw(\013)2022 1400 y Ft(j)2057 1388 y Fw(v)2097 1400 y Ft(b)2131 1388 y Fv(\()p Fw(x)2210 1400 y Ft(j)2245 1388 y Fw(;)14 b(\025)p Fv(\))p 1969 1425 395 4 v 2065 1501 a Fw(W)e Fv(\()p Fw(\025)p Fv(\))2373 1444 y Fw(C)2438 1405 y Ft(j)2432 1465 y Fo(\000)2488 1444 y Fw(;)97 b(j)28 b Fv(=)23 b(0)p Fw(;)14 b Fu(\001)g(\001)g(\001)27 b Fw(;)14 b(N)t(;)389 b FO(\(5.7\))109 1659 y(as)27 b(w)n(ell)g(as)700 1786 y Fw(c)736 1751 y Fr(0)736 1806 y Ft(b)796 1786 y Fv(=)945 1729 y Fw(\013)998 1741 y Ft(a)p 894 1767 197 4 v 894 1843 a Fw(W)12 b Fv(\()p Fw(z)t Fv(\))1100 1786 y Fw(C)1165 1751 y Ft(a)1159 1806 y Fo(\000)1216 1786 y Fw(;)97 b(c)1372 1746 y Ft(j)1372 1811 y(b)1430 1786 y Fv(=)22 b Fw(c)1553 1746 y Ft(j)s Fr(+1)1553 1811 y Ft(b)1691 1786 y Fu(\000)1784 1729 y Fw(\013)1837 1741 y Ft(j)1872 1729 y Fw(v)1912 1741 y Ft(a)1952 1729 y Fv(\()p Fw(x)2031 1741 y Ft(j)2067 1729 y Fw(;)14 b(\025)p Fv(\))p 1784 1767 401 4 v 1883 1843 a Fw(W)e Fv(\()p Fw(\025)p Fv(\))2194 1786 y Fw(C)2259 1746 y Ft(j)2253 1806 y Fo(\000)2310 1786 y Fw(;)97 b(j)28 b Fv(=)22 b(1)p Fw(;)14 b Fu(\001)g(\001)g(\001)27 b Fw(;)14 b(N)27 b Fv(+)18 b(1)p Fw(:)420 b FO(\(5.8\))109 1977 y(Inserting)26 b(equations)h(\(5.7\))g(and)h(\(5.8\))f(in)h (\(5.4\))f(\034nally)h(yields)1425 2148 y Fw(\036)p Fv(\()p Fw(x;)14 b(\025)p Fv(\))25 b(=)e(\()p Fw(T)1864 2160 y Fo(\003)1902 2148 y Fv(\()p Fw(\025)p Fv(\))2014 2113 y Fo(\003)2053 2148 y Fw(C)2112 2160 y Fo(\000)2168 2148 y Fv(\)\()p Fw(x)p Fv(\))p Fw(:)1147 b FO(\(5.9\))109 2318 y(By)27 b(inserting)h(\(5.9\))f(in)h(\(5.6\))f(w)n(e)h(\034nd)1394 2489 y Fw(C)1453 2501 y Fr(+)1532 2489 y Fv(=)22 b(\002)1684 2501 y Ft(H)1747 2489 y Fv(\()p Fw(\025)p Fv(\))1859 2455 y Fo(\003)1898 2489 y Fw(C)1957 2501 y Fo(\000)2014 2489 y Fw(;)96 b(\025)24 b Fu(2)f Fq(R)p Fw(:)109 2660 y FO(Therefore)j(w)n(e)h(get)868 2809 y Fw(~)864 2831 y(\036)913 2797 y Ft(C)961 2805 y Fc(\000)1014 2831 y Fv(\()p Fw(x;)14 b(\025)p Fv(\))24 b(:=)f Fw(e)1384 2797 y Ft(ix\025)1488 2831 y Fw(C)1547 2843 y Fo(\000)1622 2831 y Fu(\010)18 b Fv(\()p Fw(T)1786 2843 y Fo(\003)1824 2831 y Fv(\()p Fw(\025)p Fv(\))1936 2797 y Fo(\003)1975 2831 y Fw(C)2034 2843 y Fo(\000)2091 2831 y Fv(\)\()p Fw(x)p Fv(\))i Fu(\010)e Fw(e)2376 2797 y Ft(ix\025)2480 2831 y Fv(\002)2545 2843 y Ft(H)2607 2831 y Fv(\()p Fw(\025)p Fv(\))2719 2797 y Fo(\003)2759 2831 y Fw(C)2818 2843 y Fo(\000)2874 2831 y Fw(;)109 3002 y(x)23 b Fu(2)267 2981 y Fv(^)258 3002 y(\012)p FO(,)k Fw(\025)d Fu(2)f Fq(R)p FO(.)109 3149 y(A)28 b(calculation)e(as)h(in)h([KNR01b)o(])g (sho)n(ws)e(that)861 3253 y Fs(\022)992 3313 y Fv(1)p 932 3350 161 4 v 932 3367 a Fu(p)p 1002 3367 92 4 v 1002 3436 a Fv(2)p Fw(\031)1108 3348 y(~)1103 3370 y(\036)1152 3335 y Ft(C)1200 3343 y Fc(\000)1254 3370 y Fv(\()p Fu(\001)p Fw(;)14 b(\025)p Fv(\))p Fw(;)1533 3313 y Fv(1)p 1474 3350 161 4 v 1474 3367 a Fu(p)p 1543 3367 92 4 v 69 x Fv(2)p Fw(\031)1649 3348 y(~)1644 3370 y(\036)1693 3335 y Ft(C)1745 3310 y Fc(0)1741 3352 y(\000)1795 3370 y Fv(\()p Fu(\001)p Fw(;)g(\025)1935 3335 y Fo(0)1959 3370 y Fv(\))1991 3253 y Fs(\023)2052 3453 y Fo(K)2130 3370 y Fv(=)23 b Fw(\016)s Fv(\()p Fw(\025)c Fu(\000)f Fw(\025)2488 3335 y Fo(0)2512 3370 y Fv(\))p Fu(h)p Fw(C)2635 3382 y Fo(\000)2692 3370 y Fw(;)c(C)2794 3335 y Fo(0)2788 3390 y(\000)2844 3370 y Fu(i)p Fw(:)109 3589 y FO(In)n(tro)r(ducing)26 b(the)i(notions)1058 3773 y Fw(~)1054 3795 y(\036)p Fv(\()p Fu(\001)p Fw(;)14 b(\025;)g(j)5 b Fv(\))24 b(:=)1555 3739 y(1)p 1496 3776 161 4 v 1496 3792 a Fu(p)p 1565 3792 92 4 v 69 x Fv(2)p Fw(\031)1671 3773 y(~)1667 3795 y(\036)1716 3760 y Ft(e)1747 3768 y Fm(j)1783 3795 y Fv(\()p Fu(\001)p Fw(;)14 b(\025)p Fv(\))p Fw(;)98 b(j)28 b Fv(=)22 b(0)p Fw(;)14 b Fu(\001)g(\001)g(\001)27 b Fw(;)14 b(N)27 b Fv(+)18 b(1)p Fw(;)733 b FO(\(5.10\))109 4009 y(where)27 b(w)n(e)g(ha)n(v)n(e)f(set)i Fw(e)831 4021 y Ft(a)894 4009 y Fv(=)22 b Fw(e)1020 4021 y Fr(0)1085 4009 y FO(and)27 b Fw(e)1285 4021 y Ft(b)1341 4009 y Fv(=)c Fw(e)1468 4021 y Ft(N)6 b Fr(+1)1615 4009 y FO(,)27 b(w)n(e)h(get)f(the)h(follo)n(wing)e(theorem.)109 4213 y Fy(Theorem)k(5.1)40 b FA(Assume)25 b Fw(V)42 b Fu(2)24 b Fp(L)1197 4182 y Fr(2)1234 4213 y Fv(\()p Fw(a;)14 b(b)p Fv(\))p FA(,)27 b Fv(Im\()p Fw(V)19 b Fv(\))24 b(=)e(0)p FA(,)27 b(and)f Fw(\024)2107 4225 y Ft(a)2147 4213 y Fw(;)14 b(\024)2232 4225 y Fr(1)2269 4213 y Fw(;)g Fu(\001)g(\001)g(\001)27 b Fw(;)14 b(\024)2515 4225 y Ft(N)2578 4213 y Fw(;)g(\024)2663 4225 y Ft(b)2719 4213 y Fu(2)23 b Fq(C)2851 4225 y Fr(+)2913 4213 y FA(.)37 b(Then)26 b(the)g(functions)1021 4394 y Fu(f)1068 4372 y Fw(~)1063 4394 y(\036)p Fv(\()p Fu(\001)p Fw(;)14 b(\025;)g(a)p Fv(\))p Fw(;)1407 4372 y(~)1402 4394 y(\036)q Fv(\()p Fu(\001)p Fw(;)g(\025;)g Fv(1\))p Fw(;)g Fu(\001)g(\001)g(\001)27 b Fw(;)1906 4372 y(~)1901 4394 y(\036)q Fv(\()p Fu(\001)p Fw(;)14 b(\025;)g(N)9 b Fv(\))p Fw(;)2278 4372 y(~)2273 4394 y(\036)q Fv(\()p Fu(\001)p Fw(;)14 b(\025;)g(b)p Fv(\))p Fu(g)2610 4406 y Ft(\025)p Fo(2)p Fk(R)109 4565 y FA(p)l(erform)30 b(a)h(c)l(omplete)f(orthonormal)h(system)f(of)g (gener)l(alize)l(d)h(eigenfunctions)g(of)f Fw(K)6 b FA(,)30 b(i.e.)515 4664 y Fs(\020)570 4734 y Fw(~)565 4756 y(\036)q Fv(\()p Fu(\001)p Fw(;)14 b(\025;)g(\034)9 b Fv(\))p Fw(;)911 4734 y(~)906 4756 y(\036)q Fv(\()p Fu(\001)p Fw(;)14 b(\025)1096 4722 y Fo(0)1120 4756 y Fw(;)g(\034)1202 4722 y Fo(0)1226 4756 y Fv(\))1258 4664 y Fs(\021)1308 4814 y Fo(K)1386 4756 y Fv(=)23 b Fw(\016)s Fv(\()p Fw(\025)c Fu(\000)f Fw(\025)1744 4722 y Fo(0)1768 4756 y Fv(\))p Fw(\016)1837 4768 y Ft(\034)7 b(\034)1913 4752 y Fc(0)1938 4756 y Fw(;)99 b(\025;)14 b(\025)2193 4722 y Fo(0)2240 4756 y Fu(2)23 b Fq(R)p Fw(;)105 b(\034)5 b(;)14 b(\034)2623 4722 y Fo(0)2670 4756 y Fv(=)23 b Fw(a;)14 b Fv(1)p Fw(;)g Fu(\001)g(\001)g(\001)27 b Fw(;)14 b(N)t(;)g(b;)109 4951 y FA(and)30 b(their)g(line)l(ar)h(sp)l(an,)f(i.e.)40 b(elements)29 b(of)i(the)f(form)1173 5160 y Fw(f)9 b Fv(\()p Fw(x)p Fv(\))24 b(=)1446 5047 y Fs(Z)1492 5235 y Fk(R)1687 5081 y Fs(X)1552 5260 y Ft(\034)7 b Fr(=)p Ft(a;)p Fr(1)p Ft(;)p Fo(\001\001\001)i Ft(;N)s(;b)1961 5138 y Fw(~)1957 5160 y(\036)p Fv(\()p Fu(\001)p Fw(;)14 b(\025;)g(\034)9 b Fv(\))p Fw(g)2303 5125 y Ft(\034)2346 5160 y Fv(\()p Fw(\025)p Fv(\))14 b Fw(d\025;)109 5416 y FA(wher)l(e)30 b Fw(g)386 5386 y Ft(\034)457 5416 y FA(,)p Fw(\034)j Fv(=)23 b Fw(a;)14 b Fv(1)p Fw(;)g Fu(\001)g(\001)g (\001)26 b Fw(;)14 b(N)t(;)g(b)p FA(,)30 b(ar)l(e)g(smo)l(oth)g (functions)g(with)g(c)l(omp)l(act)g(supp)l(ort,)h(is)f(dense)g(in)g Fu(K)q FA(.)109 5620 y FO(W)-7 b(e)41 b(sa)n(y)f(that)h Fu(f)p Fw(e)696 5590 y Ft(a)735 5620 y Fv(\()p Fw(\025)p Fv(\))p Fw(;)14 b(e)923 5590 y Fr(1)961 5620 y Fv(\()p Fw(\025)p Fv(\))p Fw(;)g Fu(\001)g(\001)g(\001)28 b Fw(;)14 b(e)1311 5590 y Ft(N)1374 5620 y Fv(\()p Fw(\025)p Fv(\))p Fw(;)g(e)1562 5590 y Ft(b)1596 5620 y Fv(\()p Fw(\025)p Fv(\))p Fu(g)1750 5632 y Ft(\025)p Fo(2)p Fk(R)1922 5620 y FO(is)40 b(a)h(measurable)e(family)i(of)g(orthogonal)e(bases)h(in)109 5720 y Fq(C)163 5690 y Ft(N)6 b Fr(+2)316 5720 y FO(,)21 b(if)g(the)f(comp)r(onen)n(ts)f(of)h(the)h(v)n(ectors)d Fw(e)1549 5690 y Ft(\034)1590 5720 y Fv(\()p Fw(\025)p Fv(\))p FO(,)23 b Fw(\034)32 b Fv(=)23 b Fw(a;)14 b Fv(1)p Fw(;)g Fu(\001)g(\001)g(\001)27 b Fw(;)14 b(N)t(;)g(b)p FO(,)21 b(are)e(Leb)r(esgue)g(measurable)g(functions)109 5820 y(suc)n(h)27 b(that)h Fu(h)p Fw(e)547 5789 y Ft(\034)588 5820 y Fv(\()p Fw(\025)p Fv(\))p Fw(;)14 b(e)776 5789 y Ft(\030)813 5820 y Fv(\()p Fw(\025)p Fv(\))p Fu(i)24 b Fv(=)f Fw(\016)1106 5832 y Ft(\034)s(;\030)1223 5820 y FO(for)k(a.e.)37 b Fw(\025)23 b Fu(2)h Fq(R)p FO(.)43 b(Th)n(us)27 b(w)n(e)g(get)p eop %%Page: 14 16 14 15 bop 109 254 a FO(14)2477 b Fx(M.)31 b(Bar)n(o,)f(H.)h(Neidhardt) 109 503 y Fy(Corollary)g(5.2)41 b FA(Supp)l(ose)30 b(that)g Fw(V)42 b Fu(2)24 b Fp(L)1397 472 y Fr(2)1434 503 y Fv(\()p Fw(a;)14 b(b)p Fv(\))p FA(,)30 b Fv(Im\()p Fw(V)20 b Fv(\))j(=)g(0)p FA(,)30 b Fw(\024)2157 515 y Ft(a)2197 503 y Fw(;)14 b(\024)2282 515 y Fr(1)2319 503 y Fw(;)g Fu(\001)g(\001)g(\001)27 b Fw(;)14 b(\024)2565 515 y Ft(N)2628 503 y Fw(;)g(\024)2713 515 y Ft(b)2769 503 y Fu(2)24 b Fq(C)2902 515 y Fr(+)2963 503 y FA(,)30 b(and)h Fw(x)3227 515 y Fr(1)3265 503 y Fw(;)14 b Fu(\001)g(\001)g(\001)27 b Fw(;)14 b(x)3510 515 y Ft(N)3597 503 y Fu(2)109 602 y Fv(\()p Fw(a;)g(b)p Fv(\))34 b FA(with)g Fw(x)555 614 y Fr(1)624 602 y Fw(<)c(x)766 614 y Fr(2)834 602 y Fw(<)h Fu(\001)14 b(\001)g(\001)30 b Fw(<)h(x)1200 614 y Ft(N)1263 602 y FA(.)51 b(If)35 b Fu(f)p Fw(e)1512 572 y Ft(a)1551 602 y Fv(\()p Fw(\025)p Fv(\))p Fw(;)14 b(e)1739 572 y Fr(1)1777 602 y Fv(\()p Fw(\025)p Fv(\))p Fw(;)g Fu(\001)g(\001)g (\001)28 b Fw(;)14 b(e)2127 572 y Ft(N)2190 602 y Fv(\()p Fw(\025)p Fv(\))p Fw(;)g(e)2378 572 y Ft(b)2412 602 y Fv(\()p Fw(\025)p Fv(\))p Fu(g)2566 614 y Ft(\025)p Fo(2)p Fk(R)2730 602 y FA(is)35 b(a)f(me)l(asur)l(able)g(family)i(of)109 702 y(ortho)l(gonal)31 b(b)l(ases)f(in)g Fq(C)883 672 y Ft(N)6 b Fr(+2)1036 702 y FA(,)30 b(then)f(the)h(system)g(of)g (eigenfunctions)717 864 y Fu(f)764 842 y Fw(~)759 864 y(\036)p Fv(\()p Fu(\001)p Fw(;)14 b(\025;)g(e)1024 830 y Ft(a)1065 864 y Fv(\()p Fw(\025)p Fv(\)\))p Fw(;)1251 842 y(~)1246 864 y(\036)q Fv(\()p Fu(\001)p Fw(;)g(\025;)g(e)1512 830 y Fr(1)1550 864 y Fv(\()p Fw(\025)p Fv(\)\))p Fw(;)g Fu(\001)g(\001)g(\001)28 b Fw(;)1898 842 y(~)1893 864 y(\036)q Fv(\()p Fu(\001)p Fw(;)14 b(\025;)g(e)2159 830 y Ft(N)2222 864 y Fv(\()p Fw(\025)p Fv(\)\))p Fw(;)2409 842 y(~)2403 864 y(\036)r Fv(\()p Fu(\001)p Fw(;)g(\025;)g(e)2670 830 y Ft(b)2703 864 y Fv(\()p Fw(\025)p Fv(\)\))p Fu(g)2889 876 y Ft(\025)p Fo(2)p Fk(R)3020 864 y Fw(;)109 1027 y FA(wher)l(e)1022 1123 y Fw(~)1017 1145 y(\036)q Fv(\()p Fu(\001)p Fw(;)g(\025;)g(e)1283 1110 y Ft(\034)1324 1145 y Fv(\()p Fw(\025)p Fv(\)\))25 b(:=)1673 1089 y(1)p 1613 1126 161 4 v 1613 1142 a Fu(p)p 1682 1142 92 4 v 69 x Fv(2)p Fw(\031)1789 1123 y(~)1784 1145 y(\036)1833 1110 y Ft(e)1864 1085 y Fm(j)1896 1110 y Fr(\()p Ft(\025)p Fr(\))1992 1145 y Fv(\()p Fu(\001)p Fw(;)14 b(\025)p Fv(\))p Fw(;)99 b(\034)33 b Fv(=)23 b Fw(a;)14 b Fu(\001)g(\001)g(\001) 27 b Fw(;)14 b(b;)109 1325 y FA(p)l(erforms)30 b(a)h(c)l(omplete)f (orthonormal)h(system)f(of)g(gener)l(alize)l(d)h(eigenfunctions)g(of)f Fw(K)6 b FA(.)109 1522 y FO(The)23 b(generalized)e(eigenfunctions)i Fw(\036)p Fv(\()p Fu(\001)p Fw(;)14 b(\025;)g(\034)9 b Fv(\))25 b FO(are)e(usually)f(called)h(the)g(incoming)g (eigenfunctions.)35 b(By)23 b(the)g(use)109 1622 y(of)k(the)h(incoming) f(eigenfunctions,)h(one)f(de\034nes)g(a)g(transform)g Fu(F)k Fv(:)23 b Fu(K)h(!)2489 1601 y Fv(^)2472 1622 y Fu(K)h Fv(:=)e Fp(L)2716 1592 y Fr(2)2753 1622 y Fv(\()p Fq(R)p Fw(;)14 b Fq(C)2936 1592 y Ft(N)6 b Fr(+2)3089 1622 y Fv(\))28 b FO(b)n(y)1294 2012 y Fv(\()p Fu(F)5 b Fw(~)-39 b(g)s Fv(\)\()p Fw(\025)p Fv(\))25 b(=:)g(^)-45 b Fw(g)s Fv(\()p Fw(\025)p Fv(\))24 b(=)1982 1720 y Fs(0)1982 1866 y(B)1982 1916 y(B)1982 1966 y(B)1982 2016 y(B)1982 2066 y(B)1982 2119 y(@)2114 1784 y Fv(^)-45 b Fw(g)2154 1754 y Ft(b)2187 1784 y Fv(\()p Fw(\025)p Fv(\))2099 1884 y(^)h Fw(g)2140 1854 y Ft(N)2202 1884 y Fv(\()p Fw(\025)p Fv(\))2194 1972 y FO(.)2194 2006 y(.)2194 2039 y(.)2112 2138 y Fv(^)g Fw(g)2153 2108 y Fr(1)2189 2138 y Fv(\()p Fw(\025)p Fv(\))2111 2238 y(^)f Fw(g)2151 2208 y Ft(a)2191 2238 y Fv(\()p Fw(\025)p Fv(\))2357 1720 y Fs(1)2357 1866 y(C)2357 1916 y(C)2357 1966 y(C)2357 2016 y(C)2357 2066 y(C)2357 2119 y(A)2443 2012 y Fw(;)109 2396 y FO(where)1254 2518 y Fv(^)g Fw(g)1294 2483 y Ft(\034)1335 2518 y Fv(\()p Fw(\025)p Fv(\))24 b(:=)1582 2405 y Fs(Z)1635 2579 y Fr(^)1628 2594 y(\012)1693 2425 y Fs(\020)1739 2518 y Fw(~)-38 b(g)r Fv(\()p Fw(x)p Fv(\))p Fw(;)1939 2496 y(~)1933 2518 y(\036)r Fv(\()p Fw(x;)14 b(\025;)g(\034)9 b Fv(\))2262 2425 y Fs(\021)2313 2576 y Fo(K)2396 2518 y Fw(dx:)109 2744 y Fu(F)36 b FO(is)28 b(a)g(unitary)g(op)r(erator)e (and)i(called)g(the)h(incoming)e(F)-7 b(ourier)28 b(transform.)37 b(The)29 b(in)n(v)n(erse)d(incoming)i(F)-7 b(ourier)109 2843 y(transform)26 b Fu(F)558 2813 y Fo(\000)p Fr(1)675 2843 y FO(is)h(giv)n(en)g(b)n(y)889 2976 y Fs(\000)927 3044 y Fu(F)995 3009 y Fo(\000)p Fr(1)1087 3044 y Fv(^)-45 b Fw(g)1127 2976 y Fs(\001)1179 3044 y Fv(\()p Fw(x)p Fv(\))24 b(=)1401 2931 y Fs(Z)1448 3119 y Fk(R)1643 2965 y Fs(X)1507 3143 y Ft(\034)7 b Fr(=)p Ft(a;)p Fr(1)p Ft(;)p Fo(\001\001\001)j Ft(;N)s(;b)1917 3022 y Fw(~)1913 3044 y(\036)p Fv(\()p Fu(\001)p Fw(;)k(\025;)g(\034)9 b Fv(\))s(^)-45 b Fw(g)2259 3009 y Ft(\034)2302 3044 y Fv(\()p Fw(\025)p Fv(\))14 b Fw(d\025;)101 b Fv(^)-45 b Fw(g)26 b Fu(2)2800 3023 y Fv(^)2784 3044 y Fu(K)q Fw(:)109 3291 y FO(W)-7 b(e)28 b(note)f(that)1620 3391 y Fu(F)8 b Fw(K)e Fu(F)1833 3357 y Fo(\000)p Fr(1)1945 3391 y Fv(=)22 b Fw(M)t(;)1300 b FO(\(5.11\))109 3528 y(where)27 b Fw(M)36 b FO(is)27 b(the)h(m)n(ultiplication)g(op)r (erator)e(on)1691 3507 y Fv(^)1675 3528 y Fu(K)j FO(giv)n(en)e(b)n(y) 1387 3691 y Fp(D)p Fv(\()p Fw(M)9 b Fv(\))23 b(:=)f Fu(f)s Fv(^)-45 b Fw(g)26 b Fu(2)1936 3670 y Fv(^)1920 3691 y Fu(K)15 b(j)f Fw(\025)s Fv(^)-45 b Fw(g)s Fv(\()p Fw(\025)p Fv(\))24 b Fu(2)2357 3670 y Fv(^)2341 3691 y Fu(K)q(g)p Fw(;)1291 3815 y Fv(\()p Fw(M)12 b Fv(^)-45 b Fw(g)r Fv(\)\()p Fw(\025)p Fv(\))25 b(:=)d Fw(\025)s Fv(^)-45 b Fw(g)s Fv(\()p Fw(\025)p Fv(\))p Fw(;)101 b Fv(^)-45 b Fw(g)26 b Fu(2)d Fp(D)p Fv(\()p Fw(M)9 b Fv(\))p Fw(:)109 4133 y Fz(6)131 b(Carrier)45 b(densit)l(y)109 4363 y FO(A)n(ccording)36 b(to)j([KNR01c)n(])g(w)n(e)f(call)f Fw(\045)k Fv(:)g Fu(K)h(!)f(K)f FO(a)d(densit)n(y)h(matrix,)j(if)d Fw(\045)h FO(is)f(a)g(b)r(ounded,)j(non-negativ)n(e)109 4462 y(op)r(erator.)c(A)29 b(densit)n(y)g(matrix)f Fw(\045)g FO(is)g(called)h(a)f(steady)g(state,)g(if)h(it)g(comm)n(utes)f(with)h Fw(K)6 b FO(.)39 b(F)-7 b(or)28 b(steady)g(states)g Fw(\045)109 4562 y FO(there)f(exists)g(a)g(function)h Fw(\045)p Fv(\()p Fu(\001)p Fv(\))c Fu(2)f Fp(L)1221 4532 y Fo(1)1292 4562 y Fv(\()p Fq(R)p Fw(;)14 b Fn(B)p Fv(\()q Fq(C)1581 4532 y Ft(N)6 b Fr(+2)1734 4562 y Fv(\)\))28 b FO(suc)n(h)f(that)h(the)g(m)n (ultiplication)f(op)r(erator)34 b Fv(^)-50 b Fw(\045)28 b FO(de\034ned)g(b)n(y)1356 4724 y Fp(D)p Fv(\()8 b(^)-50 b Fw(\045)p Fv(\))24 b(:=)e Fu(f)s Fv(^)-45 b Fw(g)26 b Fu(2)d(K)15 b(j)f Fw(\045)p Fv(\()p Fw(\025)p Fv(\))s(^)-45 b Fw(g)t Fv(\()p Fw(\025)p Fv(\))24 b Fu(2)f(K)q(g)p Fw(;)1261 4849 y Fv(\()8 b(^)-50 b Fw(\045)s Fv(^)-45 b Fw(g)r Fv(\)\()p Fw(\025)p Fv(\))25 b(:=)d Fw(\045)p Fv(\()p Fw(\025)p Fv(\))s(^)-45 b Fw(g)t Fv(\()p Fw(\025)p Fv(\))p Fw(;)101 b Fv(^)-45 b Fw(g)25 b Fu(2)f Fp(D)p Fv(\()8 b(^)-50 b Fw(\045)p Fv(\))p Fw(;)109 5011 y FO(is)27 b(unitarily)g(equiv)-5 b(alen)n(t)28 b(to)f Fw(\045)p FO(,)h(i.e.)1658 5111 y Fw(\045)23 b Fv(=)g Fu(F)1880 5077 y Fo(\000)p Fr(1)1976 5111 y Fv(^)-49 b Fw(\045)p Fu(F)8 b Fw(:)109 5248 y FO(Since)27 b Fw(\045)c Fu(\025)g Fv(0)k FO(w)n(e)g(get)h Fw(\045)p Fv(\()p Fw(\025)p Fv(\))c Fu(\025)e Fv(0)28 b FO(for)f(a.e.)36 b Fw(\025)23 b Fu(2)h Fq(R)p FO(.)109 5395 y(The)j(time)h(ev)n(olution)f(of)h(a)f(giv)n(en)f (densit)n(y)i(matrix)f Fw(\045)g FO(is)h(giv)n(en)f(b)n(y)1379 5558 y Fw(\045)p Fv(\()p Fw(t)p Fv(\))c(:=)g Fw(e)1689 5523 y Fo(\000)p Ft(itK)1853 5558 y Fw(\045e)1935 5523 y Ft(itK)2047 5558 y Fw(;)97 b(t)23 b Fu(2)h Fq(R)p Fw(;)109 5720 y FO(cf.)44 b([LL71)o(].)g(If)30 b Fw(\045)g FO(comm)n(utes)g (with)g Fw(K)6 b FO(,)30 b(w)n(e)g(ha)n(v)n(e)f Fw(\045)p Fv(\()p Fw(t)p Fv(\))e(=)g Fw(\045)j FO(for)f(all)h Fw(t)d Fu(2)g Fq(R)p FO(.)50 b(This)30 b(justi\034es)h(the)f(de\034nition)g (of)109 5820 y(steady)d(states.)p eop %%Page: 15 17 15 16 bop 109 254 a FH(Dissipativ)n(e)27 b(Sc)n(hr\366)r(dinger-t)n(yp) r(e)e(op)r(erator)2084 b FO(15)109 503 y Fy(De\034nition)30 b(6.1)41 b FO(A)33 b(b)r(ounded)f(selfadjoin)n(t)h(op)r(erator)d Fw(A)j FO(on)f(a)g(Hilb)r(ert)h(space)f Fu(K)i FO(is)e(called)g(an)g (observ)-5 b(able.)109 602 y(W)e(e)28 b(sa)n(y)e(that)i(the)g(observ)-5 b(able)26 b Fw(A)210 825 y FO(1.)41 b(is)28 b(admissible)f(with)h(resp) r(ect)f(to)h Fw(\045)f FO(if)h Fw(\045A)g FO(is)g(a)f(trace)f(class)h (op)r(erator,)f(i.e)i Fw(\045A)23 b Fu(2)g Fn(B)2959 837 y Fr(1)2997 825 y Fv(\()p Fu(K)q Fv(\))p FO(,)210 1010 y(2.)41 b(is)24 b(admissible)f(with)h(resp)r(ect)f(to)h Fw(K)29 b FO(if)24 b Fw(E)1590 1022 y Ft(K)1654 1010 y Fv(\(\001\))p Fw(A)h Fu(2)e Fn(B)2025 1022 y Fr(1)2063 1010 y Fv(\()p Fu(K)q Fv(\))h FO(for)f(eac)n(h)g(b)r(ounded)h(in)n (terv)-5 b(al)23 b Fv(\001)g Fu(\032)g Fq(R)p FO(,)31 b(where)316 1110 y Fw(E)377 1122 y Ft(K)442 1110 y Fv(\(\001\))d FO(denotes)f(the)h(sp)r(ectral)f(pro)5 b(jection)27 b(of)g Fw(K)33 b FO(on)27 b Fv(\001)p FO(.)109 1333 y(If)h(the)h(observ)-5 b(able)26 b Fw(A)j FO(is)f(admissible)f(with)i(resp)r(ect)f(to)g Fw(\045)p FO(,)g(then)h(its)f(exp)r(ectation)g(v)-5 b(alue)28 b Fq(E)3006 1345 y Ft(\045)3051 1333 y Fv(\()p Fw(A)p Fv(\))h FO(with)f(resp)r(ect)109 1432 y(to)f(the)h(densit)n(y)f(matrix) g Fw(\045)h FO(is)f(de\034ned)h(b)n(y)1574 1625 y Fq(E)1624 1637 y Ft(\045)1668 1625 y Fv(\()p Fw(A)p Fv(\))c(:=)f(tr\()p Fw(\045A)p Fv(\))p Fw(:)109 1864 y FO(T)-7 b(o)27 b(calculate)g(the)h (carrier)d(densit)n(y)j(w)n(e)f(consider)f(the)i(observ)-5 b(able)26 b Fw(U)9 b Fv(\()p Fw(!)s Fv(\))28 b FO(giv)n(en)f(b)n(y)833 2067 y Fv(\()p Fw(U)9 b Fv(\()p Fw(!)s Fv(\))1061 2045 y Fw(~)1050 2067 y(f)f Fv(\)\()p Fw(x)p Fv(\))25 b(=)d(0)c Fu(\010)g Fw(\037)1549 2079 y Ft(!)1597 2067 y Fv(\()p Fw(x)p Fv(\))p Fw(f)9 b Fv(\()p Fw(x)p Fv(\))20 b Fu(\010)e Fv(0)p Fw(;)2145 2045 y(~)2134 2067 y(f)32 b Fv(=)22 b Fw(f)2335 2079 y Fo(\000)2409 2067 y Fu(\010)c Fw(f)27 b Fu(\010)18 b Fw(f)2684 2079 y Fr(+)2762 2067 y Fu(2)24 b(K)q Fw(;)109 2259 y FO(for)j(an)n(y)f(Borel)h(subset)g Fw(!)f Fu(\022)d Fv([)p Fw(a;)14 b(b)p Fv(])p FO(.)36 b(W)-7 b(e)28 b(remark)e(that)i Fw(U)9 b Fv(\()p Fw(!)s Fv(\))28 b FO(is)f(a)g(pro)5 b(jection)27 b(on)g Fu(K)i FO(with)f Fv(ran)o(\()p Fw(U)9 b Fv(\()p Fw(!)s Fv(\)\))24 b Fu(\022)f(H)q FO(.)109 2406 y(Let)k(us)h(in)n(tro)r(duce)f(some)g (more)g(notions:)36 b(W)-7 b(e)28 b(set)1312 2705 y Fv(\010\()p Fw(x;)14 b(\025)p Fv(\))25 b(=)1680 2513 y Fs(0)1680 2659 y(B)1680 2712 y(@)1795 2577 y Fw(\036)p Fv(\()p Fw(x;)14 b(\025;)g(N)28 b Fv(+)18 b(1\))2034 2665 y FO(.)2034 2698 y(.)2034 2731 y(.)1883 2831 y Fw(\036)p Fv(\()p Fw(x;)c(\025;)g Fv(0\))2339 2513 y Fs(1)2339 2659 y(C)2339 2712 y(A)2425 2705 y Fw(;)109 3041 y FO(and)p 270 2968 258 4 v 27 w Fv(\010\()p Fw(x;)g(\025)p Fv(\))29 b FO(denotes)e(the)h (the)g(v)n(ector)e Fv(\010\()p Fw(x;)14 b(\025)p Fv(\))29 b FO(with)f(eac)n(h)f(elemen)n(t)h(complex)f(conjugated.)109 3188 y(As)g(in)h([KNR01c)o(])g(one)f(pro)n(v)n(es)f(the)h(follo)n(wing) g(lemma:)109 3410 y Fy(Lemma)i(6.2)41 b FA(Assume)30 b(that)h Fw(m)20 b Fv(+)1293 3378 y Fr(1)p 1280 3392 59 4 v 1280 3439 a Ft(m)1375 3410 y Fu(2)26 b Fp(L)1501 3380 y Fo(1)1571 3410 y Fv(\()p Fw(a;)14 b(b)p Fv(\))p FA(,)32 b Fw(V)45 b Fu(2)27 b Fp(L)2029 3380 y Fr(2)2066 3410 y Fv(\()p Fw(a;)14 b(b)p Fv(\))p FA(,)32 b Fv(Im\()p Fw(V)19 b Fv(\))26 b(=)g(0)p FA(,)32 b Fw(\024)2798 3422 y Ft(a)2838 3410 y Fw(;)14 b(\024)2923 3422 y Fr(1)2960 3410 y Fw(;)g Fu(\001)g(\001)g(\001)27 b Fw(;)k(\024)3223 3422 y Ft(N)3286 3410 y Fw(;)14 b(\024)3371 3422 y Ft(b)3430 3410 y Fu(2)26 b Fq(C)3565 3422 y Fr(+)3627 3410 y FA(,)109 3510 y(and)k Fw(x)317 3522 y Fr(1)355 3510 y Fw(;)14 b Fu(\001)g(\001)g(\001)27 b Fw(;)14 b(x)600 3522 y Ft(n)669 3510 y Fu(2)23 b Fv(\()p Fw(a;)14 b(b)p Fv(\))30 b FA(such)f(that)h Fw(x)1363 3522 y Fr(1)1424 3510 y Fw(<)23 b(x)1559 3522 y Fr(2)1619 3510 y Fw(<)g Fu(\001)14 b(\001)g(\001)23 b Fw(<)f(x)1961 3522 y Ft(N)2025 3510 y FA(.)207 3733 y(1.)42 b(The)34 b(observable)g Fw(U)9 b Fv(\()p Fw(!)s Fv(\))32 b FA(is)h(admissible)h(with)f(r)l(esp)l(e)l(ct)f(to)h(the)f (minimal)h(selfadjoint)i(dilation)f Fw(K)k FA(of)33 b(the)316 3832 y(maximal)f(dissip)l(ative)h(op)l(er)l(ator)f Fw(H)38 b FA(for)32 b(any)f(Bor)l(el)h(set)f Fw(!)d Fu(\022)d Fv([)p Fw(a;)14 b(b)p Fv(])p FA(.)42 b(F)-6 b(urthermor)l(e,)32 b(we)f(get)g(the)g(r)l(epr)l(e-)316 3932 y(sentation)1205 4068 y Fv(tr\()p Fw(\045U)9 b Fv(\()p Fw(!)s Fv(\))p Fw(E)1591 4080 y Ft(K)1656 4068 y Fv(\(\001\)\))24 b(=)1932 3955 y Fs(Z)1978 4143 y Fr(\001)2051 4068 y Fv(tr\()p Fw(\045)p Fv(\()p Fw(\025)p Fv(\))p Fw(U)9 b Fv(\()p Fw(!)s Fv(\)\()p Fw(\025)p Fv(\)\))14 b Fw(d\025;)316 4273 y FA(for)31 b(any)f(b)l(ounde)l(d)g(Bor)l(el)h(set)e Fv(\001)23 b Fu(\032)g Fq(R)36 b FA(and)30 b(any)g(ste)l(ady)g(state)g Fw(\045)p FA(,)g(wher)l(e)1516 4503 y Fw(U)9 b Fv(\()p Fw(!)s Fv(\)\()p Fw(\025)p Fv(\))24 b(:=)1948 4390 y Fs(Z)1994 4579 y Ft(!)2056 4503 y Fw(D)r Fv(\()p Fw(x;)14 b(\025)p Fv(\))g Fw(dx;)316 4737 y FA(with)1493 4851 y Fw(D)r Fv(\()p Fw(x;)g(\025)p Fv(\))24 b(:=)p 1895 4779 258 4 v 23 w(\010\()p Fw(x;)14 b(\025)p Fv(\))h(\010\()p Fw(x;)f(\025)p Fv(\))2422 4816 y Ft(T)207 5057 y FA(2.)42 b(If)30 b(the)g(ste)l(ady)h(state)e Fw(\045)h FA(satis\034es)f(the)h(c) l(ondition)1274 5262 y Fw(C)1339 5274 y Fr(^)-39 b Ft(\045)1395 5262 y Fv(:=)23 b(sup)1506 5333 y Ft(\025)p Fo(2)p Fk(R)1646 5181 y Fs(p)p 1729 5181 229 4 v 81 x Fw(\025)1777 5239 y Fr(2)1833 5262 y Fv(+)18 b(1)o Fu(k)p Fw(\045)p Fv(\()p Fw(\025)p Fv(\))p Fu(k)2196 5279 y Fb(B)p Fr(\()p Fk(C)2317 5262 y Fm(N)5 b Fl(+2)2447 5279 y Fr(\))2500 5262 y Fw(<)23 b Fu(1)p Fw(;)787 b FO(\(6.1\))316 5509 y FA(then)27 b(the)g(observable)i Fw(U)9 b Fv(\()p Fw(!)s Fv(\))26 b FA(is)i(admissible)g(with)g(r)l(esp)l(e)l(ct)e(to)h Fw(\045)g FA(for)h(any)f(Bor)l(el)h(set)e Fw(!)g Fu(\022)d Fv([)p Fw(a;)14 b(b)p Fv(])p FA(.)37 b(F)-6 b(urther-)316 5608 y(mor)l(e)30 b(we)g(have)1341 5744 y Fv(tr\()p Fw(\045U)9 b Fv(\()p Fw(!)s Fv(\)\))23 b(=)1809 5631 y Fs(Z)1855 5820 y Fk(R)1915 5744 y Fv(tr\()p Fw(\045)p Fv(\()p Fw(\025)p Fv(\))p Fw(U)9 b Fv(\()p Fw(!)s Fv(\)\()p Fw(\025)p Fv(\)\))14 b Fw(d\025:)p eop %%Page: 16 18 16 17 bop 109 254 a FO(16)2477 b Fx(M.)31 b(Bar)n(o,)f(H.)h(Neidhardt) 109 503 y FO(W)-7 b(e)28 b(set)1383 602 y Fw(u)1431 614 y Ft(\045)1469 602 y Fv(\()p Fw(x;)14 b(\025)p Fv(\))25 b(:=)e(tr)o(\()p Fw(\045)p Fv(\()p Fw(\025)p Fv(\))p Fw(D)r Fv(\()p Fw(x;)14 b(\025)p Fv(\)\))p Fw(;)109 738 y FO(for)27 b Fw(x)c Fu(2)h Fv([)p Fw(a;)14 b(b)p Fv(])27 b FO(and)g Fw(\025)d Fu(2)f Fq(R)p FO(.)43 b(Note)28 b(that)1246 897 y Fw(u)1294 909 y Ft(\045)1332 897 y Fv(\()p Fw(x;)14 b(\025)p Fv(\))24 b(=)1640 829 y Fs(\012)1679 897 y Fw(\045)p Fv(\()p Fw(\025)p Fv(\))1834 862 y Ft(T)1887 897 y Fv(\010\()p Fw(x;)14 b(\025)p Fv(\))p Fw(;)g Fv(\010\()p Fw(x;)g(\025)p Fv(\))2436 829 y Fs(\013)2492 897 y Fw(;)966 b FO(\(6.2\))109 1056 y(where)35 b Fw(\045)p Fv(\()p Fw(\025)p Fv(\))512 1025 y Ft(T)601 1056 y FO(denotes)g(the)h(transp)r (osed)e(matrix)h(of)h Fw(\045)p Fv(\()p Fw(\025)p Fv(\))p FO(.)62 b(Since)36 b Fw(\045)p Fv(\()p Fw(\025)p Fv(\))g FO(is)g(non-negativ)n(e,)g(w)n(e)f(get)g(b)n(y)h(the)109 1155 y(represen)n(tation)f(\(6.2\))i(that)g Fw(u)1106 1167 y Ft(\045)1144 1155 y Fv(\()p Fw(x;)14 b(\025)p Fv(\))40 b Fu(\025)f Fv(0)d FO(for)h Fw(x)i Fu(2)g Fv([)p Fw(a;)14 b(b)p Fv(])37 b FO(and)g(a.e.)65 b Fw(\025)39 b Fu(2)g Fq(R)p FO(.)72 b(If)37 b(the)h(condition)e(\(6.1\))h(is)109 1255 y(satis\034ed,)27 b(w)n(e)g(get)g(b)n(y)h(Lemma)f(6.2)1288 1452 y Fq(E)1337 1464 y Ft(\045)1382 1452 y Fv(\()p Fw(U)9 b Fv(\()p Fw(!)s Fv(\)\))23 b(=)1742 1339 y Fs(Z)1788 1527 y Fk(R)1848 1339 y Fs(Z)1894 1527 y Ft(!)1956 1452 y Fw(u)2004 1464 y Ft(\045)2042 1452 y Fv(\()p Fw(x;)14 b(\025)p Fv(\))g Fw(dx)g(d\025;)109 1657 y FO(for)27 b(an)n(y)f(Borel)h(set)g Fw(!)f Fu(\022)d Fv([)p Fw(a;)14 b(b)p Fv(])p FO(.)36 b(Hence)28 b(b)n(y)f(F)-7 b(ubini's)28 b(Theorem)f(w)n(e)g(get)1436 1859 y Fq(E)1486 1871 y Ft(\045)1530 1859 y Fv(\()p Fw(U)9 b Fv(\()p Fw(!)s Fv(\)\))24 b(=)1891 1746 y Fs(Z)1937 1934 y Ft(!)1999 1859 y Fw(u)2047 1871 y Ft(\045)2085 1859 y Fv(\()p Fw(x)p Fv(\))14 b Fw(dx;)1158 b FO(\(6.3\))109 2059 y(where)1164 2181 y Fw(u)1212 2193 y Ft(\045)1250 2181 y Fv(\()p Fw(x)p Fv(\))24 b(:=)1495 2068 y Fs(Z)1542 2256 y Fk(R)1602 2181 y Fw(u)1650 2193 y Ft(\045)1688 2181 y Fv(\()p Fw(x;)14 b(\025)p Fv(\))g Fw(d\025)25 b Fu(\025)d Fv(0)p Fw(;)97 b(x)23 b Fu(2)g Fv([)p Fw(a;)14 b(b)p Fv(])p Fw(;)884 b FO(\(6.4\))109 2368 y(th)n(us)26 b Fw(u)338 2380 y Ft(\045)399 2368 y Fu(2)d Fp(L)522 2338 y Fr(1)559 2368 y Fv(\()p Fw(a;)14 b(b)p Fv(\))p FO(.)37 b(Clearly)24 b Fq(E)1136 2380 y Ft(\045)1180 2368 y Fv(\()p Fw(U)9 b Fv(\()p Fu(\001)p Fv(\)\))27 b FO(de\034nes)f(a)f(measure)g(whic)n(h)h(is)f(absolutely)g (con)n(tin)n(uous)g(with)h(resp)r(ect)109 2467 y(to)21 b(the)i(Leb)r(esgue)e(measure.)34 b(Since)22 b Fq(E)1309 2479 y Ft(\045)1354 2467 y Fv(\()p Fw(U)9 b Fv(\()p Fw(!)s Fv(\)\))22 b FO(can)g(b)r(e)g(in)n(terpreted)f(as)h(the)g(n)n(um)n(b)r (er)f(of)h(carriers)e(in)i Fw(!)k Fu(\022)c Fv([)p Fw(a;)14 b(b)p Fv(])p FO(,)109 2567 y(its)27 b(Radon-Nik)n(o)r(dym)g(deriv)-5 b(ativ)n(e)26 b(can)h(b)r(e)h(view)n(ed)f(as)g(the)h(carrier)d(densit)n (y)j(of)f(the)h(system)f(describ)r(ed)g(b)n(y)g Fw(K)6 b FO(.)109 2667 y(Since)29 b Fw(K)36 b FO(is)29 b(the)h(dilation)f (corresp)r(onding)f(to)h(the)h(maximal)f(dissipativ)n(e)g(op)r(erator)f Fw(H)36 b FO(and)30 b Fw(U)9 b Fv(\()p Fw(!)s Fv(\))29 b FO(acts)g(only)109 2766 y(on)e Fu(H)q FO(,)h(w)n(e)f(iden)n(tify)h Fw(u)818 2778 y Ft(\045)884 2766 y FO(with)g(the)g(carrier)d(densit)n (y)i(of)h(the)g(system)f(describ)r(ed)h(b)n(y)f Fw(H)7 b FO(.)109 2913 y(Note)27 b(that)1413 3013 y Fw(u)1461 3025 y Ft(\045)1499 3013 y Fv(\()p Fw(x;)14 b(\025)p Fv(\))24 b(=)f Fq(E)1856 3028 y Ft(\045)q Fr(\()p Ft(\025)p Fr(\))1992 3013 y Fv(\()p Fw(D)r Fv(\()p Fw(x;)14 b(\025)p Fv(\)\))p Fw(;)1135 b FO(\(6.5\))109 3148 y(for)24 b Fw(x)f Fu(2)h Fv([)p Fw(a;)14 b(b)p Fv(])24 b FO(and)g Fw(\025)g Fu(2)f Fq(R)p FO(.)42 b(W)-7 b(e)25 b(call)f(the)h(matrix)f Fw(D)r Fv(\()p Fw(x;)14 b(\025)p Fv(\))26 b FO(the)f(carrier)e(densit)n (y)h(observ)-5 b(able)23 b(and)i Fw(u)3277 3160 y Ft(\045)3315 3148 y Fv(\()p Fw(x;)14 b(\025)p Fv(\))26 b FO(the)109 3248 y(carrier)e(densit)n(y)i(of)g(the)h(system)f(describ)r(ed)f(b)n(y) h Fw(H)33 b FO(at)27 b(p)r(oin)n(t)f Fw(x)d Fu(2)h Fv([)p Fw(a;)14 b(b)p Fv(])26 b FO(and)g(energy)f Fw(\025)e Fu(2)h Fq(R)p FO(.)42 b(This)26 b(is)g(justi\034ed)109 3347 y(since)h(\(6.5\))g(can)g(b)r(e)h(seen)g(as)f(exp)r(ectation)g(v) -5 b(alue)27 b(of)h(the)g(carrier)d(densit)n(y)j(observ)-5 b(able.)109 3494 y(F)e(urthermore)26 b(w)n(e)h(ha)n(v)n(e)1187 3615 y Fw(u)1235 3627 y Ft(\045)1273 3615 y Fv(\()p Fw(x)p Fv(\))d(=)1495 3502 y Fs(Z)1542 3691 y Fk(R)1601 3615 y Fq(E)1651 3630 y Ft(\045)p Fr(\()p Ft(\025)q Fr(\))1787 3615 y Fv(\()p Fw(D)r Fv(\()p Fw(x;)14 b(\025)p Fv(\)\))p Fw(;)99 b(x)23 b Fu(2)g Fv([)p Fw(a;)14 b(b)p Fv(])p Fw(;)109 3792 y FO(i.e.)38 b(the)28 b(carrier)e(densit)n(y)i(for)f(eac) n(h)h(p)r(oin)n(t)g Fw(x)g FO(is)g(the)g(sum)g(of)g(the)h(exp)r (ectation)e(v)-5 b(alues)28 b(of)g(the)g(carrier)e(densit)n(y)109 3892 y(observ)-5 b(able)26 b(for)h(the)h(p)r(oin)n(t)g Fw(x)g FO(o)n(v)n(er)d(all)j(energies.)109 4039 y(F)-7 b(or)23 b(real-v)-5 b(alued)23 b Fw(h)g Fu(2)g Fp(L)867 4009 y Fo(1)937 4039 y Fv(\()p Fw(a;)14 b(b)p Fv(\))25 b FO(w)n(e)e(de\034ne)i(the)f(m)n(ultiplication)g(op)r(erator)e Fw(M)9 b Fv(\()p Fw(h)p Fv(\))25 b FO(on)e(the)i(Hilb)r(ert)f(space)g Fu(K)h FO(b)n(y)850 4208 y Fv(\()p Fw(M)9 b Fv(\()p Fw(h)p Fv(\))1095 4187 y Fw(~)1084 4208 y(f)g Fv(\)\()p Fw(x)p Fv(\))24 b(=)f(0)18 b Fu(\010)g Fw(h)p Fv(\()p Fw(x)p Fv(\))p Fw(f)9 b Fv(\()p Fw(x)p Fv(\))20 b Fu(\010)e Fv(0)p Fw(;)2128 4187 y(~)2116 4208 y(f)32 b Fv(=)23 b Fw(f)2318 4220 y Fo(\000)2392 4208 y Fu(\010)18 b Fw(f)27 b Fu(\010)18 b Fw(f)2667 4220 y Fr(+)2745 4208 y Fu(2)23 b(K)q Fw(:)109 4367 y FO(W)-7 b(e)28 b(note)f(that)h Fw(M)9 b Fv(\()p Fw(\037)790 4379 y Ft(!)838 4367 y Fv(\))23 b(=)g Fw(U)9 b Fv(\()p Fw(!)s Fv(\))p FO(,)27 b(in)h(particular)e Fw(M)9 b Fv(\()p Fw(\037)1870 4379 y Fr(\012)1922 4367 y Fv(\))23 b(=)g Fw(U)9 b Fv(\(\012\))23 b(=)g Fw(P)2431 4337 y Fo(K)2419 4390 y(H)2486 4367 y FO(.)36 b(Since)1149 4527 y Fw(\045M)9 b Fv(\()p Fw(h)p Fv(\))23 b(=)g Fw(\045U)9 b Fv(\(\012\))p Fw(M)g Fv(\()p Fw(h)p Fv(\))p Fw(;)180 b(h)23 b Fu(2)g Fp(L)2337 4492 y Fo(1)2407 4527 y Fv(\()p Fw(a;)14 b(b)p Fv(\))p Fw(;)109 4686 y FO(w)n(e)27 b(get)g(that)h(the)g (observ)-5 b(able)26 b Fw(M)9 b Fv(\()p Fw(h)p Fv(\))28 b FO(is)f(admissible)g(with)h(resp)r(ect)g(to)f Fw(\045)p FO(.)109 4880 y Fy(Lemma)i(6.3)41 b FA(Assume)35 b(that)h Fw(m)23 b Fv(+)1310 4847 y Fr(1)p 1297 4861 59 4 v 1297 4909 a Ft(m)1401 4880 y Fu(2)35 b Fp(L)1536 4850 y Fo(1)1606 4880 y FA(,)j Fw(V)54 b Fu(2)35 b Fp(L)1906 4850 y Fr(2)1943 4880 y Fv(\()p Fw(a;)14 b(b)p Fv(\))37 b FA(r)l(e)l(al-value)l(d,)i (and)d Fw(\024)2820 4892 y Ft(a)2860 4880 y Fw(;)14 b(\024)2945 4892 y Fr(1)2996 4880 y Fu(\001)g(\001)g(\001)28 b Fw(;)14 b(\024)3206 4892 y Ft(N)3268 4880 y Fw(;)g(\024)3353 4892 y Ft(b)3421 4880 y Fu(2)35 b Fq(C)3565 4892 y Fr(+)3627 4880 y FA(.)109 4980 y(If)c(the)h(ste)l(ady)f(state)g Fw(\045)g FA(satis\034es)h(the)f(c)l(ondition)39 b FO(\(6.1\))o FA(,)32 b(then)f(the)h(c)l(arrier)g(density)f Fw(u)2849 4992 y Ft(\045)2919 4980 y FA(de\034ne)l(d)g(by)39 b FO(\(6.4\))31 b FA(is)h(a)109 5079 y(non-ne)l(gative)d Fp(L)639 5049 y Fr(1)677 5079 y FA(-function)g(such)h(that)1330 5296 y Fv(tr\()p Fw(\045M)9 b Fv(\()p Fw(h)p Fv(\)\))23 b(=)1815 5183 y Fs(Z)1898 5203 y Ft(b)1861 5372 y(a)1945 5296 y Fw(u)1993 5308 y Ft(\045)2031 5296 y Fv(\()p Fw(x)p Fv(\))p Fw(h)p Fv(\()p Fw(x)p Fv(\))14 b Fw(dx;)1053 b FO(\(6.6\))109 5502 y FA(for)30 b(any)g(r)l(e)l(al-value)l(d)h (function)e Fw(h)23 b Fu(2)h Fp(L)1332 5471 y Fo(1)1402 5502 y Fv(\([)p Fw(a;)14 b(b)p Fv(]\))p FA(.)39 b(In)29 b(p)l(articular)i(one)f(has)1045 5661 y Fu(k)p Fw(u)1135 5673 y Ft(\045)1173 5661 y Fu(k)1215 5676 y Fa(L)1249 5659 y Fl(1)1280 5676 y Fr(\()p Ft(!)r Fr(\))1403 5661 y Fv(=)23 b(tr\()p Fw(\045U)9 b Fv(\()p Fw(!)s Fv(\)\))23 b Fu(\024)g Fw(C)2024 5673 y Fr(^)-39 b Ft(\045)2057 5661 y Fu(k)p Fv(\()p Fw(K)24 b Fu(\000)18 b Fw(i)p Fv(\))2370 5626 y Fo(\000)p Fr(1)2459 5661 y Fw(P)2524 5626 y Fo(K)2512 5681 y(H)2578 5661 y Fu(k)2620 5673 y Fb(B)2679 5681 y Fl(1)3481 5661 y FO(\(6.7\))109 5820 y FA(for)30 b(e)l(ach)h(Bor)l (el)g(set)e Fw(!)d Fu(\022)c Fv([)p Fw(a;)14 b(b)p Fv(])p FA(.)p eop %%Page: 17 19 17 18 bop 109 254 a FH(Dissipativ)n(e)27 b(Sc)n(hr\366)r(dinger-t)n(yp) r(e)e(op)r(erator)2084 b FO(17)109 503 y Fy(Pro)s(of.)67 b FO(By)38 b(\(6.3\))g(w)n(e)f(get)h(that)g(\(6.6\))f(holds)h(for)f Fw(h)j Fv(=)g Fw(\037)2074 515 y Fr(\012)2125 503 y FO(.)68 b(By)37 b(linearit)n(y)h(\(6.6\))g(can)f(b)r(e)i(extended)f(for)109 602 y(arbitrary)31 b(step)i(functions)h Fw(h)p FO(.)54 b(Since)33 b Fw(u)1408 614 y Ft(\045)1479 602 y FO(is)h(from)f Fp(L)1816 572 y Fr(1)1853 602 y Fv(\()p Fw(a;)14 b(b)p Fv(\))33 b FO(and)g Fw(\045U)9 b Fv(\(\012\))34 b FO(is)f(of)g(trace)g (class,)g(\(6.6\))g(admits)g(a)109 702 y(con)n(tin)n(uation)26 b(to)i(all)f Fp(L)850 672 y Fo(1)920 702 y FO(-functions)h Fw(h)p FO(,)f(whic)n(h)h(pro)n(v)n(es)d(the)j(\034rst)g(part)f(of)g (the)h(lemma.)109 849 y(Since)e Fv(\()p Fw(K)21 b Fu(\000)14 b Fw(i)p Fv(\))588 819 y Fo(\000)p Fr(1)677 849 y Fw(P)742 819 y Fo(K)730 872 y(H)823 849 y FO(is)26 b(a)g(trace)f(class)g(op)r (erator)f(and)h(since)h Fw(\045)p Fv(\()p Fw(K)21 b Fu(\000)15 b Fw(i)p Fv(\))26 b FO(is)g(a)f(b)r(ounded)h(op)r(erator)e(whose)i (norm)109 948 y(can)h(b)r(e)h(estimated)f(b)n(y)h Fw(C)933 960 y Fr(^)-39 b Ft(\045)966 948 y FO(,)27 b(w)n(e)h(obtain)938 1114 y Fv(tr\()p Fw(\045)p Fv(\()p Fw(K)c Fu(\000)18 b Fw(i)p Fv(\)\()p Fw(K)24 b Fu(\000)18 b Fw(i)p Fv(\))1620 1079 y Fo(\000)p Fr(1)1709 1114 y Fw(U)9 b Fv(\()p Fw(!)s Fv(\)\))24 b Fu(\024)e Fw(C)2102 1126 y Fr(^)-39 b Ft(\045)2135 1114 y Fu(k)p Fv(\()p Fw(K)24 b Fu(\000)18 b Fw(i)p Fv(\))2448 1079 y Fo(\000)p Fr(1)2537 1114 y Fw(P)2602 1079 y Fo(H)2590 1134 y(K)2663 1114 y Fu(k)2705 1126 y Fb(B)2764 1134 y Fl(1)2800 1114 y Fw(:)109 1279 y FO(This)27 b(v)n(eri\034es)g (\(6.7\))o(.)2818 b Fj(\003)109 1426 y FO(Supp)r(ose)23 b(that)h Fu(f)p Fw(\045)p Fv(\()p Fw(\025)p Fv(\))p Fu(g)845 1438 y Ft(\025)p Fo(2)p Fk(R)999 1426 y FO(is)g(a)f(measurable)g (family)h(of)f(non-negativ)n(e)f(selfadjoin)n(t)i(op)r(erators)e(on)h Fq(C)3325 1396 y Ft(N)6 b Fr(+2)3478 1426 y FO(.)36 b(W)-7 b(e)109 1525 y(can)27 b(\034nd)h(a)f(family)h(of)f(unitary)g(op)r (erators)f Fu(f)p Fw(V)18 b Fv(\()p Fw(\025)p Fv(\))p Fu(g)1767 1537 y Ft(\025)p Fo(2)p Fk(R)1926 1525 y FO(on)27 b Fq(C)2095 1495 y Ft(N)6 b Fr(+2)2275 1525 y FO(suc)n(h)28 b(that)774 1918 y Fw(\045)p Fv(\()p Fw(\025)p Fv(\))c(=)f Fw(V)c Fv(\()p Fw(\025)p Fv(\))1234 1626 y Fs(0)1234 1772 y(B)1234 1822 y(B)1234 1872 y(B)1234 1922 y(B)1234 1971 y(B)1234 2025 y(@)1349 1690 y Fw(\026)1399 1702 y Ft(b)1432 1690 y Fv(\()p Fw(\025)p Fv(\))176 b(0)f Fu(\001)14 b(\001)g(\001)142 b(\001)14 b(\001)g(\001)222 b Fv(0)1426 1789 y(0)160 b Fw(\026)1678 1801 y Ft(N)1741 1789 y Fv(\()p Fw(\025)p Fv(\))119 b(0)162 b Fu(\001)14 b(\001)g(\001)222 b Fv(0)1435 1878 y FO(.)1435 1911 y(.)1435 1944 y(.)1949 1886 y(.)1981 1911 y(.)2013 1936 y(.)2504 1878 y(.)2504 1911 y(.)2504 1944 y(.)1426 2044 y Fv(0)c Fu(\001)14 b(\001)g(\001)189 b Fv(0)117 b Fw(\026)2181 2056 y Fr(1)2218 2044 y Fv(\()p Fw(\025)p Fv(\))165 b(0)1426 2144 y(0)218 b Fu(\001)14 b(\001)g(\001)154 b(\001)14 b(\001)g(\001)176 b Fv(0)162 b Fw(\026)2464 2156 y Ft(a)2504 2144 y Fv(\()p Fw(\025)p Fv(\))2659 1626 y Fs(1)2659 1772 y(C)2659 1822 y(C)2659 1872 y(C)2659 1922 y(C)2659 1971 y(C)2659 2025 y(A)2745 1918 y Fw(V)19 b Fv(\()p Fw(\025)p Fv(\))2924 1883 y Fo(\003)2963 1918 y Fw(;)109 2310 y FO(for)34 b(a.e.)57 b Fw(\025)35 b Fu(2)g Fq(R)p FO(,)42 b(where)34 b Fw(\026)1014 2322 y Ft(\034)1056 2310 y Fv(\()p Fw(\025)p Fv(\))p FO(,)j Fw(\034)44 b Fv(=)34 b Fw(a;)14 b Fv(1)p Fw(;)g Fu(\001)g(\001)g(\001)27 b Fw(;)14 b(N)t(;)g(b)p FO(,)36 b(are)e(the)g(non-negativ)n(e)f(eigen)n (v)-5 b(alues)33 b(of)i(the)g(matrix)109 2409 y Fw(\045)p Fv(\()p Fw(\025)p Fv(\))p FO(.)j(Hence)27 b(from)g(equation)g(\(6.2\))h (w)n(e)f(obtain)631 2702 y Fw(u)679 2714 y Ft(\045)717 2702 y Fv(\()p Fw(x;)14 b(\025)p Fv(\))24 b(=)1025 2560 y Fs(*)1092 2510 y(0)1092 2656 y(B)1092 2709 y(@)1206 2574 y Fw(\026)1256 2586 y Ft(b)1289 2574 y Fv(\()p Fw(\025)p Fv(\))84 b Fu(\001)14 b(\001)g(\001)178 b Fv(0)1293 2662 y FO(.)1293 2695 y(.)1293 2729 y(.)1497 2670 y(.)1529 2695 y(.)1561 2721 y(.)1769 2662 y(.)1769 2695 y(.)1769 2729 y(.)1283 2828 y Fv(0)160 b Fu(\001)14 b(\001)g(\001)97 b Fw(\026)1729 2840 y Ft(a)1769 2828 y Fv(\()p Fw(\025)p Fv(\))1924 2510 y Fs(1)1924 2656 y(C)1924 2709 y(A)2010 2702 y Fw(V)19 b Fv(\()p Fw(\025)p Fv(\))2189 2667 y Ft(T)2242 2702 y Fv(\010\()p Fw(x;)14 b(\025)p Fv(\))p Fw(;)g(V)21 b Fv(\()p Fw(\025)p Fv(\))2716 2667 y Ft(T)2769 2702 y Fv(\010\()p Fw(x;)14 b(\025)p Fv(\))3025 2560 y Fs(+)3107 2702 y Fw(:)109 3003 y FO(In)n(tro)r(ducing)40 b(the)i(measurable)e(family)i(of)g(orthogonal)d(bases)h Fu(f)p Fw(e)2295 2972 y Ft(a)2335 3003 y Fv(\()p Fw(\025)p Fv(\))p Fw(;)14 b Fu(\001)g(\001)g(\001)28 b Fw(;)14 b(e)2685 2972 y Ft(b)2718 3003 y Fv(\()p Fw(\025)p Fv(\))p Fu(g)2872 3015 y Ft(\025)p Fo(2)p Fk(R)3003 3003 y FO(,)45 b(where)c Fw(e)3364 2972 y Ft(\034)3405 3003 y Fv(\()p Fw(\025)p Fv(\))47 b(:=)109 3102 y Fw(V)19 b Fv(\()p Fw(\025)p Fv(\))p Fw(e)327 3114 y Ft(\034)369 3102 y FO(,)27 b(w)n(e)h(\034nd)1151 3218 y Fw(u)1199 3230 y Ft(\045)1237 3218 y Fv(\()p Fw(x;)14 b(\025)p Fv(\))24 b(=)1617 3139 y Fs(X)1545 3318 y Ft(\034)7 b Fr(=)p Ft(a;)p Fo(\001\001\001)i Ft(;b)1822 3218 y Fw(\026)1872 3230 y Ft(\034)1914 3218 y Fv(\()p Fw(\025)p Fv(\))p Fu(j)p Fw(\036)p Fv(\()p Fw(x;)14 b(\025;)g(e)2338 3184 y Ft(\034)2381 3218 y Fv(\()p Fw(\025)p Fv(\)\))p Fu(j)2548 3184 y Fr(2)2587 3218 y Fw(;)109 3448 y FO(for)27 b Fw(x)c Fu(2)h Fv([)p Fw(a;)14 b(b)p Fv(])27 b FO(and)g Fw(\025)d Fu(2)f Fq(R)p FO(,)34 b(where)27 b(w)n(e)g(used)h(the)f(fact)h(that)1210 3740 y Fw(V)1277 3706 y Ft(T)1329 3740 y Fv(\()p Fw(\025)p Fv(\)\010\()p Fw(x;)14 b(\025)p Fv(\))25 b(=)1810 3549 y Fs(0)1810 3695 y(B)1810 3748 y(@)1928 3612 y Fw(\036)p Fv(\()p Fw(x;)14 b(\025;)g(e)2217 3624 y Ft(b)2251 3612 y Fv(\()p Fw(\025)p Fv(\)\))2150 3701 y FO(.)2150 3734 y(.)2150 3767 y(.)1924 3867 y Fw(\036)p Fv(\()p Fw(x;)g(\025;)g(e)2213 3879 y Ft(a)2254 3867 y Fv(\()p Fw(\025)p Fv(\)\))2441 3549 y Fs(1)2441 3695 y(C)2441 3748 y(A)2528 3740 y Fw(:)930 b FO(\(6.8\))109 4157 y Fz(7)131 b(Curren)l(t)45 b(densit)l(y)109 4386 y FO(W)-7 b(e)26 b(de\034ne)h(the)f(curren)n(t)f(densit)n(y)h Fw(j)1236 4398 y Ft(\045)1275 4386 y Fv(\()p Fw(x;)14 b(\025)p Fv(\))28 b FO(at)e(p)r(oin)n(t)g Fw(x)h FO(and)f(energy)f Fw(\025)f Fu(2)f Fq(R)32 b FO(of)26 b(the)h(system)f(describ)r(ed)g(b)n (y)g(the)109 4485 y(dissipativ)n(e)g(op)r(erator)g Fw(H)35 b FO(b)n(y)771 4727 y Fw(j)805 4739 y Ft(\045)844 4727 y Fv(\()p Fw(x;)14 b(\025)p Fv(\))24 b(:=)1175 4624 y Ft(N)6 b Fr(+1)1186 4648 y Fs(X)1185 4824 y Ft(\034)h Fr(=0)1331 4727 y Fw(\026)1381 4739 y Ft(\034)1423 4727 y Fv(\()p Fw(\025)p Fv(\)Im)1649 4610 y Fs(\022)1792 4671 y Fv(1)p 1720 4708 185 4 v 1720 4784 a Fw(m)p Fv(\()p Fw(x)p Fv(\))1915 4727 y Fw(\036)p Fv(\()p Fw(x;)14 b(\025;)g(e)2204 4693 y Ft(\034)2246 4727 y Fv(\()p Fw(\025)p Fv(\)\))2390 4693 y Fo(0)p 2415 4655 477 4 v 2415 4727 a Fw(\036)p Fv(\()p Fw(x;)g(\025;)g(e)2704 4703 y Ft(\034)2747 4727 y Fv(\()p Fw(\025)p Fv(\)\))2892 4610 y Fs(\023)2967 4727 y Fw(:)491 b FO(\(7.1\))109 4969 y(See)22 b([LL71)o(])g(for)f(the) i(motiv)-5 b(ation)21 b(for)h(this)g(de\034nition.)36 b(W)-7 b(e)22 b(note)g(that)g Fw(\036)p Fv(\()p Fw(x;)14 b(\025;)g(e)2633 4939 y Ft(\034)2676 4969 y Fv(\()p Fw(\025)p Fv(\)\))2820 4939 y Fo(0)2867 4969 y FO(exists)21 b(only)h(for)f Fw(x)j Fu(6)p Fv(=)e Fw(x)3593 4981 y Ft(j)3629 4969 y FO(,)109 5082 y Fw(j)28 b Fv(=)22 b(1)p Fw(;)14 b Fu(\001)g(\001)g (\001)27 b Fw(;)14 b(N)9 b FO(.)37 b(Therefore,)26 b(the)i (de\034nition)g(\(7.1\))f(mak)n(es)g(only)g(sense)g(for)g Fw(x)c Fu(2)2664 5020 y Fs(S)2733 5041 y Ft(N)2733 5107 y(j)s Fr(=0)2852 5082 y Fv(\()p Fw(x)2931 5094 y Ft(j)2967 5082 y Fw(;)14 b(x)3051 5094 y Ft(j)s Fr(+1)3170 5082 y Fv(\))p FO(.)109 5229 y(Equation)27 b(\(7.1\))g(can)h(b)r(e)g (rewritten)f(as)998 5444 y Fw(j)1032 5456 y Ft(\045)1070 5444 y Fv(\()p Fw(x;)14 b(\025)p Fv(\))25 b(=)d(Im)1491 5327 y Fs(\022)q(\034)1615 5444 y Fw(\045)p Fv(\()p Fw(\025)p Fv(\))1770 5410 y Ft(T)1905 5388 y Fv(1)p 1833 5425 185 4 v 1833 5501 a Fw(m)p Fv(\()p Fw(x)p Fv(\))2028 5444 y(\010\()p Fw(x;)14 b(\025)p Fv(\))2284 5410 y Fo(0)2309 5444 y Fw(;)g Fv(\010\()p Fw(x;)g(\025)p Fv(\))2602 5327 y Fs(\035)q(\023)2740 5444 y Fw(:)718 b FO(\(7.2\))109 5654 y(Finally)27 b(this)h(can)f(b)r(e)h(expressed)e(as)1404 5820 y Fw(j)1438 5832 y Ft(\045)1477 5820 y Fv(\()p Fw(x;)14 b(\025)p Fv(\))25 b(=)d(tr\()p Fw(\045)p Fv(\()p Fw(\025)p Fv(\))p Fw(C)6 b Fv(\()p Fw(x;)14 b(\025)p Fv(\)\))p Fw(;)p eop %%Page: 18 20 18 19 bop 109 254 a FO(18)2477 b Fx(M.)31 b(Bar)n(o,)f(H.)h(Neidhardt) 109 503 y FO(where)161 896 y Fw(C)6 b Fv(\()p Fw(x;)14 b(\025)p Fv(\))25 b(:=)e(Im)671 629 y Fs(0)671 775 y(B)671 825 y(B)671 875 y(B)671 925 y(B)671 978 y(@)853 661 y Fr(1)p 795 675 149 4 v 795 722 a Ft(m)p Fr(\()p Ft(x)p Fr(\))953 693 y Fw(\036)p Fv(\()p Fw(x;)14 b(\025;)g(N)29 b Fv(+)18 b(1\))1456 663 y Fo(0)p 1479 621 503 4 v 1479 693 a Fw(\036)p Fv(\()p Fw(x;)c(\025;)g(N)29 b Fv(+)18 b(1\))83 b Fu(\001)14 b(\001)g(\001)2326 661 y Fr(1)p 2268 675 149 4 v 2268 722 a Ft(m)p Fr(\()p Ft(x)p Fr(\))2427 693 y Fw(\036)p Fv(\()p Fw(x;)g(\025;)g Fv(0\))2751 663 y Fo(0)p 2775 621 503 4 v 2775 693 a Fw(\036)p Fv(\()p Fw(x;)g(\025;)g(N)29 b Fv(+)18 b(1\))924 783 y Fr(1)p 866 797 149 4 v 866 844 a Ft(m)p Fr(\()p Ft(x)p Fr(\))1025 816 y Fw(\036)p Fv(\()p Fw(x;)c(\025;)g(N)28 b Fv(+)18 b(1\))1527 785 y Fo(0)p 1551 743 360 4 v 1551 816 a Fw(\036)p Fv(\()p Fw(x;)c(\025;)g(N)9 b Fv(\))156 b Fu(\001)14 b(\001)g(\001)2397 783 y Fr(1)p 2340 797 149 4 v 2340 844 a Ft(m)p Fr(\()p Ft(x)p Fr(\))2498 816 y Fw(\036)p Fv(\()p Fw(x;)g(\025;)g Fv(0\))2822 785 y Fo(0)p 2847 743 360 4 v 2847 816 a Fw(\036)p Fv(\()p Fw(x;)g(\025;)g(N)9 b Fv(\))1372 917 y FO(.)1372 950 y(.)1372 984 y(.)2756 917 y(.)2756 950 y(.)2756 984 y(.)941 1060 y Fr(1)p 883 1074 149 4 v 883 1121 a Ft(m)p Fr(\()p Ft(x)p Fr(\))1042 1092 y Fw(\036)p Fv(\()p Fw(x;)14 b(\025;)g(N)29 b Fv(+)18 b(1\))1545 1062 y Fo(0)p 1568 1020 326 4 v 1568 1092 a Fw(\036)p Fv(\()p Fw(x;)c(\025;)g Fv(0\))173 b Fu(\001)14 b(\001)g(\001)2414 1060 y Fr(1)p 2357 1074 149 4 v 2357 1121 a Ft(m)p Fr(\()p Ft(x)p Fr(\))2515 1092 y Fw(\036)p Fv(\()p Fw(x;)g(\025;)g Fv(0\))2839 1062 y Fo(0)p 2864 1020 326 4 v 2864 1092 a Fw(\036)p Fv(\()p Fw(x;)g(\025;)g Fv(0\))3319 629 y Fs(1)3319 775 y(C)3319 825 y(C)3319 875 y(C)3319 925 y(C)3319 978 y(A)3406 896 y Fw(:)52 b FO(\(7.3\))109 1352 y(The)32 b(curren)n(t)g(densit)n(y)g(is)g (strongly)f(related)h(to)g(the)h(c)n(haracteristic)e(function)i(of)f (the)h(op)r(erator)d Fw(H)7 b FO(.)52 b(This)32 b(is)109 1452 y(sho)n(wn)26 b(in)i(the)g(next)g(theorem.)109 1713 y Fy(Theorem)i(7.1)40 b FA(Assume)c(that)g Fw(m)23 b Fv(+)1376 1680 y Fr(1)p 1363 1694 59 4 v 1363 1741 a Ft(m)1467 1713 y Fu(2)35 b Fp(L)1602 1682 y Fo(1)1672 1713 y Fv(\()p Fw(a;)14 b(b)p Fv(\))p FA(,)38 b Fw(V)54 b Fu(2)35 b Fp(L)2153 1682 y Fr(2)2190 1713 y Fv(\()p Fw(a;)14 b(b)p Fv(\))p FA(,)38 b(with)f Fv(Im\()p Fw(V)19 b Fv(\))35 b(=)f(0)p FA(,)k(and)f Fw(\024)3306 1725 y Ft(a)3345 1713 y Fw(;)14 b(\024)3430 1725 y Fr(1)3467 1713 y Fw(;)g Fu(\001)g(\001)g(\001)28 b Fw(;)109 1812 y(\024)157 1824 y Ft(N)219 1812 y Fw(;)14 b(\024)304 1824 y Ft(b)361 1812 y Fu(2)25 b Fq(C)495 1824 y Fr(+)586 1812 y FA(as)30 b(wel)t(l)h(as)g Fw(x)1016 1824 y Fr(1)1054 1812 y Fw(;)14 b Fu(\001)g(\001)g(\001)27 b Fw(;)14 b(x)1299 1824 y Ft(N)1386 1812 y Fu(2)24 b Fv(\()p Fw(a;)14 b(b)p Fv(\))31 b FA(such)f(that)g Fw(x)2083 1824 y Fr(1)2145 1812 y Fw(<)23 b(x)2280 1824 y Fr(2)2342 1812 y Fw(<)g Fu(\001)14 b(\001)g(\001)24 b Fw(<)f(x)2686 1824 y Ft(N)2750 1812 y FA(.)39 b(L)l(et)30 b Fw(\045)g FA(b)l(e)h(a)f(ste)l(ady)h (state.)109 1912 y(Then)f(the)g(curr)l(ent)f(density)i Fw(j)1073 1924 y Ft(\045)1111 1912 y Fv(\()p Fw(x;)14 b(\025)p Fv(\))p FA(,)32 b Fw(x)24 b Fu(2)g Fv([)p Fw(a;)14 b(b)p Fv(])p FA(,)30 b Fw(x)23 b Fu(6)p Fv(=)g Fw(x)1937 1924 y Ft(j)1973 1912 y FA(,)30 b Fw(\025)24 b Fu(2)g Fq(R)p FA(,)36 b(is)31 b(indep)l(endent)f(of)h Fw(x)f FA(for)h Fw(x)24 b Fu(2)g Fv(\()p Fw(x)3381 1924 y Ft(j)3416 1912 y Fw(;)14 b(x)3500 1924 y Ft(j)s Fr(+1)3620 1912 y Fv(\))109 2011 y FA(and)30 b(admits)g(the)g(r)l(epr)l(esentation)1404 2142 y Fw(j)1438 2154 y Ft(\045)1477 2142 y Fv(\()p Fw(x;)14 b(\025)p Fv(\))25 b(=)d(tr\()p Fw(\045)p Fv(\()p Fw(\025)p Fv(\))p Fw(C)6 b Fv(\()p Fw(x;)14 b(\025)p Fv(\)\))p Fw(;)1128 b FO(\(7.4\))109 2323 y FA(wher)l(e)1147 2514 y Fw(C)6 b Fv(\()p Fw(x;)14 b(\025)p Fv(\))25 b(:=)1574 2410 y Ft(N)1544 2435 y Fs(X)1546 2612 y Ft(j)s Fr(=0)1677 2514 y Fw(E)1738 2526 y Ft(j)1774 2514 y Fv(\()p Fw(\025)p Fv(\)\002)1951 2526 y Ft(H)2014 2514 y Fv(\()p Fw(\025)p Fv(\))2126 2479 y Fo(\003)2165 2514 y Fw(\037)2217 2529 y Fr(\()p Ft(x)2281 2537 y Fm(j)2312 2529 y Ft(;x)2370 2537 y Fm(j)r Fl(+1)2471 2529 y Fr(\))2501 2514 y Fv(\()p Fw(x)p Fv(\))869 b FO(\(7.5\))109 2778 y FA(and)1087 2927 y Fw(E)1148 2939 y Ft(j)1184 2927 y Fv(\()p Fw(\025)p Fv(\))24 b(:=)1466 2870 y(1)p 1440 2907 92 4 v 1440 2983 a(2)p Fw(\031)1556 2834 y Fs(\020)1606 2927 y Fw(P)1671 2887 y Ft(j)1659 2947 y Fo(\000)1715 2927 y Fv(\002)1780 2939 y Ft(H)1842 2927 y Fv(\()p Fw(\025)p Fv(\))p Fw(P)2019 2887 y Ft(j)2007 2947 y Fr(+)2082 2927 y Fu(\000)18 b Fw(P)2230 2887 y Ft(j)2218 2947 y Fr(+)2274 2927 y Fv(\002)2339 2939 y Ft(H)2401 2927 y Fv(\()p Fw(\025)p Fv(\))p Fw(P)2578 2887 y Ft(j)2566 2947 y Fo(\000)2624 2834 y Fs(\021)3481 2927 y FO(\(7.6\))109 3130 y FA(with)945 3324 y Fw(P)1010 3284 y Ft(j)998 3344 y Fo(\000)1077 3324 y Fv(:=)1232 3216 y Ft(j)1188 3245 y Fs(X)1189 3421 y Ft(z)r Fr(=0)1322 3324 y Fw(P)1375 3336 y Ft(z)1413 3324 y Fw(;)99 b(P)1600 3284 y Ft(j)1588 3344 y Fr(+)1666 3324 y Fv(:=)1806 3220 y Ft(N)6 b Fr(+1)1817 3245 y Fs(X)1777 3422 y Ft(z)r Fr(=)p Ft(j)s Fr(+1)1991 3324 y Fw(P)2044 3336 y Ft(z)2082 3324 y Fw(;)184 b(P)2342 3336 y Ft(z)2404 3324 y Fv(:=)22 b Fu(h\001)p Fw(;)14 b(e)2645 3336 y Ft(z)2684 3324 y Fu(i)p Fw(e)2755 3336 y Ft(z)2793 3324 y Fw(:)109 3598 y FA(Mor)l(e)l(over,)31 b(if)g Fv(tr\()p Fw(\045)p Fv(\()p Fu(\001)p Fv(\)\))24 b Fu(2)f Fp(L)986 3568 y Fr(1)1023 3598 y Fv(\()p Fq(R)q Fv(\))p FA(,)36 b(then)30 b(the)f(total)h(curr)l (ent)f Fw(j)2041 3610 y Ft(\045)2080 3598 y Fv(\()p Fw(x)p Fv(\))h FA(at)g(p)l(oint)g Fw(x)24 b Fu(2)f Fv(\()p Fw(x)2759 3610 y Ft(j)2795 3598 y Fw(;)14 b(x)2879 3610 y Ft(j)s Fr(+1)2998 3598 y Fv(\))p FA(,)1469 3847 y Fw(j)1503 3859 y Ft(\045)1541 3847 y Fv(\()p Fw(x)p Fv(\))25 b(:=)1787 3734 y Fs(Z)1833 3922 y Fk(R)1893 3847 y Fw(j)1927 3859 y Ft(\045)1966 3847 y Fv(\()p Fw(x;)14 b(\025)p Fv(\))g Fw(d\025;)109 4091 y FA(is)30 b(\034nite)f(and)h(c)l(an)g(b)l(e)g (estimate)l(d)g(by)1387 4333 y Fu(j)p Fw(j)1444 4345 y Ft(\045)1483 4333 y Fv(\()p Fw(x)p Fv(\))p Fu(j)24 b(\024)1763 4276 y Fv(1)p 1738 4313 V 1738 4390 a(2)p Fw(\031)1854 4220 y Fs(Z)1900 4408 y Fk(R)1960 4333 y Fv(tr\()p Fw(\045)p Fv(\()p Fw(\025)p Fv(\)\))14 b Fw(d\025:)1109 b FO(\(7.7\))109 4625 y Fy(Pro)s(of.)36 b FO(W)-7 b(e)28 b(ha)n(v)n(e)1404 4755 y Fw(j)1438 4767 y Ft(\045)1477 4755 y Fv(\()p Fw(x;)14 b(\025)p Fv(\))25 b(=)d(tr\()p Fw(\045)p Fv(\()p Fw(\025)p Fv(\))p Fw(C)6 b Fv(\()p Fw(x;)14 b(\025)p Fv(\)\))p Fw(;)1128 b FO(\(7.8\))109 4936 y(where)27 b Fw(C)6 b Fv(\()p Fw(x;)14 b(\025)p Fv(\))29 b FO(is)e(giv)n(en)g(b)n(y)i(\(7.3\))o(.)37 b Fw(C)6 b Fv(\()p Fw(x;)14 b(\025)p Fv(\))29 b FO(can)f(b)r(e)g (rewritten)f(as)230 5330 y Fw(C)6 b Fv(\()p Fw(x;)14 b(\025)p Fv(\))25 b(=)614 5274 y(1)p 614 5311 42 4 v 620 5387 a Fw(i)679 5088 y Fs(0)679 5234 y(B)679 5284 y(B)679 5334 y(B)679 5387 y(@)793 5147 y Fw(W)12 b Fv(\()p 915 5075 479 4 v Fw(\036)p Fv(\()p Fu(\001)p Fw(;)i(\025;)g(N)28 b Fv(+)18 b(1\))p Fw(;)c(\036)p Fv(\()p Fu(\001)p Fw(;)g(\025;)g(N)28 b Fv(+)18 b(1\)\)\()p Fw(x)p Fv(\))85 b Fu(\001)14 b(\001)g(\001)96 b Fw(W)12 b Fv(\()p 2451 5075 446 4 v Fw(\036)p Fv(\()p Fu(\001)p Fw(;)i(\025;)g(N)29 b Fv(+)18 b(1)o Fw(;)c(\036)p Fv(\()p Fu(\001)p Fw(;)g(\025;)g Fv(0\)\)\()p Fw(x)p Fv(\))864 5256 y Fw(W)e Fv(\()p 986 5184 336 4 v Fw(\036)p Fv(\()p Fu(\001)p Fw(;)i(\025;)g(N)9 b Fv(\))r Fw(;)14 b(\036)p Fv(\()p Fu(\001)p Fw(;)g(\025;)g(N)28 b Fv(+)18 b(1\)\)\()p Fw(x)p Fv(\))156 b Fu(\001)14 b(\001)g(\001)152 b Fw(W)12 b Fv(\()p 2507 5184 V Fw(\036)p Fv(\()p Fu(\001)p Fw(;)i(\025;)g(N)9 b Fv(\))q Fw(;)14 b(\036)p Fv(\()p Fu(\001)p Fw(;)g(\025;)g Fv(0\)\)\()p Fw(x)p Fv(\))1411 5345 y FO(.)1411 5378 y(.)1411 5411 y(.)2843 5345 y(.)2843 5378 y(.)2843 5411 y(.)882 5520 y Fw(W)e Fv(\()p 1004 5448 301 4 v Fw(\036)p Fv(\()p Fu(\001)p Fw(;)i(\025;)g Fv(0\))q Fw(;)g(\036)p Fv(\()p Fu(\001)p Fw(;)g(\025;)g(N)28 b Fv(+)18 b(1\)\)\()p Fw(x)p Fv(\))173 b Fu(\001)14 b(\001)g(\001)169 b Fw(W)12 b Fv(\()p 2524 5448 V Fw(\036)p Fv(\()p Fu(\001)p Fw(;)i(\025;)g Fv(0\))q Fw(;)g(\036)p Fv(\()p Fu(\001)p Fw(;)g(\025;)g Fv(0\)\)\()p Fw(x)p Fv(\))3421 5088 y Fs(1)3421 5234 y(C)3421 5284 y(C)3421 5334 y(C)3421 5387 y(A)3507 5330 y Fw(:)3481 5620 y FO(\(7.9\))109 5720 y(Clearly)29 b Fw(W)12 b Fv(\()p 522 5648 286 4 v Fw(\036)p Fv(\()p Fu(\001)p Fw(;)i(\025;)g(l)r Fv(\))q Fw(;)g(\036)p Fv(\()p Fu(\001)p Fw(;)g(\025;)g(m)p Fv(\)\)\()p Fw(x)p Fv(\))33 b FO(is)e(constan)n(t)f(for)g(ev)n(ery)f Fw(x)i FO(in)g(eac)n(h)f(subin)n(terv)-5 b(al)30 b Fv(\()p Fw(x)3004 5732 y Ft(j)3040 5720 y Fw(;)14 b(x)3124 5732 y Ft(j)s Fr(+1)3243 5720 y Fv(\))p FO(,)32 b(for)e(ev)n(ery)109 5820 y Fw(l)r(;)14 b(m)22 b Fv(=)h(0)p Fw(;)14 b Fu(\001)g(\001)g(\001) 26 b Fw(;)14 b(N)28 b Fv(+)18 b(1)p FO(.)36 b(Hence,)28 b Fw(j)1178 5832 y Ft(\045)1217 5820 y Fv(\()p Fw(x;)14 b(\025)p Fv(\))1413 5835 y Fo(j)p Fr(\()p Ft(x)1497 5843 y Fm(j)1528 5835 y Ft(;x)1586 5843 y Fm(j)r Fl(+1)1688 5835 y Fr(\))1745 5820 y FO(is)28 b(constan)n(t.)p eop %%Page: 19 21 19 20 bop 109 254 a FH(Dissipativ)n(e)27 b(Sc)n(hr\366)r(dinger-t)n(yp) r(e)e(op)r(erator)2084 b FO(19)109 503 y(Let)32 b(us)g(compute)f Fw(C)6 b Fv(\()p Fw(x;)14 b(\025)p Fv(\))974 518 y Fo(j)p Fr(\()p Ft(x)1058 526 y Fm(j)1090 518 y Ft(;x)1148 526 y Fm(j)r Fl(+1)1250 518 y Fr(\))1280 503 y FO(.)50 b(Since)32 b Fw(\036)p Fv(\()p Fw(x;)14 b(\025;)g(k)s Fv(\))33 b FO(has)f(to)f(ful\034l)i(the)f(b)r(oundary)f(conditions)h(\(5.5\))f (and)109 602 y(\(5.6\))c(\(see)g(also)g(\034gure)g(1\),)g(w)n(e)g(get) 206 794 y Fw(W)12 b Fv(\()p 328 722 286 4 v Fw(\036)p Fv(\()p Fu(\001)p Fw(;)i(\025;)g(l)r Fv(\))q Fw(;)g(\036)p Fv(\()p Fu(\001)p Fw(;)g(\025;)g(m)p Fv(\)\)\()p Fw(x)p Fv(\))1125 809 y Fo(j)p Fr(\()p Ft(x)1209 817 y Fm(j)1242 809 y Ft(;x)1300 817 y Fm(j)r Fl(+1)1401 809 y Fr(\))1454 794 y Fv(=)p Fw(W)e Fv(\()p 1641 722 V Fw(\036)p Fv(\()p Fu(\001)p Fw(;)i(\025;)g(l)r Fv(\))q Fw(;)g(\036)p Fv(\()p Fu(\001)p Fw(;)g(\025;)g(m)p Fv(\)\)\()p Fw(x)2406 806 y Ft(j)2462 794 y Fv(+)k(0\))1454 938 y(=)p Fw(W)12 b Fv(\()p 1641 866 V Fw(\036)p Fv(\()p Fu(\001)p Fw(;)i(\025;)g(l)r Fv(\))q Fw(;)g(\036)p Fv(\()p Fu(\001)p Fw(;)g(\025;)g(m)p Fv(\)\)\()p Fw(x)2406 950 y Ft(j)2462 938 y Fu(\000)k Fv(0\))1537 1087 y Fu(\000)g Fw(i\013)1702 1052 y Fr(2)1702 1107 y Ft(j)p 1740 1014 346 4 v 1740 1087 a Fw(\036)p Fv(\()p Fw(x)1868 1099 y Ft(j)1904 1087 y Fw(;)c(\025;)g(l)r Fv(\))p Fw(\036)p Fv(\()p Fw(x)2213 1099 y Ft(j)2249 1087 y Fw(;)g(\025;)g(m)p Fv(\))19 b(+)f Fw(\016)2615 1099 y Ft(j)s(m)2741 1031 y Fw(\013)2794 1043 y Ft(m)p 2719 1068 161 4 v 2719 1084 a Fu(p)p 2788 1084 92 4 v 69 x Fv(2)p Fw(\031)p 2890 1014 374 4 v 2890 1087 a(\036)p Fv(\()p Fw(x)3018 1099 y Ft(m)3082 1087 y Fw(;)c(\025;)g(l)r Fv(\))1454 1276 y(=)p Fw(W)e Fv(\()p 1641 1204 286 4 v Fw(\036)p Fv(\()p Fu(\001)p Fw(;)i(\025;)g(l)r Fv(\))q Fw(;)g(\036)p Fv(\()p Fu(\001)p Fw(;)g(\025;)g(m)p Fv(\)\)\()p Fw(x)2406 1288 y Ft(j)s Fo(\000)p Fr(1)2546 1276 y Fv(+)19 b(0\))1537 1425 y Fu(\000)f Fw(i\013)1702 1391 y Fr(2)1702 1445 y Ft(j)p 1740 1353 346 4 v 1740 1425 a Fw(\036)p Fv(\()p Fw(x)1868 1437 y Ft(j)1904 1425 y Fw(;)c(\025;)g(l)r Fv(\))p Fw(\036)p Fv(\()p Fw(x)2213 1437 y Ft(j)2249 1425 y Fw(;)g(\025;)g(m)p Fv(\))19 b(+)f Fw(\016)2615 1437 y Ft(j)s(m)2741 1369 y Fw(\013)2794 1381 y Ft(m)p 2719 1406 161 4 v 2719 1422 a Fu(p)p 2788 1422 92 4 v 69 x Fv(2)p Fw(\031)p 2890 1353 374 4 v 2890 1425 a(\036)p Fv(\()p Fw(x)3018 1437 y Ft(m)3082 1425 y Fw(;)c(\025;)g(l)r Fv(\))1496 1594 y FO(.)1496 1627 y(.)1496 1660 y(.)1454 1872 y Fv(=)1578 1764 y Ft(j)1534 1793 y Fs(X)1533 1969 y Ft(\034)7 b Fr(=0)1668 1755 y Fs(\032)1730 1872 y Fu(\000)p Fw(i\013)1877 1837 y Fr(2)1877 1892 y Ft(\034)p 1918 1799 352 4 v 1918 1872 a Fw(\036)p Fv(\()p Fw(x)2046 1884 y Ft(\034)2089 1872 y Fw(;)14 b(\025;)g(l)r Fv(\))p Fw(\036)p Fv(\()p Fw(x)2398 1884 y Ft(\034)2440 1872 y Fw(;)g(\025;)g(m)p Fv(\))19 b(+)f Fw(\016)2806 1884 y Ft(\034)s(;m)2966 1815 y Fw(\013)3019 1827 y Ft(\034)p 2933 1853 161 4 v 2933 1869 a Fu(p)p 3002 1869 92 4 v 69 x Fv(2)p Fw(\031)p 3104 1799 352 4 v 3104 1872 a(\036)p Fv(\()p Fw(x)3232 1884 y Ft(\034)3274 1872 y Fw(;)c(\025;)g(l)r Fv(\))3455 1755 y Fs(\033)3531 1872 y Fw(;)109 2140 y FO(for)39 b Fw(l)r(;)14 b(m)42 b Fv(=)h(0)p Fw(;)14 b Fu(\001)g(\001)g(\001)27 b Fw(;)14 b(N)35 b Fv(+)26 b(1)39 b FO(and)g Fw(l)45 b(>)d(j)5 b FO(,)43 b(where)c Fw(\016)1793 2152 y Ft(\034)s(;m)1952 2140 y Fv(=)k Fu(h)p Fw(e)2131 2152 y Ft(\034)2172 2140 y Fw(;)14 b(e)2248 2152 y Ft(m)2311 2140 y Fu(i)p FO(.)73 b(Similar)39 b(w)n(e)h(obtain)f(for)g(all)g Fw(l)r(;)14 b(m)42 b Fv(=)109 2240 y(0)p Fw(;)14 b Fu(\001)g(\001)g (\001)27 b Fw(;)14 b(N)27 b Fv(+)18 b(1)p FO(,)27 b Fw(l)d Fu(\024)f Fw(j)5 b FO(,)147 2432 y Fw(W)12 b Fv(\()p 269 2360 286 4 v Fw(\036)p Fv(\()p Fu(\001)p Fw(;)i(\025;)g(l)r Fv(\))q Fw(;)g(\036)p Fv(\()p Fu(\001)p Fw(;)g(\025;)g(m)p Fv(\)\)\()p Fw(x)p Fv(\))1066 2447 y Fo(j)p Fr(\()p Ft(x)1150 2455 y Fm(j)1183 2447 y Ft(;x)1241 2455 y Fm(j)r Fl(+1)1342 2447 y Fr(\))1396 2432 y Fv(=)o Fw(W)e Fv(\()p 1582 2360 311 4 v Fw(\036)p Fv(\()p Fw(x;)i(\025;)g(l)r Fv(\))q Fw(;)g(\036)p Fv(\()p Fw(x;)g(\025;)g(m)p Fv(\)\)\()p Fw(x)2395 2444 y Ft(j)s Fr(+1)2535 2432 y Fu(\000)k Fv(0\))1396 2575 y(=)o Fw(W)12 b Fv(\()p 1582 2503 286 4 v Fw(\036)p Fv(\()p Fu(\001)p Fw(;)i(\025;)g(l)r Fv(\))q Fw(;)g(\036)p Fv(\()p Fu(\001)p Fw(;)g(\025;)g(m)p Fv(\)\)\()p Fw(x)2347 2587 y Ft(j)s Fr(+1)2487 2575 y Fv(+)k(0\))1479 2724 y(+)g Fw(i\013)1644 2736 y Ft(j)s Fr(+1)p 1762 2652 430 4 v 1762 2724 a Fw(\036)p Fv(\()p Fw(x)1890 2736 y Ft(j)s Fr(+1)2011 2724 y Fw(;)c(\025;)g(l)r Fv(\))o Fw(\036)p Fv(\()p Fw(x)2319 2736 y Ft(j)s Fr(+1)2440 2724 y Fw(;)g(\025;)g(m)p Fv(\))19 b Fu(\000)f Fw(\016)2806 2736 y Ft(j)s Fr(+1)p Ft(;m)3036 2668 y Fw(\013)3089 2680 y Ft(m)p 3013 2705 161 4 v 3013 2722 a Fu(p)p 3082 2722 92 4 v 68 x Fv(2)p Fw(\031)p 3184 2652 430 4 v 3184 2724 a(\036)p Fv(\()p Fw(x)3312 2736 y Ft(j)s Fr(+1)3432 2724 y Fw(;)c(\025;)g(l)r Fv(\))1396 2913 y(=)o Fw(W)e Fv(\()p 1582 2841 286 4 v Fw(\036)p Fv(\()p Fu(\001)p Fw(;)i(\025;)g(l)r Fv(\))q Fw(;)g(\036)p Fv(\()p Fu(\001)p Fw(;)g(\025;)g(m)p Fv(\)\)\()p Fw(x)2347 2925 y Ft(j)s Fr(+2)2487 2913 y Fu(\000)k Fv(0\))1479 3062 y(+)g Fw(i\013)1644 3074 y Ft(j)s Fr(+1)p 1762 2990 430 4 v 1762 3062 a Fw(\036)p Fv(\()p Fw(x)1890 3074 y Ft(j)s Fr(+1)2011 3062 y Fw(;)c(\025;)g(l)r Fv(\))o Fw(\036)p Fv(\()p Fw(x)2319 3074 y Ft(j)s Fr(+1)2440 3062 y Fw(;)g(\025;)g(m)p Fv(\))19 b Fu(\000)f Fw(\016)2806 3074 y Ft(j)s Fr(+1)p Ft(;m)3036 3006 y Fw(\013)3089 3018 y Ft(m)p 3013 3043 161 4 v 3013 3060 a Fu(p)p 3082 3060 92 4 v 68 x Fv(2)p Fw(\031)p 3184 2990 430 4 v 3184 3062 a(\036)p Fv(\()p Fw(x)3312 3074 y Ft(j)s Fr(+1)3432 3062 y Fw(;)c(\025;)g(l)r Fv(\))1437 3231 y FO(.)1437 3264 y(.)1437 3298 y(.)1396 3506 y Fv(=)1504 3403 y Ft(N)6 b Fr(+1)1516 3427 y Fs(X)1474 3604 y Ft(\034)h Fr(=)p Ft(j)s Fr(+1)1691 3389 y Fs(\032)1753 3506 y Fw(i\013)1835 3472 y Fr(2)1835 3527 y Ft(\034)p 1876 3434 352 4 v 1876 3506 a Fw(\036)p Fv(\()p Fw(x)2004 3518 y Ft(\034)2047 3506 y Fw(;)14 b(\025;)g(l)r Fv(\))p Fw(\036)p Fv(\()p Fw(x)2356 3518 y Ft(\034)2399 3506 y Fw(;)g(\025;)g(m)p Fv(\))19 b Fu(\000)f Fw(\016)2765 3518 y Ft(\034)s(;m)2924 3450 y Fw(\013)2977 3462 y Ft(\034)p 2891 3487 161 4 v 2891 3504 a Fu(p)p 2960 3504 92 4 v 69 x Fv(2)p Fw(\031)p 3062 3434 352 4 v 3062 3506 a(\036)p Fv(\()p Fw(x)3190 3518 y Ft(\034)3233 3506 y Fw(;)c(\025;)g(l)r Fv(\))3414 3389 y Fs(\033)3490 3506 y Fw(:)109 3782 y FO(Note)27 b(that)1268 3858 y Fv(1)p 1208 3895 161 4 v 1208 3912 a Fu(p)p 1277 3912 92 4 v 68 x Fv(2)p Fw(\031)1379 3914 y Fu(h)p Fw(\013T)1513 3926 y Fo(\003)1552 3914 y Fv(\()p Fw(\025)p Fv(\))1664 3880 y Fo(\003)1703 3914 y Fw(e)1742 3926 y Ft(\030)1778 3914 y Fw(;)14 b(e)1854 3926 y Ft(\034)1895 3914 y Fu(i)23 b Fv(=)g Fu(\000)p Fw(\013)2156 3926 y Ft(\034)2197 3914 y Fw(\036)p Fv(\()p Fw(x)2325 3926 y Ft(\034)2368 3914 y Fw(;)14 b(\025;)g(\030)t Fv(\))109 4122 y FO(for)27 b(all)g Fw(\034)5 b(;)14 b(\030)27 b Fv(=)c(0)p Fw(;)14 b Fu(\001)g(\001)g(\001)27 b Fw(;)14 b(N)27 b Fv(+)18 b(1)p FO(.)36 b(Hence)28 b(w)n(e)f(\034nd)155 4393 y Fw(W)12 b Fv(\()p 277 4321 286 4 v Fw(\036)p Fv(\()p Fu(\001)p Fw(;)i(\025;)g(l)r Fv(\))q Fw(;)g(\036)p Fv(\()p Fu(\001)p Fw(;)g(\025;)g(m)p Fv(\)\)\()p Fw(x)p Fv(\))1074 4408 y Fo(j)p Fr(\()p Ft(x)1158 4416 y Fm(j)1191 4408 y Ft(;x)1249 4416 y Fm(j)r Fl(+1)1350 4408 y Fr(\))1403 4393 y Fv(=)1561 4337 y(1)p 1501 4374 161 4 v 1501 4390 a Fu(p)p 1570 4390 92 4 v 69 x Fv(2)p Fw(\031)1731 4285 y Ft(j)1686 4314 y Fs(X)1686 4490 y Ft(\034)7 b Fr(=0)1821 4301 y Fs(n)1876 4393 y Fu(\000)p Fw(i)1970 4320 y Fu(p)p 2039 4320 V 73 x Fv(2)p Fw(\031)r(\013)2183 4405 y Ft(\034)2225 4393 y Fw(\036)p Fv(\()p Fw(x)2353 4405 y Ft(\034)2396 4393 y Fw(;)14 b(\025;)g(m)p Fv(\))19 b(+)f Fw(\016)2762 4405 y Ft(\034)s(;m)2878 4301 y Fs(o)2947 4393 y Fw(\013)3000 4405 y Ft(\034)p 3042 4321 352 4 v 3042 4393 a Fw(\036)p Fv(\()p Fw(x)3170 4405 y Ft(\034)3212 4393 y Fw(;)c(\025;)g(l)r Fv(\))47 b FO(\(7.10\))1403 4686 y Fv(=)23 b Fu(\000)1591 4630 y Fv(1)p 1566 4667 92 4 v 1566 4743 a(2)p Fw(\031)1726 4578 y Ft(j)1682 4607 y Fs(X)1681 4783 y Ft(\034)7 b Fr(=0)1816 4686 y Fu(f)p Fw(i)p Fu(h)p Fw(\013T)2021 4698 y Fo(\003)2059 4686 y Fv(\()p Fw(\025)p Fv(\))2171 4652 y Fo(\003)2210 4686 y Fw(e)2249 4698 y Ft(m)2312 4686 y Fw(;)14 b(e)2388 4698 y Ft(\034)2429 4686 y Fu(i)k Fv(+)g Fu(h)p Fw(e)2633 4698 y Ft(m)2696 4686 y Fw(;)c(e)2772 4698 y Ft(\034)2814 4686 y Fu(ig)p 2901 4614 538 4 v 13 w(h)p Fw(\013T)3035 4698 y Fo(\003)3074 4686 y Fv(\()p Fw(\025)p Fv(\))3186 4662 y Fo(\003)3225 4686 y Fw(e)3264 4698 y Ft(l)3289 4686 y Fw(;)g(e)3365 4698 y Ft(\034)3406 4686 y Fu(i)109 4954 y FO(for)27 b Fw(l)d(>)f(j)5 b FO(.)37 b(Using)27 b(equation)g(\(4.9\))g(w)n(e)g(get)401 5230 y Fw(W)12 b Fv(\()p 523 5158 286 4 v Fw(\036)p Fv(\()p Fu(\001)p Fw(;)i(\025;)g(l)r Fv(\))q Fw(;)g(\036)p Fv(\()p Fu(\001)p Fw(;)g(\025;)g(m)p Fv(\)\)\()p Fw(x)p Fv(\))1320 5245 y Fo(j)p Fr(\()p Ft(x)1404 5253 y Fm(j)1437 5245 y Ft(;x)1495 5253 y Fm(j)r Fl(+1)1596 5245 y Fr(\))1649 5230 y Fv(=)23 b Fu(\000)1837 5174 y Fv(1)p 1812 5211 92 4 v 1812 5287 a(2)p Fw(\031)1972 5123 y Ft(j)1928 5151 y Fs(X)1927 5327 y Ft(\034)7 b Fr(=0)2048 5230 y Fu(h)p Fv(\002)2145 5242 y Ft(H)2208 5230 y Fv(\()p Fw(\025)p Fv(\))2320 5196 y Fo(\003)2359 5230 y Fw(e)2398 5242 y Ft(m)2461 5230 y Fw(;)14 b(e)2537 5242 y Ft(\034)2578 5230 y Fu(i)p 2610 5158 538 4 v(h)p Fw(\013T)2744 5242 y Fo(\003)2783 5230 y Fv(\()p Fw(\025)p Fv(\))2895 5206 y Fo(\003)2934 5230 y Fw(e)2973 5242 y Ft(l)2998 5230 y Fw(;)g(e)3074 5242 y Ft(\034)3115 5230 y Fu(i)293 b FO(\(7.11\))1649 5478 y Fv(=)23 b Fu(\000)1837 5422 y Fv(1)p 1812 5459 92 4 v 1812 5535 a(2)p Fw(\031)1913 5478 y Fu(h)p Fw(P)2010 5439 y Ft(j)1998 5499 y Fo(\000)2055 5478 y Fv(\002)2120 5490 y Ft(H)2182 5478 y Fv(\()p Fw(\025)p Fv(\))2294 5444 y Fo(\003)2334 5478 y Fw(e)2373 5490 y Ft(m)2435 5478 y Fw(;)14 b(\013T)2574 5490 y Fo(\003)2612 5478 y Fv(\()p Fw(\025)p Fv(\))2724 5444 y Fo(\003)2763 5478 y Fw(e)2802 5490 y Ft(l)2828 5478 y Fu(i)109 5671 y FO(or)591 5804 y Fw(W)e Fv(\()p 713 5732 286 4 v Fw(\036)p Fv(\()p Fu(\001)p Fw(;)i(\025;)g(l)r Fv(\))q Fw(;)g(\036)p Fv(\()p Fu(\001)p Fw(;)g(\025;)g(m)p Fv(\)\)\()p Fw(x)p Fv(\))1510 5819 y Fo(j)p Fr(\()p Ft(x)1594 5827 y Fm(j)1627 5819 y Ft(;x)1685 5827 y Fm(j)r Fl(+1)1786 5819 y Fr(\))1840 5804 y Fv(=)22 b Fu(\000)2033 5748 y Fw(i)p 2002 5785 92 4 v 2002 5861 a Fv(2)p Fw(\031)2104 5804 y Fu(h)p Fw(P)2201 5764 y Ft(j)2189 5825 y Fo(\000)2245 5804 y Fv(\002)2310 5816 y Ft(H)2373 5804 y Fv(\()p Fw(\025)p Fv(\))2485 5770 y Fo(\003)2524 5804 y Fw(e)2563 5816 y Ft(m)2626 5804 y Fw(;)14 b Fv(\002)2728 5816 y Ft(H)2790 5804 y Fv(\()p Fw(\025)p Fv(\))2902 5770 y Fo(\003)2941 5804 y Fw(P)3006 5764 y Ft(j)2994 5825 y Fr(+)3073 5804 y Fw(e)3112 5816 y Ft(l)3137 5804 y Fu(i)271 b FO(\(7.12\))p eop %%Page: 20 22 20 21 bop 109 254 a FO(20)2477 b Fx(M.)31 b(Bar)n(o,)f(H.)h(Neidhardt) 109 503 y FO(for)c Fw(l)d(>)f(j)5 b FO(.)37 b(If)28 b Fw(l)c Fu(\024)f Fw(j)5 b FO(,)27 b(then)245 729 y Fw(W)12 b Fv(\()p 367 657 286 4 v Fw(\036)p Fv(\()p Fu(\001)p Fw(;)i(\025;)g(l)r Fv(\))q Fw(;)g(\036)p Fv(\()p Fu(\001)p Fw(;)g(\025;)g(m)p Fv(\)\)\()p Fw(x)p Fv(\))1164 744 y Fo(j)p Fr(\()p Ft(x)1248 752 y Fm(j)1281 744 y Ft(;x)1339 752 y Fm(j)r Fl(+1)1440 744 y Fr(\))1494 729 y Fv(=)1651 673 y(1)p 1591 710 161 4 v 1591 726 a Fu(p)p 1660 726 92 4 v 69 x Fv(2)p Fw(\031)1806 625 y Ft(N)6 b Fr(+1)1818 650 y Fs(X)1776 827 y Ft(\034)h Fr(=)p Ft(j)s Fr(+1)1993 637 y Fs(n)2048 729 y Fw(i)2077 656 y Fu(p)p 2146 656 V 73 x Fv(2)p Fw(\031)s(\013)2291 741 y Ft(\034)2332 729 y Fw(\036)p Fv(\()p Fw(x)2460 741 y Ft(\034)2503 729 y Fw(;)14 b(\025;)g(m)p Fv(\))19 b Fu(\000)f Fw(\016)2869 741 y Ft(\034)s(;m)2985 637 y Fs(o)3055 729 y Fw(\013)3108 741 y Ft(\034)p 3149 657 352 4 v 3149 729 a Fw(\036)p Fv(\()p Fw(x)3277 741 y Ft(\034)3320 729 y Fw(;)c(\025;)g(l)r Fv(\))3440 917 y FO(\(7.13\))1494 1126 y Fv(=)1616 1069 y(1)p 1591 1107 92 4 v 1591 1183 a(2)p Fw(\031)1737 1022 y Ft(N)6 b Fr(+1)1748 1047 y Fs(X)1707 1224 y Ft(\034)h Fr(=)p Ft(j)s Fr(+1)1924 1126 y Fu(f)o Fw(i)p Fu(h)p Fw(T)2075 1138 y Fo(\003)2113 1126 y Fv(\()p Fw(\025)p Fv(\))2225 1091 y Fo(\003)2264 1126 y Fw(e)2303 1138 y Ft(m)2366 1126 y Fw(;)14 b(e)2442 1138 y Ft(\034)2483 1126 y Fu(i)19 b Fv(+)f Fu(h)p Fw(e)2688 1138 y Ft(m)2751 1126 y Fw(;)c(e)2827 1138 y Ft(\034)2868 1126 y Fu(ig)p 2955 1053 538 4 v 13 w(h)p Fw(\013T)3089 1138 y Fo(\003)3128 1126 y Fv(\()p Fw(\025)p Fv(\))3240 1102 y Fo(\003)3279 1126 y Fw(e)3318 1138 y Ft(l)3343 1126 y Fw(;)g(e)3419 1138 y Ft(\034)3460 1126 y Fu(i)p Fw(:)109 1364 y FO(Using)27 b(again)f(\(4.9\))i(w)n(e)f(obtain)392 1595 y Fw(W)12 b Fv(\()p 514 1522 286 4 v Fw(\036)p Fv(\()p Fu(\001)p Fw(;)i(\025;)g(l)r Fv(\))q Fw(;)g(\036)p Fv(\()p Fu(\001)p Fw(;)g(\025;)g(m)p Fv(\)\)\()p Fw(x)p Fv(\))1311 1610 y Fo(j)p Fr(\()p Ft(x)1395 1618 y Fm(j)1428 1610 y Ft(;x)1486 1618 y Fm(j)r Fl(+1)1587 1610 y Fr(\))1641 1595 y Fv(=)1763 1539 y(1)p 1738 1576 92 4 v 1738 1652 a(2)p Fw(\031)1884 1491 y Ft(N)6 b Fr(+1)1895 1516 y Fs(X)1854 1693 y Ft(\034)h Fr(=)p Ft(j)s Fr(+1)2057 1595 y Fu(h)p Fv(\002)2154 1607 y Ft(H)2217 1595 y Fv(\()p Fw(\025)p Fv(\))2329 1560 y Fo(\003)2368 1595 y Fw(e)2407 1607 y Ft(m)2470 1595 y Fw(;)14 b(e)2546 1607 y Ft(\034)2587 1595 y Fu(i)p 2619 1522 538 4 v(h)p Fw(\013T)2753 1607 y Fo(\003)2791 1595 y Fv(\()p Fw(\025)p Fv(\))2903 1571 y Fo(\003)2943 1595 y Fw(e)2982 1607 y Ft(l)3007 1595 y Fw(;)g(e)3083 1607 y Ft(\034)3124 1595 y Fu(i)284 b FO(\(7.14\))1641 1856 y Fv(=)1770 1800 y Fw(i)p 1738 1837 92 4 v 1738 1913 a Fv(2)p Fw(\031)1840 1856 y Fu(h)p Fw(P)1937 1816 y Ft(j)1925 1877 y Fr(+)1981 1856 y Fv(\002)2046 1868 y Ft(H)2108 1856 y Fv(\()p Fw(\025)p Fv(\))2220 1822 y Fo(\003)2259 1856 y Fw(e)2298 1868 y Ft(m)2361 1856 y Fw(;)14 b Fv(\002)2463 1868 y Ft(H)2526 1856 y Fv(\()p Fw(\025)p Fv(\))2638 1822 y Fo(\003)2677 1856 y Fw(e)2716 1868 y Ft(l)2741 1856 y Fu(i)109 2029 y FO(whic)n(h)27 b(yields)624 2214 y Fw(W)12 b Fv(\()p 746 2142 286 4 v Fw(\036)p Fv(\()p Fu(\001)p Fw(;)i(\025;)g(l)r Fv(\))q Fw(;)g(\036)p Fv(\()p Fu(\001)p Fw(;)g(\025;)g(m)p Fv(\)\)\()p Fw(x)p Fv(\))1543 2229 y Fo(j)p Fr(\()p Ft(x)1627 2237 y Fm(j)1659 2229 y Ft(;x)1717 2237 y Fm(j)r Fl(+1)1819 2229 y Fr(\))1872 2214 y Fv(=)2001 2158 y Fw(i)p 1970 2195 92 4 v 1970 2271 a Fv(2)p Fw(\031)2071 2214 y Fu(h)p Fw(P)2168 2174 y Ft(j)2156 2235 y Fr(+)2212 2214 y Fv(\002)2277 2226 y Ft(H)2340 2214 y Fv(\()p Fw(\025)p Fv(\))2452 2180 y Fo(\003)2491 2214 y Fw(e)2530 2226 y Ft(m)2592 2214 y Fw(;)g Fv(\002)2694 2226 y Ft(H)2757 2214 y Fv(\()p Fw(\025)p Fv(\))2869 2180 y Fo(\003)2908 2214 y Fw(P)2973 2174 y Ft(j)2961 2235 y Fo(\000)3040 2214 y Fw(e)3079 2226 y Ft(l)3105 2214 y Fu(i)303 b FO(\(7.15\))109 2392 y(for)27 b Fw(l)d Fu(\024)f Fw(j)5 b FO(.)37 b(T)-7 b(aking)26 b(in)n(to)i(accoun)n(t)e(\(7.9\),)i(\(7.12\))e(and)i(\(7.15\))f(w)n(e)g (\034nd)381 2581 y Fu(h)p Fw(C)6 b Fv(\()p Fw(x;)14 b(\025)p Fv(\))p Fw(e)713 2593 y Ft(m)778 2581 y Fw(;)g(e)854 2593 y Ft(l)879 2581 y Fv(\))p Fu(i)943 2596 y Fo(j)p Fr(\()p Ft(x)1027 2604 y Fm(j)1058 2596 y Ft(;x)1116 2604 y Fm(j)r Fl(+1)1217 2596 y Fr(\))1270 2581 y Fv(=)1393 2524 y(1)p 1368 2562 V 1368 2638 a(2)p Fw(\031)1483 2488 y Fs(D)1534 2581 y Fu(f)p Fw(P)1641 2541 y Ft(j)1629 2601 y Fo(\000)1685 2581 y Fv(\002)1750 2593 y Ft(H)1812 2581 y Fv(\()p Fw(\025)p Fv(\))p Fw(P)1989 2541 y Ft(j)1977 2601 y Fr(+)2052 2581 y Fu(\000)k Fw(P)2200 2541 y Ft(j)2188 2601 y Fr(+)2244 2581 y Fv(\002)2309 2593 y Ft(H)2371 2581 y Fv(\()p Fw(\025)p Fv(\))p Fw(P)2548 2541 y Ft(j)2536 2601 y Fo(\000)2594 2581 y Fu(g)p Fv(\002)2701 2593 y Ft(H)2763 2581 y Fv(\()p Fw(\025)p Fv(\))2875 2546 y Fo(\003)2914 2581 y Fw(e)2953 2593 y Ft(m)3016 2581 y Fw(;)c(e)3092 2593 y Ft(l)3117 2488 y Fs(E)3440 2581 y FO(\(7.16\))109 2758 y(for)27 b Fw(m;)14 b(l)24 b Fv(=)f(0)p Fw(;)14 b Fv(1)p Fw(;)g(:)g(:)g(:)e(;)i(N)27 b Fv(+)18 b(1)p FO(.)37 b(Using)27 b(notation)g(\(7.5\))g(and)h(\(7.6\))f(w)n(e)g (immediately)h(obtain)f(\(7.4\).)109 2905 y(F)-7 b(rom)27 b(the)h(ab)r(o)n(v)n(e)e(calculations)g(w)n(e)h(get)h(for)f(the)h (total)f(curren)n(t)g Fw(j)2209 2917 y Ft(\045)2247 2905 y Fv(\()p Fw(x)p Fv(\))p FO(,)i Fw(x)24 b Fu(2)f Fv(\()p Fw(x)2638 2917 y Ft(j)2674 2905 y Fw(;)14 b(x)2758 2917 y Ft(j)s Fr(+1)2877 2905 y Fv(\))p FO(,)616 3097 y Fu(j)p Fw(j)673 3109 y Ft(\045)712 3097 y Fv(\()p Fw(x)p Fv(\))p Fu(j)24 b(\024)f(k)p Fw(\045)p Fv(\()p Fw(\025)p Fv(\))p Fw(C)6 b Fv(\()p Fw(x;)14 b(\025)p Fv(\))p Fu(k)1458 3113 y Fb(B)1517 3121 y Fl(1)1551 3113 y Fr(\()p Fk(C)1613 3097 y Fm(N)5 b Fl(+2)1743 3113 y Fr(\))1796 3097 y Fu(\024)1919 3041 y Fv(1)p 1894 3078 V 1894 3154 a(2)p Fw(\031)1995 3097 y Fu(k)p Fw(\045)p Fv(\()p Fw(\025)p Fv(\))p Fu(k)2234 3113 y Fb(B)2293 3121 y Fl(1)2326 3113 y Fr(\()p Fk(C)2388 3097 y Fm(N)g Fl(+2)2518 3113 y Fr(\))2548 3097 y Fu(k)p Fw(E)2651 3109 y Ft(j)2686 3097 y Fv(\()p Fw(\025)p Fv(\))p Fu(k)2840 3113 y Fb(B)p Fr(\()p Fk(C)2961 3097 y Fm(N)g Fl(+2)3091 3113 y Fr(\))3121 3097 y Fw(:)109 3274 y FO(By)28 b(\(7.6\))f(one)h(gets)700 3424 y Fw(E)761 3436 y Ft(j)796 3424 y Fv(\()p Fw(\025)p Fv(\))908 3390 y Fo(\003)947 3424 y Fw(E)1008 3436 y Ft(j)1044 3424 y Fv(\()p Fw(\025)p Fv(\))c(=)e Fw(P)1332 3384 y Ft(j)1320 3445 y Fr(+)1376 3424 y Fv(\002)1441 3436 y Ft(H)1503 3424 y Fv(\()p Fw(\025)p Fv(\))1615 3390 y Fo(\003)1655 3424 y Fw(P)1720 3384 y Ft(j)1708 3445 y Fo(\000)1764 3424 y Fv(\002)1829 3436 y Ft(H)1891 3424 y Fv(\()p Fw(\025)p Fv(\))p Fw(P)2068 3384 y Ft(j)2056 3445 y Fr(+)2131 3424 y Fv(+)c Fw(P)2279 3384 y Ft(j)2267 3445 y Fo(\000)2324 3424 y Fv(\002)2389 3436 y Ft(H)2451 3424 y Fv(\()p Fw(\025)p Fv(\))2563 3390 y Fo(\003)2602 3424 y Fw(P)2667 3384 y Ft(j)2655 3445 y Fr(+)2711 3424 y Fv(\002)2776 3436 y Ft(H)2838 3424 y Fv(\()p Fw(\025)p Fv(\))p Fw(P)3015 3384 y Ft(j)3003 3445 y Fo(\000)109 3574 y FO(whic)n(h)27 b(yields)1564 3673 y Fw(E)1625 3685 y Ft(j)1660 3673 y Fv(\()p Fw(\025)p Fv(\))1772 3639 y Fo(\003)1811 3673 y Fw(E)1872 3685 y Ft(j)1907 3673 y Fv(\()p Fw(\025)p Fv(\))d Fu(\024)f Fw(I)7 b(:)109 3803 y FO(Hence)27 b Fu(k)p Fw(E)458 3815 y Ft(j)493 3803 y Fv(\()p Fw(\025)p Fv(\))p Fu(k)647 3819 y Fb(B)p Fr(\()p Fk(C)769 3803 y Fm(N)5 b Fl(+2)899 3819 y Fr(\))952 3803 y Fu(\024)23 b Fv(1)k FO(whic)n(h)g(v)n (eri\034es)g(\(7.7\))o(.)1770 b Fj(\003)109 3950 y FO(Let)27 b(us)h(sho)n(w)f(that)h(piecewise)f(constan)n(t)g(matrix-v)-5 b(alued)26 b(function)j Fw(C)6 b Fv(\()p Fw(x;)14 b(\025)p Fv(\))29 b FO(is)f(selfadjoin)n(t)f(for)g(eac)n(h)g Fw(x)d Fu(6)p Fv(=)e Fw(x)3593 3962 y Ft(j)3629 3950 y FO(.)109 4049 y(If)28 b Fw(x)23 b Fu(2)h Fv(\()p Fw(x)420 4061 y Ft(j)455 4049 y Fw(;)14 b(x)539 4061 y Ft(j)s Fr(+1)659 4049 y Fv(\))p FO(,)28 b(then)g(one)f(gets)599 4241 y Fw(C)6 b Fv(\()p Fw(x;)14 b(\025)p Fv(\))860 4207 y Fo(\003)923 4241 y Fv(=)22 b(\002)1075 4253 y Ft(H)1138 4241 y Fv(\()p Fw(\025)p Fv(\))p Fw(E)1311 4253 y Ft(j)1347 4241 y Fv(\()p Fw(\025)p Fv(\))1459 4207 y Fo(\003)1521 4241 y Fv(=)1644 4185 y(1)p 1619 4222 V 1619 4298 a(2)p Fw(\031)1721 4241 y Fv(\002)1786 4253 y Ft(H)1848 4241 y Fv(\()p Fw(\025)p Fv(\))p Fu(f)p Fw(P)2067 4201 y Ft(j)2055 4262 y Fo(\000)2112 4241 y Fv(\002)2177 4253 y Ft(H)2240 4241 y Fv(\()p Fw(\025)p Fv(\))2352 4207 y Fo(\003)2391 4241 y Fw(P)2456 4201 y Ft(j)2444 4262 y Fr(+)2518 4241 y Fu(\000)c Fw(P)2666 4201 y Ft(j)2654 4262 y Fr(+)2709 4241 y Fv(\002)2774 4253 y Ft(H)2837 4241 y Fv(\()p Fw(\025)p Fv(\))2949 4207 y Fo(\003)2988 4241 y Fw(P)3053 4201 y Ft(j)3041 4262 y Fo(\000)3097 4241 y Fu(g)p Fw(:)109 4414 y FO(One)27 b(has)706 4582 y Fw(C)6 b Fv(\()p Fw(x;)14 b(\025)p Fv(\))967 4547 y Fo(\003)1030 4582 y Fv(=)1130 4526 y(1)p 1105 4563 V 1105 4639 a(2)p Fw(\031)1207 4582 y Fu(f)p Fw(P)1314 4542 y Ft(j)1302 4602 y Fr(+)1357 4582 y Fv(\002)1422 4594 y Ft(H)1484 4582 y Fv(\()p Fw(\025)p Fv(\))p Fw(P)1661 4542 y Ft(j)1649 4602 y Fr(+)1706 4582 y Fv(\002)1771 4594 y Ft(H)1833 4582 y Fv(\()p Fw(\025)p Fv(\))1945 4547 y Fo(\003)1984 4582 y Fw(P)2049 4542 y Ft(j)2037 4602 y Fo(\000)2112 4582 y Fv(+)k Fw(P)2260 4542 y Ft(j)2248 4602 y Fo(\000)2304 4582 y Fv(\002)2369 4594 y Ft(H)2432 4582 y Fv(\()p Fw(\025)p Fv(\))p Fw(P)2609 4542 y Ft(j)2597 4602 y Fr(+)2653 4582 y Fv(\002)2718 4594 y Ft(H)2781 4582 y Fv(\()p Fw(\025)p Fv(\))2893 4547 y Fo(\003)2932 4582 y Fw(P)2997 4542 y Ft(j)2985 4602 y Fo(\000)1113 4750 y Fu(\000)g Fw(P)1261 4710 y Ft(j)1249 4771 y Fr(+)1305 4750 y Fv(\002)1370 4762 y Ft(H)1432 4750 y Fv(\()p Fw(\025)p Fv(\))p Fw(P)1609 4710 y Ft(j)1597 4771 y Fo(\000)1655 4750 y Fv(\002)1720 4762 y Ft(H)1782 4750 y Fv(\()p Fw(\025)p Fv(\))1894 4716 y Fo(\003)1933 4750 y Fw(P)1998 4710 y Ft(j)1986 4771 y Fr(+)2060 4750 y Fu(\000)g Fw(P)2208 4710 y Ft(j)2196 4771 y Fo(\000)2253 4750 y Fv(\002)2318 4762 y Ft(H)2380 4750 y Fv(\()p Fw(\025)p Fv(\))p Fw(P)2557 4710 y Ft(j)2545 4771 y Fo(\000)2602 4750 y Fv(\002)2667 4762 y Ft(H)2730 4750 y Fv(\()p Fw(\025)p Fv(\))2842 4716 y Fo(\003)2881 4750 y Fw(P)2946 4710 y Ft(j)2934 4771 y Fr(+)2990 4750 y Fu(g)p Fw(:)109 4915 y FO(Since)27 b Fw(P)390 4875 y Ft(j)378 4935 y Fr(+)452 4915 y Fv(+)18 b Fw(P)600 4875 y Ft(j)588 4935 y Fo(\000)668 4915 y Fv(=)k Fw(I)35 b FO(and)27 b Fv(\002)1052 4927 y Ft(H)1115 4915 y Fv(\()p Fw(\025)p Fv(\)\002)1292 4927 y Ft(H)1355 4915 y Fv(\()p Fw(\025)p Fv(\))1467 4884 y Fo(\003)1530 4915 y Fv(=)22 b Fw(I)35 b FO(w)n(e)27 b(\034nd)680 5110 y Fw(C)6 b Fv(\()p Fw(x;)14 b(\025)p Fv(\))941 5076 y Fo(\003)1004 5110 y Fv(=)1104 5054 y(1)p 1078 5091 V 1078 5167 a(2)p Fw(\031)1180 5110 y Fu(f\000)p Fw(P)1352 5070 y Ft(j)1340 5131 y Fr(+)1395 5110 y Fv(\002)1460 5122 y Ft(H)1522 5110 y Fv(\()p Fw(\025)p Fv(\))p Fw(P)1699 5070 y Ft(j)1687 5131 y Fo(\000)1745 5110 y Fv(\002)1810 5122 y Ft(H)1872 5110 y Fv(\()p Fw(\025)p Fv(\))1984 5076 y Fo(\003)2023 5110 y Fw(P)2088 5070 y Ft(j)2076 5131 y Fo(\000)2151 5110 y Fv(+)k Fw(P)2299 5070 y Ft(j)2287 5131 y Fo(\000)2344 5110 y Fv(\002)2409 5122 y Ft(H)2471 5110 y Fv(\()p Fw(\025)p Fv(\))p Fw(P)2648 5070 y Ft(j)2636 5131 y Fr(+)2692 5110 y Fv(\002)2757 5122 y Ft(H)2820 5110 y Fv(\()p Fw(\025)p Fv(\))2932 5076 y Fo(\003)2971 5110 y Fw(P)3036 5070 y Ft(j)3024 5131 y Fo(\000)1087 5279 y Fu(\000)g Fw(P)1235 5239 y Ft(j)1223 5299 y Fr(+)1278 5279 y Fv(\002)1343 5291 y Ft(H)1406 5279 y Fv(\()p Fw(\025)p Fv(\))p Fw(P)1583 5239 y Ft(j)1571 5299 y Fo(\000)1628 5279 y Fv(\002)1693 5291 y Ft(H)1756 5279 y Fv(\()p Fw(\025)p Fv(\))1868 5244 y Fo(\003)1907 5279 y Fw(P)1972 5239 y Ft(j)1960 5299 y Fr(+)2034 5279 y Fv(+)g Fw(P)2182 5239 y Ft(j)2170 5299 y Fo(\000)2226 5279 y Fv(\002)2291 5291 y Ft(H)2354 5279 y Fv(\()p Fw(\025)p Fv(\))p Fw(P)2531 5239 y Ft(j)2519 5299 y Fr(+)2575 5279 y Fv(\002)2640 5291 y Ft(H)2703 5279 y Fv(\()p Fw(\025)p Fv(\))2815 5244 y Fo(\003)2854 5279 y Fw(P)2919 5239 y Ft(j)2907 5299 y Fr(+)2962 5279 y Fu(g)109 5428 y FO(whic)n(h)27 b(yields)742 5562 y Fw(C)6 b Fv(\()p Fw(x;)14 b(\025)p Fv(\))1003 5528 y Fo(\003)1066 5562 y Fv(=)1189 5506 y(1)p 1164 5543 V 1164 5619 a(2)p Fw(\031)1265 5562 y Fu(f)p Fw(P)1372 5523 y Ft(j)1360 5583 y Fo(\000)1416 5562 y Fv(\002)1481 5574 y Ft(H)1544 5562 y Fv(\()p Fw(\025)p Fv(\))p Fw(P)1721 5523 y Ft(j)1709 5583 y Fr(+)1784 5562 y Fu(\000)k Fw(P)1932 5523 y Ft(j)1920 5583 y Fr(+)1975 5562 y Fv(\002)2040 5574 y Ft(H)2103 5562 y Fv(\()p Fw(\025)p Fv(\))p Fw(P)2280 5523 y Ft(j)2268 5583 y Fo(\000)2325 5562 y Fu(g)p Fv(\002)2432 5574 y Ft(H)2494 5562 y Fv(\()p Fw(\025)p Fv(\))2606 5528 y Fo(\003)2668 5562 y Fv(=)23 b Fw(C)6 b Fv(\()p Fw(x;)14 b(\025)p Fv(\))109 5720 y FO(Since)39 b Fw(C)6 b Fv(\()p Fw(x;)14 b(\025)p Fv(\))41 b FO(is)e(selfadjoin)n(t)g(it)h(is)f(useful,)j(in)e(corresp)r(ondence)d (to)i(the)h(curren)n(t)e(densit)n(y)-7 b(,)42 b(to)d(mak)n(e)g(the)109 5820 y(follo)n(wing)26 b(de\034nition.)p eop %%Page: 21 23 21 22 bop 109 254 a FH(Dissipativ)n(e)27 b(Sc)n(hr\366)r(dinger-t)n(yp) r(e)e(op)r(erator)2084 b FO(21)109 503 y Fy(De\034nition)30 b(7.2)41 b FO(The)21 b(piecewise)g(constan)n(t)f(matrix)h Fw(C)6 b Fv(\()p Fu(\001)p Fw(;)14 b(\025)p Fv(\))22 b FO(de\034ned)g(b)n(y)g(\(7.5\))f(is)g(called)f(the)i(curren)n(t)e (densit)n(y)109 602 y(observ)-5 b(able)26 b(and)h(the)h(piecewise)f (constan)n(t)g(function)1416 777 y Fw(j)1450 789 y Ft(\045)1489 777 y Fv(\()p Fw(x;)14 b(\025)p Fv(\))24 b(=)f(tr\()p Fw(\045)p Fv(\()p Fw(\025)p Fv(\))p Fw(C)6 b Fv(\()p Fw(x;)14 b(\025)p Fv(\)\))109 951 y FO(is)27 b(called)g(the)h(curren)n (t)f(densit)n(y)g(at)h(p)r(oin)n(t)f Fw(x)d Fu(2)f Fv([)p Fw(a;)14 b(b)p Fv(])p FO(,)28 b Fw(x)23 b Fu(6)p Fv(=)g Fw(x)2031 963 y Ft(j)2066 951 y FO(,)28 b(and)f(energy)g Fw(\025)c Fu(2)h Fq(R)p FO(.)109 1158 y(The)d(de\034nition)h(is)g (justi\034ed)g(b)n(y)f(the)h(fact)g(that)g(the)g(curren)n(t)e(densit)n (y)h(is)h(the)g(exp)r(ectation)f(v)-5 b(alue)22 b(of)f(the)h(curren)n (t)109 1257 y(densit)n(y)33 b(observ)-5 b(able)32 b(at)h(energy)f Fw(\025)i Fu(2)f Fq(R)39 b FO(and)33 b(p)r(oin)n(t)h Fw(x)f Fu(2)g Fq(R)p FO(,)41 b(i.e.)55 b Fw(j)2323 1269 y Ft(\045)2361 1257 y Fv(\()p Fw(x;)14 b(\025)p Fv(\))34 b(=)f Fq(E)2738 1272 y Ft(\045)q Fr(\()p Ft(\025)p Fr(\))2874 1257 y Fv(\()p Fw(C)6 b Fv(\()p Fw(x;)14 b(\025)p Fv(\)\))p FO(.)57 b(Using)33 b(this)109 1357 y(notion)27 b(w)n(e)g(get)1359 1494 y Fw(j)1393 1506 y Ft(\045)1432 1494 y Fv(\()p Fw(x)p Fv(\))d(=)1655 1381 y Fs(Z)1701 1570 y Fk(R)1761 1494 y Fq(E)1810 1509 y Ft(\045)p Fr(\()p Ft(\025)q Fr(\))1946 1494 y Fv(\()p Fw(C)6 b Fv(\()p Fw(x;)14 b(\025)p Fv(\)\))g Fw(d\025:)109 1733 y FO(Let)31 b(us)h(consider)f(the)h(sp)r(ecial)f (case,)h(where)f(the)h(steady)f(state)g Fw(\045)h FO(is)g(a)f(function) h(of)g Fw(K)6 b FO(,)32 b(i.e.)49 b Fw(\045)30 b Fv(=)f Fw(f)9 b Fv(\()p Fw(K)d Fv(\))p FO(,)33 b(for)109 1832 y(some)28 b Fw(f)34 b Fu(2)26 b Fp(L)519 1802 y Fo(1)589 1832 y Fv(\()p Fq(R)p Fv(\))p FO(,)36 b Fw(f)e Fu(\025)25 b Fv(0)p FO(.)40 b(In)30 b(this)f(case)f Fw(\045)h FO(b)r(elong)f(to)h (the)g(bicomm)n(utan)n(t)g(of)g Fw(K)6 b FO(.)41 b(As)29 b(in)g([KNR01c)o(])g(w)n(e)f(get)109 1932 y(that)f(in)h(this)g(case)f (the)h(curren)n(t)e(densit)n(y)i(is)f(zero.)36 b(This)28 b(fact)f(is)h(pro)n(v)n(en)e(b)n(y)h(the)h(follo)n(wing)e(Corollary)-7 b(.)109 2139 y Fy(Corollary)31 b(7.3)41 b FA(Supp)l(ose)34 b(that)g Fw(m)21 b Fv(+)1394 2106 y Fr(1)p 1382 2120 59 4 v 1382 2167 a Ft(m)1481 2139 y Fu(2)31 b Fp(L)1612 2108 y Fo(1)1683 2139 y Fv(\()p Fw(a;)14 b(b)p Fv(\))p FA(,)35 b Fw(V)50 b Fu(2)31 b Fp(L)2153 2108 y Fr(2)2190 2139 y Fv(\()p Fw(a;)14 b(b)p Fv(\))p FA(,)36 b(such)e(that)g Fv(Im\()p Fw(V)19 b Fv(\))31 b(=)g(0)p FA(,)k Fw(\024)3306 2151 y Ft(a)3345 2139 y Fw(;)14 b(\024)3430 2151 y Fr(1)3467 2139 y Fw(;)g Fu(\001)g(\001)g(\001)28 b Fw(;)109 2238 y(\024)157 2250 y Ft(N)219 2238 y Fw(;)14 b(\024)304 2250 y Ft(b)365 2238 y Fu(2)27 b Fq(C)501 2250 y Fr(+)562 2238 y FA(,)33 b(as)g(wel)t(l)g(as)f Fw(x)1056 2250 y Fr(1)1094 2238 y Fw(;)14 b Fu(\001)g(\001)g(\001)27 b Fw(;)14 b(x)1339 2250 y Ft(N)1430 2238 y Fu(2)27 b Fv(\()p Fw(a;)14 b(b)p Fv(\))p FA(,)33 b(with)g Fw(x)1981 2250 y Fr(1)2045 2238 y Fw(<)27 b(x)2184 2250 y Fr(2)2249 2238 y Fw(<)g Fu(\001)14 b(\001)g(\001)27 b Fw(<)g(x)2604 2250 y Ft(N)2667 2238 y FA(.)45 b(F)-6 b(urthermor)l(e)32 b(assume)g(that)109 2338 y(the)g(ste)l(ady)g(state)f Fw(\045)h FA(is)g(of)h(the)f(form)g Fw(\045)27 b Fv(=)g Fw(f)9 b Fv(\()p Fw(K)d Fv(\))p FA(,)32 b(wher)l(e)g Fw(f)k Fu(2)27 b Fp(L)2160 2308 y Fo(1)2230 2338 y Fv(\()p Fq(R)q Fv(\))38 b FA(is)32 b(non-ne)l(gative.)45 b(Then)32 b Fw(j)3256 2350 y Ft(\045)3295 2338 y Fv(\()p Fw(x;)14 b(\025)p Fv(\))28 b(=)e(0)109 2437 y FA(for)k Fw(x)24 b Fu(6)p Fv(=)e Fw(x)446 2449 y Ft(\034)488 2437 y FA(,)30 b Fw(\034)j Fv(=)23 b Fw(a;)14 b(x)828 2449 y Fr(1)865 2437 y Fw(;)g Fu(\001)g(\001)g(\001)28 b Fw(;)14 b(x)1111 2449 y Ft(N)1174 2437 y Fw(;)g(b)p FA(,)30 b(and)g(a.e.)40 b Fw(\025)23 b Fu(2)h Fq(R)p FA(.)109 2644 y Fy(Pro)s(of.)36 b FO(Clearly)26 b(w)n(e)i(ha)n(v)n(e)e Fw(\045)p Fv(\()p Fw(\025)p Fv(\))e(=)f Fw(f)9 b Fv(\()p Fw(\025)p Fv(\))p Fw(I)1471 2661 y Fk(C)1508 2644 y Fm(N)c Fl(+2)13 b FO(,)28 b Fw(\025)23 b Fu(2)h Fq(R)p FO(.)42 b(Th)n(us)28 b(w)n(e)f(get)g(for)g Fw(x)d Fu(2)f Fv(\()p Fw(x)2788 2656 y Ft(j)2824 2644 y Fw(;)14 b(x)2908 2656 y Ft(j)s Fr(+1)3027 2644 y Fv(\))1327 2819 y Fw(j)1361 2831 y Ft(\045)1400 2819 y Fv(\()p Fw(x;)g(\025)p Fv(\))25 b(=)d Fw(f)9 b Fv(\()p Fw(\025)p Fv(\)tr)1936 2835 y Fk(C)1972 2818 y Fm(N)c Fl(+2)2106 2819 y Fv(\()p Fw(C)h Fv(\()p Fw(x;)14 b(\025)p Fv(\)\))109 2993 y FO(Using)28 b(\(7.5\))f(and)h(\(7.6\))f(w)n(e)g(obtain)309 3192 y Fw(j)343 3204 y Ft(\045)382 3192 y Fv(\()p Fw(x;)14 b(\025)p Fv(\))24 b(=)f Fw(f)9 b Fv(\()p Fw(\025)p Fv(\)tr\()p Fw(E)1010 3204 y Ft(j)1046 3192 y Fv(\()p Fw(x;)14 b(\025)p Fv(\)\002)1307 3204 y Ft(H)1371 3192 y Fv(\()p Fw(\025)p Fv(\))1483 3158 y Fo(\003)1522 3192 y Fv(\))23 b(=)g Fw(f)9 b Fv(\()p Fw(\025)p Fv(\)tr)1906 3100 y Fs(\020)o(\020)2005 3192 y Fw(P)2070 3152 y Ft(j)2058 3213 y Fo(\000)2114 3192 y Fv(\002)2179 3204 y Ft(H)2242 3192 y Fv(\()p Fw(\025)p Fv(\))p Fw(P)2419 3152 y Ft(j)2407 3213 y Fr(+)2482 3192 y Fu(\000)18 b Fw(P)2630 3152 y Ft(j)2618 3213 y Fr(+)2673 3192 y Fv(\002)2738 3204 y Ft(H)2801 3192 y Fv(\()p Fw(\025)p Fv(\))p Fw(P)2978 3152 y Ft(j)2966 3213 y Fo(\000)3023 3100 y Fs(\021)3087 3192 y Fv(\002)3152 3204 y Ft(H)3214 3192 y Fv(\()p Fw(\025)p Fv(\))3326 3158 y Fo(\003)3365 3100 y Fs(\021)3429 3192 y Fw(:)109 3386 y FO(Since)289 3561 y Fv(tr)367 3469 y Fs(\020)417 3561 y Fw(P)482 3521 y Ft(j)470 3581 y Fr(+)525 3561 y Fv(\002)590 3573 y Ft(H)653 3561 y Fv(\()p Fw(\025)p Fv(\))p Fw(P)830 3521 y Ft(j)818 3581 y Fo(\000)875 3561 y Fv(\002)940 3573 y Ft(H)1003 3561 y Fv(\()p Fw(\025)p Fv(\))1115 3526 y Fo(\003)1154 3469 y Fs(\021)1226 3561 y Fv(=)23 b(tr)1393 3469 y Fs(\020)1442 3561 y Fw(P)1507 3521 y Ft(j)1495 3581 y Fr(+)1551 3561 y Fv(\002)1616 3573 y Ft(H)1678 3561 y Fv(\()p Fw(\025)p Fv(\))p Fw(P)1855 3521 y Ft(j)1843 3581 y Fo(\000)1901 3561 y Fv(\002)1966 3573 y Ft(H)2028 3561 y Fv(\()p Fw(\025)p Fv(\))2140 3526 y Fo(\003)2179 3561 y Fw(P)2244 3521 y Ft(j)2232 3581 y Fr(+)2288 3469 y Fs(\021)2360 3561 y Fv(=)g(tr)2527 3469 y Fs(\020)2576 3561 y Fw(P)2641 3521 y Ft(j)2629 3581 y Fo(\000)2686 3561 y Fv(\002)2751 3573 y Ft(H)2813 3561 y Fv(\()p Fw(\025)p Fv(\))2925 3526 y Fo(\003)2964 3561 y Fw(P)3029 3521 y Ft(j)3017 3581 y Fr(+)3073 3561 y Fv(\002)3138 3573 y Ft(H)3200 3561 y Fv(\()p Fw(\025)p Fv(\))p Fw(P)3377 3521 y Ft(j)3365 3581 y Fo(\000)3423 3469 y Fs(\021)109 3755 y FO(and)855 3855 y Fv(tr)934 3762 y Fs(\020)984 3855 y Fw(P)1049 3815 y Ft(j)1037 3875 y Fo(\000)1093 3855 y Fv(\002)1158 3867 y Ft(H)1221 3855 y Fv(\()p Fw(\025)p Fv(\))p Fw(P)1398 3815 y Ft(j)1386 3875 y Fr(+)1442 3855 y Fv(\002)1507 3867 y Ft(H)1570 3855 y Fv(\()p Fw(\025)p Fv(\))1682 3820 y Fo(\003)1721 3762 y Fs(\021)1793 3855 y Fv(=)g(tr)1960 3762 y Fs(\020)2009 3855 y Fw(P)2074 3815 y Ft(j)2062 3875 y Fo(\000)2119 3855 y Fv(\002)2184 3867 y Ft(H)2246 3855 y Fv(\()p Fw(\025)p Fv(\))p Fw(P)2423 3815 y Ft(j)2411 3875 y Fr(+)2468 3855 y Fv(\002)2533 3867 y Ft(H)2595 3855 y Fv(\()p Fw(\025)p Fv(\))2707 3820 y Fo(\003)2746 3855 y Fw(P)2811 3815 y Ft(j)2799 3875 y Fo(\000)2856 3762 y Fs(\021)109 4019 y FO(w)n(e)k(ha)n(v)n(e) 1016 4118 y Fv(tr)1095 4026 y Fs(\020)o(\020)1194 4118 y Fw(P)1259 4079 y Ft(j)1247 4139 y Fo(\000)1303 4118 y Fv(\002)1368 4130 y Ft(H)1431 4118 y Fv(\()p Fw(\025)p Fv(\))p Fw(P)1608 4079 y Ft(j)1596 4139 y Fr(+)1671 4118 y Fu(\000)18 b Fw(P)1819 4079 y Ft(j)1807 4139 y Fr(+)1862 4118 y Fv(\002)1927 4130 y Ft(H)1990 4118 y Fv(\()p Fw(\025)p Fv(\))p Fw(p)2144 4084 y Ft(J)2144 4139 y Fo(\000)2201 4026 y Fs(\021)2264 4118 y Fv(\002)2329 4130 y Ft(H)2391 4118 y Fv(\()p Fw(\025)p Fv(\))2503 4084 y Fo(\003)2543 4026 y Fs(\021)2615 4118 y Fv(=)23 b(0)109 4288 y FO(whic)n(h)k(pro)n (v)n(es)f Fw(j)637 4300 y Ft(\045)676 4288 y Fv(\()p Fw(x;)14 b(\025)p Fv(\))24 b(=)f(0)p FO(.)2538 b Fj(\003)109 4608 y Fz(8)131 b(Curren)l(t)45 b(v)-7 b(ariations)109 4837 y FO(In)35 b(Theorem)g(7.1)f(w)n(e)h(sho)n(w)n(ed)f(that)i(the)g (curren)n(t)e(densit)n(y)i Fw(j)2108 4849 y Ft(\045)2146 4837 y Fv(\()p Fw(x;)14 b(\025)p Fv(\))37 b FO(is)f(piecewise)e (constan)n(t)h(in)h Fw(x)p FO(.)61 b(Let)35 b(us)109 4947 y(calculate)27 b(the)g(curren)n(t)g(di\033erence)h(for)f(some)h Fv(~)-43 b Fw(a)22 b Fu(2)i Fv(\()p Fw(x)1817 4959 y Ft(j)1853 4947 y Fw(;)14 b(x)1937 4959 y Ft(j)s Fr(+1)2056 4947 y Fv(\))28 b FO(and)2274 4925 y Fv(~)2277 4947 y Fw(b)23 b Fu(2)g Fv(\()p Fw(x)2493 4959 y Ft(j)s Fr(+1)2613 4947 y Fw(;)14 b(x)2697 4959 y Ft(j)s Fr(+2)2817 4947 y Fv(\))p FO(:)567 5163 y Fw(j)601 5175 y Ft(\045)640 5163 y Fv(\()669 5141 y(~)672 5163 y Fw(b;)g(\025)p Fv(\))19 b Fu(\000)f Fw(j)961 5175 y Ft(\045)1000 5163 y Fv(\()q(~)-43 b Fw(a;)14 b(\025)p Fv(\))23 b(=)1339 5107 y(1)p 1314 5144 92 4 v 1314 5220 a(2)p Fw(\031)1416 5163 y Fv(tr\()p Fw(\045)p Fv(\()p Fw(\025)p Fv(\))p Fu(f)p Fw(C)1769 5175 y Ft(j)s Fr(+1)1889 5163 y Fv(\()p Fw(\025)p Fv(\))c Fu(\000)f Fw(C)2162 5175 y Ft(j)2197 5163 y Fv(\()p Fw(\025)p Fv(\))p Fu(g)p Fv(\))1216 5363 y(=)1339 5307 y(1)p 1314 5344 V 1314 5420 a(2)p Fw(\031)1416 5363 y Fv(tr\()p Fw(\045)p Fv(\()p Fw(\025)p Fv(\))p Fu(f)p Fw(P)1775 5323 y Ft(j)s Fr(+1)1763 5384 y Fo(\000)1894 5363 y Fv(\002)1959 5375 y Ft(H)2022 5363 y Fv(\()p Fw(\025)p Fv(\))p Fw(P)2199 5323 y Ft(j)s Fr(+1)2187 5384 y Fo(\000)2337 5363 y Fu(\000)g Fw(P)2485 5323 y Ft(j)2473 5384 y Fo(\000)2529 5363 y Fv(\002)2594 5375 y Ft(H)2657 5363 y Fv(\()p Fw(\025)p Fv(\))p Fw(P)2834 5323 y Ft(j)2822 5384 y Fo(\000)2879 5363 y Fu(g)p Fv(\002)2986 5329 y Fo(\003)2986 5384 y Ft(H)3048 5363 y Fv(\()p Fw(\025)p Fv(\)\))1216 5563 y(=)1339 5507 y(1)p 1314 5544 V 1314 5620 a(2)p Fw(\031)1416 5563 y Fv(tr\()p Fw(\045)p Fv(\()p Fw(\025)p Fv(\))p Fu(f)p Fw(P)1763 5575 y Ft(j)s Fr(+1)1883 5563 y Fv(\002)1948 5575 y Ft(H)2010 5563 y Fv(\()p Fw(\025)p Fv(\))i Fu(\000)e Fv(\002)2290 5575 y Ft(H)2352 5563 y Fv(\()p Fw(\025)p Fv(\))p Fw(P)2517 5575 y Ft(j)s Fr(+1)2638 5563 y Fu(g)p Fv(\002)2745 5529 y Fo(\003)2745 5583 y Ft(H)2807 5563 y Fv(\()p Fw(\025)p Fv(\)\))1216 5763 y(=)1339 5706 y(1)p 1314 5744 V 1314 5820 a(2)p Fw(\031)1416 5763 y Fv(tr\()p Fw(P)1566 5775 y Ft(j)s Fr(+1)1685 5763 y Fu(f)p Fw(\045)p Fv(\()p Fw(\025)p Fv(\))h Fu(\000)f Fv(\002)2049 5728 y Fo(\003)2087 5763 y Fv(\()p Fw(\025)p Fv(\))p Fw(\045)p Fv(\()p Fw(\025)p Fv(\)\002)2419 5775 y Ft(H)2483 5763 y Fv(\()p Fw(\025)p Fv(\))p Fu(g)p Fw(P)2690 5775 y Ft(j)s Fr(+1)2810 5763 y Fv(\))p Fw(:)616 b FO(\(8.1\))p eop %%Page: 22 24 22 23 bop 109 254 a FO(22)2477 b Fx(M.)31 b(Bar)n(o,)f(H.)h(Neidhardt) 109 503 y FO(More)26 b(general)g(w)n(e)i(get)f(for)h Fv(^)-43 b Fw(a)23 b Fu(2)g Fv(\()p Fw(x)1221 515 y Ft(j)1257 503 y Fw(;)14 b(x)1341 515 y Ft(j)s Fr(+1)1460 503 y Fv(\))28 b FO(and)1679 481 y Fv(^)1682 503 y Fw(b)23 b Fu(2)g Fv(\()p Fw(x)1898 515 y Ft(j)s Fr(+)p Ft(k)q Fr(+1)2105 503 y Fw(;)14 b(x)2189 515 y Ft(j)s Fr(+)p Ft(k)q Fr(+2)2396 503 y Fv(\))825 708 y Fw(j)859 720 y Ft(\045)898 708 y Fv(\()927 686 y(^)930 708 y Fw(b;)g(\025)p Fv(\))19 b Fu(\000)f Fw(j)1219 720 y Ft(\045)1258 708 y Fv(\()q(^)-43 b Fw(a;)14 b(\025)p Fv(\))24 b(=)1598 652 y(1)p 1572 689 92 4 v 1572 765 a(2)p Fw(\031)1674 708 y Fv(tr\()p Fw(Q)p Fu(f)p Fw(\045)p Fv(\()p Fw(\025)p Fv(\))19 b Fu(\000)f Fv(\002)2201 720 y Ft(H)2263 708 y Fv(\()p Fw(\025)p Fv(\))p Fw(\045)p Fv(\()p Fw(\025)p Fv(\)\002)2595 673 y Fo(\003)2595 728 y Ft(H)2660 708 y Fv(\()p Fw(\025)p Fv(\))p Fu(g)p Fw(Q)p Fv(\))p Fw(;)546 b FO(\(8.2\))109 919 y(where)27 b Fw(Q)22 b Fv(=)525 856 y Fs(P)613 877 y Ft(j)s Fr(+)p Ft(k)613 943 y(\034)7 b Fr(=)p Ft(j)750 919 y Fw(P)803 931 y Ft(\034)845 919 y FO(.)109 1065 y(F)-7 b(rom)30 b(equation)g(\(8.2\))g(w)n(e)h(see)f (that)h(the)g(curren)n(t)f(di\033erence)g(dep)r(ends)h(essen)n(tially)f (on)g(the)h(densit)n(y)g(matrix)109 1165 y Fw(\045)p Fv(\()p Fw(\025)p Fv(\))p FO(.)51 b(F)-7 b(or)31 b(example,)i(if)g Fw(\045)p Fv(\()p Fw(\025)p Fv(\))g FO(comm)n(utes)f(with)g Fv(\002)1767 1177 y Ft(H)1830 1165 y Fv(\()p Fw(\025)p Fv(\))h FO(w)n(e)f(get)f(that)i(the)f(di\033erence)g(is)g(zero.)49 b(Hence,)34 b(the)109 1265 y(curren)n(t)18 b(is)g(constan)n(t.)34 b(In)19 b(the)g(case)f(that)h(the)h(di\033erence)e(in)h(\(8.2\))g(is)g (p)r(ositiv)n(e,)h(the)f(system)g(is)g(\021losing\021)24 b(electrons)109 1364 y(b)r(et)n(w)n(een)h(the)g(p)r(oin)n(ts)h Fv(~)-43 b Fw(a)25 b FO(and)1039 1342 y Fv(~)1042 1364 y Fw(b)o FO(.)37 b(If)25 b(the)g(di\033erence)g(in)h(\(8.2\))e(is)h (negativ)n(e,)g(the)g(system)g(\021gains\021)30 b(electrons.)35 b(The)109 1464 y(e\033ect)28 b(of)f(losing)g(and)g(gaining)g(electrons) f(is)i(closely)e(related)h(to)h(a)f(recom)n(bination-generation)d(pro)r (cess.)109 1611 y(In)33 b(the)g(follo)n(wing)f(w)n(e)g(are)g(going)g (to)h(construct)f(a)h(densit)n(y)f(matrix)h Fw(\045)p Fv(\()p Fu(\001)p Fv(\))p FO(,)i(suc)n(h)d(that)h(the)h(r.h.s.)52 b(of)40 b(\(8.2\))32 b(is)109 1710 y(expressed)26 b(in)i(terms)f(of)h (the)g(densit)n(y)f Fw(u)1381 1722 y Ft(\045)1419 1710 y Fv(\()p Fw(x)1498 1722 y Ft(\030)1535 1710 y Fw(;)14 b(\025)p Fv(\))28 b FO(and)g(the)g Fw(\013)2038 1722 y Ft(j)2073 1710 y FO('s.)109 1857 y(Let)d Fu(R)325 1869 y Ft(\025)364 1877 y Fl(0)402 1857 y Fw(;)14 b Fu(G)488 1869 y Ft(\025)527 1877 y Fl(0)587 1857 y Fu(\022)22 b(f)p Fv(1)p Fw(;)14 b Fu(\001)g(\001)g(\001)27 b Fw(;)14 b(N)9 b Fu(g)22 b Fv(:=)h Fu(T)47 b FO(suc)n(h)25 b(that)h Fu(R)1732 1869 y Ft(\025)1771 1877 y Fl(0)1822 1857 y Fu(\\)15 b(G)1941 1869 y Ft(\025)1980 1877 y Fl(0)2040 1857 y Fv(=)23 b Fu(;)p FO(,)i Fu(R)2288 1869 y Ft(\025)2327 1877 y Fl(0)2379 1857 y Fu([)14 b(G)2497 1869 y Ft(\025)2536 1877 y Fl(0)2597 1857 y Fv(=)22 b Fu(T)f FO(,)27 b(and)e Fw(u)p Fv(\()p Fw(\025)3087 1869 y Fr(0)3124 1857 y Fv(\))f Fu(2)f Fq(C)3312 1827 y Ft(N)6 b Fr(+2)3491 1857 y FO(with)636 2020 y Fu(h)p Fw(u)p Fv(\()p Fw(\025)796 2032 y Fr(0)834 2020 y Fv(\))p Fw(;)14 b(e)942 2032 y Ft(\034)983 2020 y Fu(i)24 b Fv(=)e(0)p Fw(;)97 b Fu(8)p Fw(\034)32 b Fu(2)23 b(R)1551 2032 y Ft(\025)1590 2040 y Fl(0)1627 2020 y Fw(;)42 b FO(and)27 b Fu(h)p Fw(u)p Fv(\()p Fw(\025)2013 2032 y Fr(0)2051 2020 y Fv(\))p Fw(;)14 b Fv(\002)2185 2032 y Ft(H)2248 2020 y Fv(\()p Fw(\025)2328 2032 y Fr(0)2366 2020 y Fv(\))p Fw(e)2437 2032 y Ft(\034)2478 2020 y Fu(i)24 b Fv(=)e(0)p Fw(;)97 b Fu(8)p Fw(\034)32 b Fu(2)23 b(G)3025 2032 y Ft(\025)3064 2040 y Fl(0)3101 2020 y Fw(;)357 b FO(\(8.3\))109 2183 y(where)27 b Fw(\025)397 2195 y Fr(0)457 2183 y Fu(2)d Fq(R)33 b FO(is)28 b(\034xed.)37 b(W)-7 b(e)28 b(de\034ne)g(the)f(op)r(erator)f Fw(\045)p Fv(\()p Fw(\025)1923 2195 y Fr(0)1961 2183 y Fv(\))i FO(b)n(y)1455 2347 y Fw(\045)p Fv(\()p Fw(\025)1578 2359 y Fr(0)1616 2347 y Fv(\))1648 2312 y Ft(T)1724 2347 y Fv(:=)p 1834 2274 198 4 v 22 w Fw(u)p Fv(\()p Fw(\025)1962 2359 y Fr(0)2000 2347 y Fv(\))p Fw(u)p Fv(\()p Fw(\025)2160 2359 y Fr(0)2198 2347 y Fv(\))2230 2312 y Ft(T)2282 2347 y Fw(:)109 2557 y FO(The)f(function)h Fw(f)9 b Fv(\()p Fw(x;)14 b(\025)818 2569 y Fr(0)856 2557 y Fv(\))28 b FO(giv)n(en)f(b)n(y)1303 2655 y Fs(0)1303 2801 y(B)1303 2851 y(B)1303 2901 y(B)1303 2954 y(@)1539 2719 y Fv(0)1548 2808 y FO(.)1548 2841 y(.)1548 2874 y(.)1539 2974 y Fv(0)1418 3073 y Fw(f)9 b Fv(\()p Fw(x;)14 b(\025)1632 3085 y Fr(0)1670 3073 y Fv(\))1743 2655 y Fs(1)1743 2801 y(C)1743 2851 y(C)1743 2901 y(C)1743 2954 y(A)1839 2897 y Fv(=)23 b Fw(u)p Fv(\()p Fw(\025)2055 2909 y Fr(0)2092 2897 y Fv(\))2124 2863 y Ft(T)2177 2897 y Fv(\010\()p Fw(x;)14 b(\025)p Fv(\))p Fw(;)109 3232 y FO(satis\034es)26 b(the)i(follo)n(wing)f(b)r (oundary)g(conditions:)1070 3395 y Fv(\001\()p Fw(f)9 b Fv(\()p Fu(\001)p Fw(;)14 b(\025)1361 3407 y Fr(0)1399 3395 y Fv(\)\)\()p Fw(x)1542 3407 y Ft(\034)1585 3395 y Fv(\))24 b(=)e Fu(\000)p Fw(\024)1841 3407 y Ft(\034)1882 3395 y Fw(f)9 b Fv(\()p Fw(x)2011 3407 y Ft(\034)2053 3395 y Fw(;)14 b(\025)2138 3407 y Fr(0)2175 3395 y Fv(\))p Fw(;)98 b Fu(8)p Fw(\034)31 b Fu(2)24 b(R)2591 3407 y Ft(\025)2630 3415 y Fl(0)2667 3395 y Fw(;)791 b FO(\(8.4\))1076 3520 y Fv(\001\()p Fw(f)9 b Fv(\()p Fu(\001)p Fw(;)14 b(\025)1367 3532 y Fr(0)1405 3520 y Fv(\)\)\()p Fw(x)1548 3532 y Ft(\030)1585 3520 y Fv(\))24 b(=)e Fu(\000)p 1793 3474 85 4 v Fw(\024)1841 3532 y Ft(\030)1877 3520 y Fw(f)9 b Fv(\()p Fw(x)2006 3532 y Ft(\030)2043 3520 y Fw(;)14 b(\025)2128 3532 y Fr(0)2165 3520 y Fv(\))p Fw(;)97 b Fu(8)p Fw(\030)27 b Fu(2)c(G)2554 3532 y Ft(\025)2593 3540 y Fl(0)2630 3520 y Fw(:)828 b FO(\(8.5\))109 3740 y(Let)30 b Fw(!)f Fv(=)d([)q(~)-43 b Fw(a;)533 3718 y Fv(~)536 3740 y Fw(b)p Fv(])26 b Fu(\022)h Fv(\012)i FO(b)r(e)i(a)e(giv)n(en)g(in)n(terv)-5 b(al)29 b(and)g Fu(T)1718 3752 y Ft(!)1793 3740 y Fv(:=)e Fu(f)p Fw(j)k Fu(2)c(T)35 b(j)14 b Fw(x)2261 3752 y Ft(j)2323 3740 y Fu(2)27 b Fw(!)s Fu(g)p FO(.)43 b(The)30 b(curren)n(t)e(di\033erence) i(is)g(then)109 3839 y(giv)n(en)c(b)n(y)228 4048 y Fw(j)262 4060 y Ft(\045)301 4048 y Fv(\()330 4026 y(~)333 4048 y Fw(b;)14 b(\025)454 4060 y Fr(0)492 4048 y Fv(\))k Fu(\000)g Fw(j)659 4060 y Ft(\045)698 4048 y Fv(\()q(~)-43 b Fw(a;)14 b(\025)859 4060 y Fr(0)897 4048 y Fv(\))23 b(=Im)1130 3931 y Fs(\022\034)1253 4048 y Fw(u)p Fv(\()p Fw(\025)1381 4060 y Fr(0)1419 4048 y Fv(\))1451 4013 y Ft(T)1579 3991 y Fv(1)p 1513 4028 174 4 v 1513 4116 a Fw(m)p Fv(\()1615 4094 y(~)1618 4116 y Fw(b)p Fv(\))1696 4048 y(\010\()1785 4026 y(~)1788 4048 y Fw(b;)14 b(\025)1909 4060 y Fr(0)1947 4048 y Fv(\))1979 4013 y Fo(0)2002 4048 y Fw(;)g(u)p Fv(\()p Fw(\025)2167 4060 y Fr(0)2205 4048 y Fv(\))2237 4013 y Ft(T)2289 4048 y Fv(\010\()2378 4026 y(~)2381 4048 y Fw(b;)g(\025)2502 4060 y Fr(0)2540 4048 y Fv(\))2572 3931 y Fs(\035\023)1035 4290 y Fu(\000)k Fv(Im)1231 4173 y Fs(\022\034)1355 4290 y Fw(u)p Fv(\()p Fw(\025)1483 4302 y Fr(0)1520 4290 y Fv(\))1552 4256 y Ft(T)1685 4234 y Fv(1)p 1615 4271 182 4 v 1615 4347 a Fw(m)p Fv(\()q(~)-43 b Fw(a)p Fv(\))1806 4290 y(\010\()q(~)g Fw(a;)14 b(\025)2027 4302 y Fr(0)2065 4290 y Fv(\))2097 4256 y Fo(0)2120 4290 y Fw(;)g(u)p Fv(\()p Fw(\025)2285 4302 y Fr(0)2323 4290 y Fv(\))2355 4256 y Ft(T)2407 4290 y Fv(\010\()q(~)-43 b Fw(a)q(;)14 b(\025)2629 4302 y Fr(0)2666 4290 y Fv(\))2698 4173 y Fs(\035)q(\023)952 4512 y Fv(=)1027 4456 y(1)p 1027 4493 42 4 v 1033 4569 a Fw(i)1113 4433 y Fs(X)1092 4611 y Ft(\034)7 b Fo(2T)1213 4619 y Fm(!)1268 4420 y Fs(n)1324 4512 y Fw(W)12 b Fv(\()p 1446 4440 260 4 v Fw(f)d Fv(\()p Fu(\001)p Fw(;)14 b(\025)1636 4524 y Fr(0)1673 4512 y Fv(\))q Fw(;)g(f)9 b Fv(\()p Fu(\001)p Fw(;)14 b(\025)1933 4524 y Fr(0)1970 4512 y Fv(\)\)\()p Fw(x)2113 4524 y Ft(\034)2174 4512 y Fv(+)k(0\))g Fu(\000)g Fw(W)12 b Fv(\()p 2554 4440 V Fw(f)d Fv(\()p Fu(\001)p Fw(;)14 b(\025)2744 4524 y Fr(0)2782 4512 y Fv(\))p Fw(;)g(f)9 b Fv(\()p Fu(\001)p Fw(;)14 b(\025)3041 4524 y Fr(0)3079 4512 y Fv(\)\)\()p Fw(x)3222 4524 y Ft(\034)3283 4512 y Fu(\000)k Fv(0\))3440 4420 y Fs(o)3509 4512 y Fw(:)109 4762 y FO(Using)27 b(the)h(b)r(oundary)f(conditions)g (\(8.4\))g(and)g(\(8.5\))h(w)n(e)f(obtain)591 4941 y Fw(j)625 4953 y Ft(\045)664 4941 y Fv(\()693 4919 y(~)696 4941 y Fw(b;)14 b(\025)817 4953 y Fr(0)854 4941 y Fv(\))19 b Fu(\000)f Fw(j)1022 4953 y Ft(\045)1061 4941 y Fv(\()q(~)-43 b Fw(a;)14 b(\025)1222 4953 y Fr(0)1260 4941 y Fv(\))23 b(=)1493 4863 y Fs(X)1394 5041 y Ft(\034)7 b Fo(2T)1515 5049 y Fm(!)1556 5041 y Fo(\\G)1641 5050 y Fm(\025)1676 5062 y Fl(0)1726 4941 y Fw(\013)1779 4907 y Fr(2)1779 4962 y Ft(\034)1820 4941 y Fu(j)p Fw(f)i Fv(\()p Fw(x)1972 4953 y Ft(\034)2014 4941 y Fw(;)14 b(\025)2099 4953 y Fr(0)2137 4941 y Fv(\))p Fu(j)2192 4907 y Fr(2)2248 4941 y Fu(\000)2436 4863 y Fs(X)2331 5041 y Ft(\030)r Fo(2T)2446 5049 y Fm(!)2488 5041 y Fo(\\R)2590 5050 y Fm(\025)2625 5062 y Fl(0)2675 4941 y Fw(\013)2728 4907 y Fr(2)2728 4962 y Ft(\030)2766 4941 y Fu(j)p Fw(f)9 b Fv(\()p Fw(x)2918 4953 y Ft(\030)2954 4941 y Fw(;)14 b(\025)3039 4953 y Fr(0)3077 4941 y Fv(\))p Fu(j)3132 4907 y Fr(2)1315 5191 y Fv(=)1493 5112 y Fs(X)1394 5290 y Ft(\034)7 b Fo(2T)1515 5298 y Fm(!)1556 5290 y Fo(\\G)1641 5299 y Fm(\025)1676 5311 y Fl(0)1726 5191 y Fw(\013)1779 5157 y Fr(2)1779 5212 y Ft(\034)1834 5191 y Fu(h)p Fw(V)19 b Fv(\()p Fw(\025)2013 5203 y Fr(0)2051 5191 y Fv(\)\010\()p Fw(x)2222 5203 y Ft(\034)2265 5191 y Fw(;)14 b(\025)2350 5203 y Fr(0)2387 5191 y Fv(\))p Fw(;)g(V)19 b Fv(\()p Fw(\025)2603 5203 y Fr(0)2641 5191 y Fv(\)\010\()p Fw(x)2812 5203 y Ft(\034)2855 5191 y Fw(;)14 b(\025)2940 5203 y Fr(0)2977 5191 y Fv(\))p Fu(i)1398 5440 y(\000)1586 5361 y Fs(X)1481 5540 y Ft(\030)r Fo(2T)1596 5548 y Fm(!)1638 5540 y Fo(\\R)1740 5549 y Fm(\025)1775 5561 y Fl(0)1825 5440 y Fw(\013)1878 5406 y Fr(2)1878 5461 y Ft(\030)1930 5440 y Fu(h)p Fw(V)19 b Fv(\()p Fw(\025)2109 5452 y Fr(0)2147 5440 y Fv(\)\010\()p Fw(x)2318 5452 y Ft(\030)2355 5440 y Fw(;)14 b(\025)2440 5452 y Fr(0)2478 5440 y Fv(\))p Fw(;)g(V)19 b Fv(\()p Fw(\025)2694 5452 y Fr(0)2732 5440 y Fv(\)\010\()p Fw(x)2903 5452 y Ft(\030)2940 5440 y Fw(;)14 b(\025)3025 5452 y Fr(0)3063 5440 y Fv(\))p Fu(i)1315 5690 y Fv(=)1493 5611 y Fs(X)1394 5789 y Ft(\034)7 b Fo(2T)1515 5797 y Fm(!)1556 5789 y Fo(\\G)1641 5798 y Fm(\025)1676 5810 y Fl(0)1726 5690 y Fw(\013)1779 5656 y Fr(2)1779 5711 y Ft(\034)1820 5690 y Fw(u)1868 5702 y Ft(\045)1906 5690 y Fv(\()p Fw(x)1985 5702 y Ft(\034)2028 5690 y Fw(;)14 b(\025)2113 5702 y Fr(0)2150 5690 y Fv(\))19 b Fu(\000)2389 5611 y Fs(X)2284 5790 y Ft(\030)r Fo(2T)2399 5798 y Fm(!)2441 5790 y Fo(\\R)2543 5799 y Fm(\025)2578 5811 y Fl(0)2628 5690 y Fw(\013)2681 5656 y Fr(2)2681 5711 y Ft(\030)2719 5690 y Fw(u)2767 5702 y Ft(\045)2805 5690 y Fv(\()p Fw(x)2884 5702 y Ft(\030)2921 5690 y Fw(;)14 b(\025)3006 5702 y Fr(0)3043 5690 y Fv(\))p Fw(:)p eop %%Page: 23 25 23 24 bop 109 254 a FH(Dissipativ)n(e)27 b(Sc)n(hr\366)r(dinger-t)n(yp) r(e)e(op)r(erator)2084 b FO(23)109 503 y(Clearly)40 b(a)h(v)n(ector)f Fw(u)p Fv(\()p Fw(\025)885 515 y Fr(0)922 503 y Fv(\))i FO(with)g(the)g(prop)r(erties)e(\(8.3\))h(alw)n(a)n(ys)f(exists.)78 b(In)41 b(the)h(follo)n(wing)e(w)n(e)h(will)h(con-)109 602 y(struct)36 b(a)g(v)n(ector)f Fw(u)p Fv(\()p Fw(\025)822 614 y Fr(0)859 602 y Fv(\))i FO(for)e(almost)h(ev)n(ery)f Fw(\025)1616 614 y Fr(0)1691 602 y Fu(2)j Fq(R)p FO(,)45 b(suc)n(h)36 b(that)g Fw(u)p Fv(\()p Fw(\025)2418 614 y Fr(0)2456 602 y Fv(\))g FO(is)g(determined)h(up)f(to)g(t)n(w)n(o)g (giv)n(en)109 702 y(parameters.)72 b(Let)40 b Fw(u)817 714 y Ft(a)857 702 y Fv(\()p Fw(\025)937 714 y Fr(0)975 702 y Fv(\))p Fw(;)14 b(u)1092 714 y Ft(b)1125 702 y Fv(\()p Fw(\025)1205 714 y Fr(0)1243 702 y Fv(\))43 b Fu(2)i Fq(C)60 b FO(b)r(e)41 b(arbitrary)-7 b(.)72 b(W)-7 b(e)40 b(are)f(no)n(w)g(going)g(to)h(construct)f(the)h(v)n(ector)109 801 y Fw(u)p Fv(\()p Fw(\025)237 813 y Fr(0)274 801 y Fv(\))23 b(=)g(\()p Fw(u)497 771 y Ft(a)537 801 y Fv(\()p Fw(\025)617 813 y Fr(0)655 801 y Fv(\))p Fw(;)14 b Fu(\001)g(\001)g (\001)28 b Fw(;)14 b(u)934 771 y Ft(b)967 801 y Fv(\()p Fw(\025)1047 813 y Fr(0)1085 801 y Fv(\)\))1149 771 y Ft(T)1201 801 y FO(.)109 948 y(W)-7 b(e)28 b(set)f Fw(P)434 960 y Fo(G)474 969 y Fm(\025)509 981 y Fl(0)573 948 y Fv(:=)684 886 y Fs(P)771 973 y Ft(j)s Fo(2G)886 982 y Fm(\025)921 994 y Fl(0)976 948 y Fw(P)1029 960 y Ft(j)1092 948 y FO(and)h Fw(m)1327 960 y Fo(G)1367 969 y Fm(\025)1402 981 y Fl(0)1465 948 y Fv(:=)23 b(#)p Fu(G)1694 960 y Ft(\025)1733 968 y Fl(0)1771 948 y FO(,)k(i.e.)37 b(the)28 b(n)n(um)n(b)r(er)f(of)h(elemen)n(ts)f(in)h Fu(G)2989 960 y Ft(\025)3028 968 y Fl(0)3066 948 y FO(.)109 1119 y(The)33 b(\034rst)g(condition)g(in)h(\(8.3\))f(implies)h(that)f Fw(u)1660 1089 y Ft(j)1695 1119 y Fv(\()p Fw(\025)1775 1131 y Fr(0)1813 1119 y Fv(\))g(=)f(0)h FO(for)g Fw(j)k Fu(2)d(R)2413 1131 y Ft(\025)2452 1139 y Fl(0)2489 1119 y FO(.)54 b(F)-7 b(urthermore)32 b(w)n(e)h(set)h Fw(u)3365 1089 y Ft(a)3404 1119 y Fv(\()p Fw(\025)3484 1131 y Fr(0)3522 1119 y Fv(\))f(=)109 1218 y Fw(u)157 1230 y Ft(a)196 1218 y Fv(\()p Fw(\025)276 1230 y Fr(0)314 1218 y Fv(\))28 b FO(and)g Fw(u)584 1188 y Ft(b)617 1218 y Fv(\()p Fw(\025)697 1230 y Fr(0)734 1218 y Fv(\))c(=)e Fw(u)925 1230 y Ft(b)958 1218 y Fv(\()p Fw(\025)1038 1230 y Fr(0)1076 1218 y Fv(\))p FO(.)38 b(The)27 b(second)g(condition)g(in)h(\(8.3\))f(can)h(b)r(e)g (rewritten)f(to)533 1387 y Fw(P)586 1399 y Fo(G)626 1408 y Fm(\025)661 1420 y Fl(0)702 1387 y Fv(\002)767 1399 y Ft(H)830 1387 y Fv(\()p Fw(\025)910 1399 y Fr(0)947 1387 y Fv(\))979 1352 y Fo(\003)1018 1387 y Fw(P)1071 1399 y Fo(G)1111 1408 y Fm(\025)1146 1420 y Fl(0)1187 1387 y Fw(u)22 b Fv(=)h Fu(\000)1424 1319 y Fs(\000)1462 1387 y Fw(u)1510 1399 y Ft(b)1542 1387 y Fv(\()p Fw(\025)1622 1399 y Fr(0)1660 1387 y Fv(\))p Fw(P)1745 1399 y Fo(G)1785 1408 y Fm(\025)1820 1420 y Fl(0)1862 1387 y Fv(\002)1927 1399 y Ft(H)1989 1387 y Fv(\()p Fw(\025)2069 1399 y Fr(0)2107 1387 y Fv(\))2139 1352 y Fo(\003)2178 1387 y Fw(e)2217 1399 y Ft(b)2268 1387 y Fv(+)18 b Fw(u)2399 1399 y Ft(a)2439 1387 y Fv(\()p Fw(\025)2519 1399 y Fr(0)2557 1387 y Fv(\))p Fw(P)2642 1399 y Fo(G)2682 1408 y Fm(\025)2717 1420 y Fl(0)2758 1387 y Fv(\002)2823 1399 y Ft(H)2885 1387 y Fv(\()p Fw(\025)2965 1399 y Fr(0)3003 1387 y Fv(\))3035 1352 y Fo(\003)3074 1387 y Fw(e)3113 1399 y Ft(a)3153 1319 y Fs(\001)3204 1387 y Fw(:)254 b FO(\(8.6\))109 1602 y(Since)31 b Fw(P)382 1614 y Fo(G)422 1623 y Fm(\025)457 1635 y Fl(0)497 1602 y Fv(\002)562 1614 y Ft(H)625 1602 y Fv(\()p Fw(\025)705 1614 y Fr(0)743 1602 y Fv(\))775 1572 y Fo(\003)813 1602 y Fw(P)866 1614 y Fo(G)906 1623 y Fm(\025)941 1635 y Fl(0)1013 1602 y FO(can)f(b)r(e)i(seen)e(as)g(a)g Fw(m)1718 1614 y Fo(G)1758 1623 y Fm(\025)1793 1635 y Fl(0)1854 1602 y Fu(\002)20 b Fw(m)2012 1614 y Fo(G)2052 1623 y Fm(\025)2087 1635 y Fl(0)2128 1602 y FO(-matrix,)31 b(the)g(equation)f(\(8.6\))g(has)g(a)h(solution)109 1702 y(for)c(a.e.)36 b Fw(\025)445 1714 y Fr(0)510 1702 y FO(if)1289 1802 y Fv(det)1404 1814 y Ft(m)1463 1822 y Fc(G)1498 1837 y Fm(\025)1533 1849 y Fl(0)1578 1802 y Fv(\()p Fw(P)1663 1814 y Fo(G)1703 1823 y Fm(\025)1738 1835 y Fl(0)1779 1802 y Fv(\002)1844 1814 y Ft(H)1907 1802 y Fv(\()p Fw(\025)1987 1814 y Fr(0)2025 1802 y Fv(\))2057 1767 y Fo(\003)2095 1802 y Fw(P)2148 1814 y Fo(G)2188 1823 y Fm(\025)2223 1835 y Fl(0)2264 1802 y Fv(\))24 b Fu(6)p Fv(=)e(0)p Fw(;)109 1968 y FO(where)27 b Fv(det)464 1980 y Ft(m)523 1988 y Fc(G)558 2003 y Fm(\025)593 2015 y Fl(0)638 1968 y Fv(\()p Fu(\001)p Fv(\))h FO(denotes)f(the)h (determinan)n(t)g(in)f(the)h(v)n(ector)f(space)f Fq(C)2429 1924 y Ft(m)2488 1932 y Fc(G)2524 1947 y Fm(\025)2559 1959 y Fl(0)2609 1968 y FO(.)109 2145 y(Since)31 b Fu(h)p Fv(\002)426 2157 y Ft(H)488 2145 y Fv(\()p Fu(\001)p Fv(\))p Fw(e)614 2157 y Ft(j)650 2145 y Fw(;)14 b(e)726 2157 y Ft(k)766 2145 y Fu(i)31 b FO(is)g(a)f(b)r(ounded)h(analytic)f (function)h(on)p 2096 2078 117 4 v 31 w Fq(C)2150 2157 y Fo(\000)2242 2145 y FO(\(cf.)47 b([FN70]\),)32 b(w)n(e)e(get)h(that)f Fv(det)3304 2157 y Ft(m)3363 2165 y Fc(G)3398 2180 y Fm(\025)3433 2192 y Fl(0)3478 2145 y Fv(\()p Fu(\001)p Fv(\))h FO(is)109 2275 y(also)26 b(a)h(b)r(ounded)h(analytic)f (function)h(on)p 1439 2208 V 27 w Fq(C)1493 2287 y Fo(\000)1555 2275 y FO(.)37 b(Hence)28 b Fv(det)1977 2287 y Ft(m)2036 2295 y Fc(G)2071 2310 y Fm(\025)2106 2322 y Fl(0)2151 2275 y Fv(\(\002)2248 2287 y Ft(H)2311 2275 y Fv(\()p Fw(\025)p Fv(\)\))d Fu(6)p Fv(=)d(0)27 b FO(for)g(a.e.)37 b Fw(\025)2973 2287 y Fr(0)3010 2275 y FO(.)109 2447 y(The)26 b(remaining)f(comp)r(onen)n(ts)h(of)g Fw(u)p FO(,)h(i.e.)36 b Fw(u)1501 2417 y Ft(j)1536 2447 y FO(,)27 b Fw(j)h Fu(2)23 b(G)1775 2459 y Ft(\025)1814 2467 y Fl(0)1851 2447 y FO(,)k(are)e(then)i(determined)g(as)e(the)i(solution)f (of)g(the)h(linear)109 2547 y(equation)g(\(8.6\))o(.)109 2694 y(Th)n(us)j(w)n(e)g(pro)n(v)n(ed)f(that)i(w)n(e)f(can)h(construct) f(a)g(densit)n(y)g(matrix)g Fw(\045)p Fv(\()p Fw(\025)2309 2706 y Fr(0)2347 2694 y Fv(\))h FO(for)f(a.e.)46 b Fw(\025)2759 2706 y Fr(0)2796 2694 y FO(,)32 b(suc)n(h)e(that)h(w)n(e)f(obtain)g(a) 109 2793 y(loss)f(and)h(gain)f(e\033ect)i(at)f(prede\034ned)g(p)r(oin)n (ts)g Fw(x)1644 2805 y Ft(j)1679 2793 y FO(.)45 b(F)-7 b(urthermore)29 b(w)n(e)g(sho)n(w)n(ed)g(that)i(one)e(has)h(t)n(w)n(o)f (degrees)g(of)109 2893 y(freedom)e(in)h(the)g(c)n(hoice)e(of)i Fw(\045)p Fv(\()p Fw(\025)1129 2905 y Fr(0)1167 2893 y Fv(\))p FO(,)g(namely)f Fw(u)1586 2863 y Ft(b)1619 2893 y Fv(\()p Fw(\025)1699 2905 y Fr(0)1737 2893 y Fv(\))h FO(and)f Fw(u)2006 2863 y Ft(a)2046 2893 y Fv(\()p Fw(\025)2126 2905 y Fr(0)2164 2893 y Fv(\))p FO(.)109 3040 y(By)34 b(rep)r(eating)g(the)g(ab)r(o)n(v)n(e)f(pro)r(cedure)h(for)g(ev)n(ery)f Fw(\025)i Fu(2)g Fq(R)40 b FO(w)n(e)34 b(can)g(construct)g(a)g(densit)n (y)g(matrix)g Fw(\045)p Fv(\()p Fw(\025)p Fv(\))p FO(,)k(for)109 3139 y(ev)n(ery)31 b Fw(\025)h Fu(2)g Fq(R)p FO(.)58 b(It)32 b(can)h(easily)e(b)r(e)i(seen)g(that)f(the)h(condition)g (\(6.1\))f(is)h(satis\034ed)f(if)h Fw(u)2871 3109 y Ft(a)2910 3139 y Fv(\()p Fw(\025)p Fv(\))h FO(and)f Fw(u)3271 3109 y Ft(b)3303 3139 y Fv(\()p Fw(\025)p Fv(\))h FO(deca)n(y)109 3239 y(su\036cien)n(tly)27 b(fast,)h(e.g.)36 b Fw(u)916 3209 y Ft(a)956 3239 y Fv(\()p Fw(\025)p Fv(\))p Fw(;)14 b(u)1153 3209 y Ft(b)1187 3239 y Fv(\()p Fw(\025)p Fv(\))28 b FO(ha)n(v)n(e)f(compact)g(supp)r(ort.)109 3386 y(W)-7 b(e)26 b(will)f(giv)n(e)g(some)g(examples:)35 b(Assume)26 b(that)g Fw(u)1700 3356 y Ft(b)1732 3386 y Fv(\()p Fw(\025)1812 3398 y Fr(0)1850 3386 y Fv(\))p Fw(;)14 b(u)1967 3356 y Ft(a)2007 3386 y Fv(\()p Fw(\025)2087 3398 y Fr(0)2125 3386 y Fv(\))24 b Fu(2)f Fq(C)47 b FO(are)24 b(giv)n(en)h(and)g(let)h Fu(R)3043 3398 y Ft(\025)3082 3406 y Fl(0)3143 3386 y Fv(=)c Fu(f)p Fv(1)p Fw(;)14 b Fu(\001)g(\001)g(\001)27 b Fw(;)14 b(N)9 b Fu(g)p FO(,)109 3486 y Fu(G)158 3498 y Ft(\025)197 3506 y Fl(0)257 3486 y Fv(=)23 b Fu(;)p FO(.)36 b(Clearly)27 b(the)h(v)n(ector)1217 3654 y Fw(u)1265 3666 y Ft(R)1319 3654 y Fv(\()p Fw(\025)1399 3666 y Fr(0)1437 3654 y Fv(\))23 b(:=)g(\()p Fw(u)1683 3620 y Ft(b)1716 3654 y Fv(\()p Fw(\025)1796 3666 y Fr(0)1834 3654 y Fv(\))p Fw(;)14 b Fv(0)p Fw(;)g Fu(\001)g(\001)g(\001)27 b Fw(;)14 b Fv(0)p Fw(;)g(u)2270 3620 y Ft(a)2309 3654 y Fv(\()p Fw(\025)2389 3666 y Fr(0)2427 3654 y Fv(\)\))2491 3620 y Ft(T)109 3822 y FO(satis\034es)26 b(equation)h(\(8.3\).)37 b(In)n(tro)r(ducing)27 b(the)h(densit)n(y)f(matrix)g(b)n(y)1374 3991 y Fw(\045)1417 4003 y Ft(R)1471 3991 y Fv(\()p Fw(\025)1551 4003 y Fr(0)1589 3991 y Fv(\))1621 3957 y Ft(T)1697 3991 y Fv(:=)p 1807 3919 253 4 v 22 w Fw(u)1855 4003 y Ft(R)1909 3991 y Fv(\()p Fw(\025)1989 4003 y Fr(0)2027 3991 y Fv(\))p Fw(u)2107 4003 y Ft(R)2161 3991 y Fv(\()p Fw(\025)2241 4003 y Fr(0)2279 3991 y Fv(\))2311 3957 y Ft(T)2364 3991 y Fw(;)109 4159 y FO(w)n(e)g(get)1057 4336 y Fw(j)1091 4348 y Ft(\045)1125 4356 y Fm(R)1177 4336 y Fv(\()1206 4314 y(~)1209 4336 y Fw(b;)14 b(\025)1330 4348 y Fr(0)1368 4336 y Fv(\))k Fu(\000)g Fw(j)1535 4348 y Ft(\045)1569 4356 y Fm(R)1621 4336 y Fv(\()q(~)-43 b Fw(a;)14 b(\025)1782 4348 y Fr(0)1820 4336 y Fv(\))23 b(=)g Fu(\000)2073 4232 y Ft(N)2042 4257 y Fs(X)2042 4433 y Ft(\034)7 b Fr(=1)2176 4336 y Fw(\013)2229 4301 y Fr(2)2229 4356 y Ft(\034)2271 4336 y Fw(u)2319 4348 y Ft(\045)2353 4356 y Fm(R)2404 4336 y Fv(\()p Fw(x)2483 4348 y Ft(\034)2525 4336 y Fw(;)14 b(\025)2610 4348 y Fr(0)2648 4336 y Fv(\))p Fw(;)109 4553 y FO(i.e.)37 b(the)28 b(densit)n(y)f(matrix)g Fw(\045)994 4565 y Ft(R)1048 4553 y Fv(\()p Fw(\025)1128 4565 y Fr(0)1166 4553 y Fv(\))h FO(is)g(purely)f(recom)n(binativ)n(e.)109 4700 y(T)-7 b(o)27 b(obtain)g(a)g(densit)n(y)h(matrix)f(whic)n(h)g(is)h (purely)f(generativ)n(e)e(w)n(e)j(in)n(tro)r(duce)f(the)h(v)n(ector) 1410 4868 y Fw(u)1458 4880 y Ft(G)1513 4868 y Fv(\()p Fw(\025)1593 4880 y Fr(0)1631 4868 y Fv(\))c(:=)e(\002)1862 4880 y Ft(H)1925 4868 y Fv(\()p Fw(\025)2005 4880 y Fr(0)2043 4868 y Fv(\))p Fw(u)2123 4880 y Ft(R)2177 4868 y Fv(\()p Fw(\025)2257 4880 y Fr(0)2295 4868 y Fv(\))p Fw(;)109 5036 y FO(whic)n(h)j(satis\034es)f(equation)h(\(8.3\))f(with)i Fu(R)1436 5048 y Ft(\025)1475 5056 y Fl(0)1535 5036 y Fv(=)d Fu(;)i FO(and)g Fu(G)1898 5048 y Ft(\025)1937 5056 y Fl(0)1997 5036 y Fv(=)e Fu(f)p Fv(1)p Fw(;)14 b Fu(\001)g(\001)g(\001)26 b Fw(;)14 b(N)9 b Fu(g)p FO(.)36 b(As)25 b(ab)r(o)n(v)n(e)f(w)n(e)g(de\034ne)i(the)f(densit)n(y)109 5136 y(matrix)i(b)n(y)g Fw(\045)537 5148 y Ft(G)593 5136 y Fv(\()p Fw(\025)673 5148 y Fr(0)711 5136 y Fv(\))743 5106 y Ft(T)818 5136 y Fv(:=)p 929 5064 254 4 v 23 w Fw(u)977 5148 y Ft(G)1032 5136 y Fv(\()p Fw(\025)1112 5148 y Fr(0)1150 5136 y Fv(\))q Fw(u)1231 5148 y Ft(G)1286 5136 y Fv(\()p Fw(\025)1366 5148 y Fr(0)1404 5136 y Fv(\))1436 5106 y Ft(T)1516 5136 y FO(and)h(get)1095 5386 y Fw(j)1129 5398 y Ft(\045)1163 5406 y Fm(G)1216 5386 y Fv(\()1245 5364 y(~)1248 5386 y Fw(b;)14 b(\025)1369 5398 y Fr(0)1406 5386 y Fv(\))19 b Fu(\000)f Fw(j)1574 5398 y Ft(\045)1608 5406 y Fm(G)1661 5386 y Fv(\()q(~)-43 b Fw(a;)14 b(\025)1822 5398 y Fr(0)1860 5386 y Fv(\))23 b(=)2034 5282 y Ft(N)2003 5307 y Fs(X)2003 5483 y Ft(\034)7 b Fr(=1)2138 5386 y Fw(\013)2191 5351 y Fr(2)2191 5406 y Ft(\034)2233 5386 y Fw(u)2281 5398 y Ft(\045)2315 5406 y Fm(G)2367 5386 y Fv(\()p Fw(x)2446 5398 y Ft(\034)2488 5386 y Fw(;)14 b(\025)2573 5398 y Fr(0)2611 5386 y Fv(\))p Fw(:)109 5630 y FO(Th)n(us)27 b Fw(\045)362 5642 y Ft(G)418 5630 y Fv(\()p Fw(\025)498 5642 y Fr(0)536 5630 y Fv(\))h FO(is)f(purely)g(generativ)n(e.)35 b(W)-7 b(e)28 b(note)g(that)1268 5799 y Fw(\045)1311 5811 y Ft(G)1367 5799 y Fv(\()p Fw(\025)1447 5811 y Fr(0)1485 5799 y Fv(\))23 b(=)g(\002)1693 5811 y Ft(H)1755 5799 y Fv(\()p Fw(\025)1835 5811 y Fr(0)1873 5799 y Fv(\))p Fw(\045)1948 5811 y Ft(R)2003 5799 y Fv(\()p Fw(\025)2083 5811 y Fr(0)2121 5799 y Fv(\)\002)2218 5811 y Ft(H)2281 5799 y Fv(\()p Fw(\025)2361 5811 y Fr(0)2399 5799 y Fv(\))2431 5765 y Fo(\003)2469 5799 y Fw(:)989 b FO(\(8.7\))p eop %%Page: 24 26 24 25 bop 109 254 a FO(24)2477 b Fx(M.)31 b(Bar)n(o,)f(H.)h(Neidhardt) 109 503 y Fz(9)131 b(Remarks)109 739 y FO(The)25 b(de\034nition)g(of)g (carrier)f(and)h(curren)n(t)f(densit)n(y)h(w)n(as)f(form)n(ulated)g(in) i(terms)f(of)g(the)g(complete)g(orthonormal)109 839 y(system)i(of)g (eigenfunctions)h Fu(f)1068 817 y Fw(~)1063 839 y(\036)p Fv(\()p Fu(\001)p Fw(;)14 b(\025;)g(a)p Fv(\))p Fw(;)g Fu(\001)g(\001)g(\001)28 b Fw(;)1568 817 y(~)1564 839 y(\036)p Fv(\()p Fu(\001)p Fw(;)14 b(\025;)g(b)p Fv(\))p Fu(g)1900 851 y Ft(\025)p Fo(2)p Fk(R)2030 839 y FO(.)37 b(Since)28 b(the)g(eigenfunctions)f(b)r(eha)n(v)n(e)g(on)g Fq(R)3438 851 y Fo(\000)3527 839 y FO(lik)n(e)109 938 y(free)i(w)n(a)n(v)n(e,)e(they)j(are)e(called)g(the)i(incoming)f (eigenfunctions.)41 b(By)29 b(equation)f(\(5.10\))g(w)n(e)h(can)g (de\034ne)g(another)109 1038 y(system)e(of)g(orthogonal)f (eigenfunctions)h(b)n(y)937 1251 y Fw(~)933 1273 y(\036)982 1285 y Fr(out)1082 1273 y Fv(\()p Fu(\001)p Fw(;)14 b(\025;)g(j)5 b Fv(\))24 b(:=)1534 1217 y(1)p 1475 1254 161 4 v 1475 1270 a Fu(p)p 1544 1270 92 4 v 69 x Fv(2)p Fw(\031)1650 1251 y(~)1645 1273 y(\036)1694 1239 y Fr(\002\()p Ft(\025)p Fr(\))p Ft(e)1867 1247 y Fm(j)1904 1273 y Fv(\()p Fu(\001)p Fw(;)14 b(\025)p Fv(\))p Fw(;)98 b(j)28 b Fv(=)22 b(0)p Fw(;)14 b Fu(\001)g(\001)g(\001)27 b Fw(;)14 b(N)27 b Fv(+)18 b(1)p Fw(;)109 1521 y FO(whic)n(h)27 b(are)g(called)g(the)h (outgoing)e(eigenfunctions.)37 b(By)27 b(equation)g(\(6.8\))g(the)h (follo)n(wing)e(relation)h(holds)961 1849 y Fv(\002)1026 1861 y Ft(H)1089 1849 y Fv(\()p Fw(\025)p Fv(\)\010)1261 1861 y Fr(in)1321 1849 y Fv(\()p Fw(x;)14 b(\025)p Fv(\))25 b(=)1630 1657 y Fs(0)1630 1803 y(B)1630 1856 y(@)1748 1721 y Fw(\036)1797 1733 y Fr(out)1897 1721 y Fv(\()p Fw(x;)14 b(\025;)g(b)p Fv(\))1946 1809 y FO(.)1946 1842 y(.)1946 1875 y(.)1744 1975 y Fw(\036)1793 1987 y Fr(out)1893 1975 y Fv(\()p Fw(x;)g(\025;)g(a)p Fv(\))2213 1657 y Fs(1)2213 1803 y(C)2213 1856 y(A)2308 1849 y Fv(:=)23 b(\010)2479 1861 y Fr(out)2579 1849 y Fv(\()p Fw(x;)14 b(\025)p Fv(\))p Fw(;)683 b FO(\(9.1\))109 2176 y(where)27 b(w)n(e)g(ha)n(v)n(e)f(set)i Fv(\010)852 2188 y Fr(in)911 2176 y Fv(\()p Fw(x;)14 b(\025)p Fv(\))25 b(=)d(\010\()p Fw(x;)14 b(\025)p Fv(\))p FO(.)109 2323 y(Using)27 b(the)h(outgoing)e (eigenfunction)i(w)n(e)f(de\034ne)h(the)g(\021outgoing\021)k(carrier)26 b(and)h(curren)n(t)g(densit)n(y)g(b)n(y)929 2523 y Fw(u)977 2489 y Fr(out)977 2544 y Ft(\045)1076 2523 y Fv(\()p Fw(x;)14 b(\025)p Fv(\))24 b(:=)1407 2456 y Fs(\012)1446 2523 y Fw(\045)p Fv(\()p Fw(\025)p Fv(\))1601 2489 y Ft(T)1654 2523 y Fv(\010)1714 2535 y Fr(out)1814 2523 y Fv(\()p Fw(x;)14 b(\025)p Fv(\))p Fw(;)g Fv(\010)2107 2535 y Fr(out)2208 2523 y Fv(\()p Fw(x;)g(\025)p Fv(\))2404 2456 y Fs(\013)2459 2523 y Fw(;)937 2709 y(j)976 2674 y Fr(out)971 2729 y Ft(\045)1076 2709 y Fv(\()p Fw(x;)g(\025)p Fv(\))24 b(:=)f(Im)1520 2592 y Fs(\034)1582 2709 y Fw(\045)p Fv(\()p Fw(\025)p Fv(\))1737 2674 y Ft(T)1872 2652 y Fv(1)p 1800 2690 185 4 v 1800 2766 a Fw(m)p Fv(\()p Fw(x)p Fv(\))1995 2709 y(\010)2055 2721 y Fr(out)2155 2709 y Fv(\()p Fw(x;)14 b(\025)p Fv(\))2351 2674 y Fo(0)2376 2709 y Fw(;)g Fv(\010)2473 2721 y Fr(out)2573 2709 y Fv(\()p Fw(x;)g(\025)p Fv(\))2769 2592 y Fs(\035)109 3006 y FO(Using)28 b(\(9.1\))f(w)n(e)h(get)872 3206 y Fw(u)920 3172 y Fr(out)920 3227 y Ft(\045)1020 3206 y Fv(\()p Fw(x;)14 b(\025)p Fv(\))24 b(=)f Fw(u)1376 3172 y Fr(in)1376 3227 y(\002)1427 3235 y Fm(H)1480 3227 y Ft(\045)p Fr(\002)1565 3207 y Fc(\003)1565 3246 y Fm(H)1622 3206 y Fv(\()p Fw(x;)14 b(\025)p Fv(\))p Fw(;)99 b(j)1979 3172 y Fr(out)1974 3227 y Ft(\045)2078 3206 y Fv(\()p Fw(x;)14 b(\025)p Fv(\))25 b(=)e Fw(j)2426 3172 y Fr(in)2421 3227 y(\002)2472 3235 y Fm(H)2525 3227 y Ft(\045)p Fr(\002)2610 3207 y Fc(\003)2610 3246 y Fm(H)2668 3206 y Fv(\()p Fw(x;)14 b(\025)p Fv(\))p Fw(;)109 3424 y FO(where)28 b Fw(u)398 3394 y Fr(in)398 3445 y Ft(\045)457 3424 y Fv(\()p Fw(x;)14 b(\025)p Fv(\))p FO(,)31 b Fw(j)746 3394 y Fr(in)741 3445 y Ft(\045)805 3424 y Fv(\()p Fw(x;)14 b(\025)p Fv(\))30 b FO(denotes)f(the)g(carrier,)e(resp)r(ectiv)n(ely)h(curren)n(t,)g (densit)n(y)h(de\034ned)g(in)g(the)g(previous)109 3524 y(sections.)109 3671 y(The)g(incoming)g(and)g(the)h(outgoing)e(carrier) f(and)i(curren)n(t)g(densities)g(coincide,)h(if)g Fw(\045)p Fv(\()p Fw(\025)p Fv(\))g FO(comm)n(utes)f(with)h(the)109 3771 y(c)n(haracteristic)20 b(function)i Fv(\002)998 3783 y Ft(H)1061 3771 y Fv(\()p Fw(\025)p Fv(\))p FO(.)36 b(As)22 b(already)f(men)n(tioned)h(ab)r(o)n(v)n(e)f(in)h(this)g(case)g (neither)g(loss)f(nor)g(gain)h(e\033ects)109 3870 y(will)27 b(o)r(ccur.)109 4017 y(In)i(the)h(previous)e(section)h(w)n(e)g(ga)n(v)n (e)e(an)i(example)g(for)g(a)g(purely)g(recom)n(binativ)n(e)f(densit)n (y)h(matrix)f Fw(\045)3321 4029 y Ft(R)3376 4017 y Fv(\()p Fw(\025)p Fv(\))i FO(and)109 4117 y(an)d(example)g(for)g(a)g(strict)h (generativ)n(e)d(matrix)i Fw(\045)1672 4129 y Ft(G)1728 4117 y Fv(\()p Fw(\025)p Fv(\))p FO(.)38 b(Using)27 b(the)h(relations)f (\(9.1\))g(and)g(\(8.7\))h(w)n(e)f(get)1486 4317 y Fw(j)1525 4283 y Fr(in)1520 4337 y Ft(\045)1554 4345 y Fm(G)1607 4317 y Fv(\()p Fw(x;)14 b(\025)p Fv(\))25 b(=)d Fw(j)1954 4283 y Fr(out)1949 4337 y Ft(\045)1983 4345 y Fm(R)2054 4317 y Fv(\()p Fw(x;)14 b(\025)p Fv(\))p Fw(:)109 4517 y FO(i.e.)48 b(the)31 b(generativ)n(e)f(matrix)g Fw(\045)1126 4529 y Ft(G)1214 4517 y FO(in)h(the)h(incoming)e(represen)n(tation)g(b) r(ecomes)h(a)g(recom)n(binativ)n(e)e(matrix)i(in)109 4617 y(the)d(outgoing)e(represen)n(tation.)109 4764 y(In)n(tro)r (ducing)e(the)i(notions)f Fw(P)1051 4734 y Fr(in)1039 4785 y Ft(j)1134 4764 y Fv(=)d Fu(h\001)p Fw(;)14 b(e)1352 4776 y Ft(j)1387 4764 y Fu(i)p Fw(e)1458 4776 y Ft(j)1519 4764 y FO(and)25 b Fw(P)1743 4734 y Fr(out)1731 4785 y Ft(j)1866 4764 y Fv(=)d Fu(h\001)p Fw(;)14 b(E)2106 4776 y Ft(j)2142 4764 y Fu(i)p Fw(E)2235 4776 y Ft(j)2270 4764 y FO(,)27 b(where)d Fw(E)2618 4776 y Ft(j)2677 4764 y Fv(=)e(\002)2829 4734 y Fo(\003)2829 4787 y Ft(H)2892 4764 y Fv(\()p Fw(\025)p Fv(\))p Fw(e)3043 4776 y Ft(j)3079 4764 y FO(,)k(w)n(e)f(can)g(rewrite)109 4863 y(\(8.1\))i(as)g(follo)n (ws)1065 5028 y Fw(j)1099 5040 y Ft(\045)1138 5028 y Fv(\()1167 5006 y(~)1170 5028 y Fw(b;)14 b(\025)p Fv(\))19 b Fu(\000)f Fw(j)1459 5040 y Ft(\045)1497 5028 y Fv(\()q(~)-43 b Fw(a)q(;)14 b(\025)p Fv(\))23 b(=)1837 4972 y(1)p 1812 5009 92 4 v 1812 5085 a(2)p Fw(\031)1914 5028 y Fv(tr\()p Fw(\045)p Fv(\()p Fw(\025)p Fv(\))p Fu(f)p Fw(P)2273 4994 y Fr(in)2261 5049 y Ft(j)2351 5028 y Fu(\000)18 b Fw(P)2499 4994 y Fr(out)2487 5049 y Ft(j)2599 5028 y Fu(g)p Fv(\))p Fw(:)109 5274 y FO(Let)36 b(us)g(consider)e(t)n(w)n(o) h(di\033eren)n(t)h(dissipativ)n(e)f(Sc)n(hr\366)r(dinger)g(op)r(erator) f Fw(H)2506 5286 y Ft(n)2587 5274 y FO(and)i Fw(H)2826 5286 y Ft(p)2900 5274 y FO(for)g(the)g(t)n(w)n(o)f(di\033eren)n(t)109 5374 y(sp)r(ecies)26 b(of)h(particles)f(\(electron)h(and)f(holes\),)h (with)h Fw(m)1840 5386 y Ft(n)1885 5374 y FO(,)f Fw(m)2008 5386 y Ft(p)2046 5374 y FO(,)h Fw(V)2145 5386 y Ft(n)2190 5374 y FO(,)f Fw(V)2288 5386 y Ft(p)2327 5374 y FO(,)g Fw(\024)2425 5386 y Ft(j;n)2517 5374 y FO(,)h Fw(\024)2616 5386 y Ft(j;p)2701 5374 y FO(,)f Fw(j)h Fv(=)23 b(1)p Fw(;)14 b Fu(\001)g(\001)g(\001)27 b Fw(;)14 b(N)9 b FO(,)26 b(and)h(let)g Fw(x)3593 5386 y Ft(j)3629 5374 y FO(,)109 5473 y Fw(j)j Fv(=)25 b(1)p Fw(;)14 b Fu(\001)g(\001)g(\001) 27 b Fw(;)14 b(N)9 b FO(,)30 b(b)r(e)f(n)n(um)n(b)r(ers)g(within)g(the) h(in)n(terv)-5 b(al)28 b Fv(\()p Fw(a;)14 b(b)p Fv(\))30 b FO(suc)n(h)e(that)i Fw(a)25 b(<)g(x)2575 5485 y Fr(1)2639 5473 y Fw(<)g Fu(\001)14 b(\001)g(\001)25 b Fw(<)g(x)2988 5485 y Ft(N)3077 5473 y Fw(<)h(b)p FO(.)41 b(W)-7 b(e)29 b(denote)109 5573 y(b)n(y)d Fw(u)271 5585 y Ft(\045)305 5594 y Fm(k)345 5573 y Fv(\()p Fw(x;)14 b(\025)p Fv(\))p FO(,)28 b(resp)r(ectiv)n(ely)e Fw(j)1078 5585 y Ft(\045)1112 5594 y Fm(k)1153 5573 y Fv(\()p Fw(x;)14 b(\025)p Fv(\))p FO(,)28 b(the)f(carrier)d(densit)n(y)-7 b(,)27 b(resp)r(ectiv)n(ely)e (curren)n(t)h(densit)n(y)-7 b(,)26 b(corresp)r(onding)109 5673 y(to)h(the)h(densit)n(y)f(matrix)g Fw(\045)952 5685 y Ft(k)1021 5673 y FO(and)g(the)h(op)r(erator)e Fw(H)1729 5685 y Ft(k)1770 5673 y FO(,)i Fw(k)d Fv(=)e Fw(n;)14 b(p)p FO(.)109 5820 y(Assume)38 b(that)g(the)h(sets)f Fu(G)993 5832 y Ft(\025)1078 5820 y Fv(=)i Fu(G)k FO(and)38 b Fu(R)1518 5832 y Ft(\025)1602 5820 y Fv(=)j Fu(R)d FO(are)g(giv)n(en.)68 b(Let)38 b Fw(\045)2449 5832 y Ft(n)2494 5820 y Fv(\()p Fw(\025)p Fv(\))h FO(and)f Fw(\045)2860 5832 y Ft(p)2899 5820 y Fv(\()p Fw(\025)p Fv(\))h FO(densit)n(y)f (matrices)p eop %%Page: 25 27 25 26 bop 109 254 a FH(Dissipativ)n(e)27 b(Sc)n(hr\366)r(dinger-t)n(yp) r(e)e(op)r(erator)2084 b FO(25)109 503 y(satisfying)26 b(equation)h(\(6.1\))h(suc)n(h)f(that)179 687 y Fw(j)213 699 y Ft(\045)247 707 y Fm(n)292 687 y Fv(\()321 665 y(~)324 687 y Fw(b;)14 b(\025)p Fv(\))19 b Fu(\000)f Fw(j)613 699 y Ft(\045)647 707 y Fm(n)693 687 y Fv(\()q(~)-43 b Fw(a;)14 b(\025)p Fv(\))24 b(=)1069 608 y Fs(X)997 786 y Ft(\034)7 b Fo(2R\\T)1220 794 y Fm(!)1275 687 y Fw(\013)1328 652 y Fr(2)1328 707 y Ft(\034)s(;n)1427 687 y Fw(u)1475 699 y Ft(\045)1509 707 y Fm(n)1554 687 y Fv(\()p Fw(x)1633 699 y Ft(\034)1675 687 y Fw(;)14 b(\025)p Fv(\))19 b Fu(\000)1957 608 y Fs(X)1894 787 y Ft(\030)r Fo(2G)s(\\T)2097 795 y Fm(!)2153 687 y Fw(\013)2206 652 y Fr(2)2206 707 y Ft(\030)r(;n)2303 687 y Fw(u)2351 699 y Ft(\045)2385 707 y Fm(n)2430 687 y Fv(\()p Fw(x)2509 699 y Ft(\030)2546 687 y Fw(;)14 b(\025)p Fv(\))p Fw(;)181 b FO(for)27 b(a.e.)82 b Fw(\025)24 b Fu(2)f Fq(R)76 b FO(\(9.2\))191 1010 y Fw(j)225 1022 y Ft(\045)259 1030 y Fm(p)298 1010 y Fv(\()327 988 y(~)330 1010 y Fw(b;)14 b(\025)p Fv(\))19 b Fu(\000)f Fw(j)619 1022 y Ft(\045)653 1030 y Fm(p)693 1010 y Fv(\()q(~)-43 b Fw(a;)14 b(\025)p Fv(\))24 b(=)e Fu(\000)1076 843 y Fs(0)1076 992 y(@)1220 931 y(X)1149 1109 y Ft(\034)7 b Fo(2R\\T)1372 1117 y Fm(!)1426 1010 y Fw(\013)1479 975 y Fr(2)1479 1030 y Ft(\034)s(;p)1571 1010 y Fw(u)1619 1022 y Ft(\045)1653 1030 y Fm(p)1692 1010 y Fv(\()p Fw(x)1771 1022 y Ft(\034)1813 1010 y Fw(;)14 b(\025)p Fv(\))19 b Fu(\000)2095 931 y Fs(X)2032 1109 y Ft(\030)r Fo(2G)s(\\T)2235 1117 y Fm(!)2291 1010 y Fw(\013)2344 975 y Fr(2)2344 1030 y Ft(\030)r(;p)2435 1010 y Fw(u)2483 1022 y Ft(\045)2517 1030 y Fm(p)2556 1010 y Fv(\()p Fw(x)2635 1022 y Ft(\030)2672 1010 y Fw(;)14 b(\025)p Fv(\))2789 843 y Fs(1)2789 992 y(A)2876 1010 y Fw(;)180 b FO(for)27 b(a.e.)82 b Fw(\025)24 b Fu(2)f Fq(R)p Fw(:)3481 1209 y FO(\(9.3\))109 1371 y(By)32 b(the)h (consideration)e(of)h(section)g(8,)i(suc)n(h)e(densit)n(y)g(matrices)g (exists.)51 b(In)n(tegrating)33 b(\(9.2\))f(and)g(\(9.3\))h(with)109 1470 y(resp)r(ect)27 b(to)g Fw(\025)h FO(yields)691 1650 y Fw(j)725 1662 y Ft(\045)759 1670 y Fm(n)804 1650 y Fv(\()833 1628 y(~)836 1650 y Fw(b)p Fv(\))19 b Fu(\000)f Fw(j)1040 1662 y Ft(\045)1074 1670 y Fm(n)1120 1650 y Fv(\()q(~)-43 b Fw(a)p Fv(\))23 b(=)1411 1571 y Fs(X)1339 1749 y Ft(\034)7 b Fo(2R\\T)1562 1757 y Fm(!)1617 1650 y Fw(\013)1670 1616 y Fr(2)1670 1670 y Ft(\034)s(;n)1768 1650 y Fw(u)1816 1662 y Ft(\045)1850 1670 y Fm(n)1895 1650 y Fv(\()p Fw(x)1974 1662 y Ft(\034)2016 1650 y Fv(\))19 b Fu(\000)2213 1571 y Fs(X)2150 1750 y Ft(\030)r Fo(2G)s(\\T)2353 1758 y Fm(!)2409 1650 y Fw(\013)2462 1616 y Fr(2)2462 1670 y Ft(\030)r(;n)2559 1650 y Fw(u)2607 1662 y Ft(\045)2641 1670 y Fm(n)2686 1650 y Fv(\()p Fw(x)2765 1662 y Ft(\030)2802 1650 y Fw(;)14 b Fv(\))p Fw(;)587 b FO(\(9.4\))703 1973 y Fw(j)737 1985 y Ft(\045)771 1993 y Fm(p)810 1973 y Fv(\()839 1951 y(~)842 1973 y Fw(b)p Fv(\))19 b Fu(\000)f Fw(j)1046 1985 y Ft(\045)1080 1993 y Fm(p)1120 1973 y Fv(\()q(~)-43 b Fw(a)p Fv(\))23 b(=)g Fu(\000)1418 1806 y Fs(0)1418 1955 y(@)1562 1894 y(X)1490 2072 y Ft(\034)7 b Fo(2R\\T)1713 2080 y Fm(!)1768 1973 y Fw(\013)1821 1938 y Fr(2)1821 1993 y Ft(\034)s(;p)1913 1973 y Fw(u)1961 1985 y Ft(\045)1995 1993 y Fm(p)2034 1973 y Fv(\()p Fw(x)2113 1985 y Ft(\034)2155 1973 y Fv(\))18 b Fu(\000)2351 1894 y Fs(X)2289 2073 y Ft(\030)r Fo(2G)s(\\T)2492 2081 y Fm(!)2547 1973 y Fw(\013)2600 1938 y Fr(2)2600 1993 y Ft(\030)r(;p)2691 1973 y Fw(u)2739 1985 y Ft(\045)2773 1993 y Fm(p)2812 1973 y Fv(\()p Fw(x)2891 1985 y Ft(\030)2928 1973 y Fv(\))2960 1806 y Fs(1)2960 1955 y(A)3047 1973 y Fw(:)411 b FO(\(9.5\))109 2287 y(In)n(tegrating)26 b(the)i(stationary)e(con)n(tin)n(uit)n(y)h(equation)g(\(cf.)37 b(\(1.3\)\))28 b(o)n(v)n(er)d Fw(!)31 b FO(giv)n(es)1205 2479 y Fw(j)1239 2491 y Ft(\045)1273 2499 y Fm(n)1318 2479 y Fv(\()1347 2458 y(~)1350 2479 y Fw(b)p Fv(\))19 b Fu(\000)f Fw(j)1554 2491 y Ft(\045)1588 2499 y Fm(n)1634 2479 y Fv(\()q(~)-43 b Fw(a)p Fv(\))23 b(=)g Fu(\000)1932 2387 y Fs(\020)1981 2479 y Fw(j)2015 2491 y Ft(\045)2049 2499 y Fm(p)2088 2479 y Fv(\()2117 2458 y(~)2120 2479 y Fw(b)p Fv(\))c Fu(\000)f Fw(j)2324 2491 y Ft(\045)2358 2499 y Fm(p)2398 2479 y Fv(\()q(~)-43 b Fw(a)p Fv(\))2506 2387 y Fs(\021)3481 2479 y FO(\(9.6\))109 2677 y(Matc)n(hing)27 b(equation)f(\(9.6\))i(with)g(\(9.4\))f(and)g(\(9.5\))h(leads)f(to)g(a) g(condition)h(on)f(the)h Fw(\013)2816 2647 y Fr(2)2816 2699 y Ft(j;n)2908 2677 y FO('s)g(and)f Fw(\013)3206 2647 y Fr(2)3206 2699 y Ft(j;p)3291 2677 y FO('s,)h(i.e.)801 2858 y Fw(\013)854 2824 y Fr(2)854 2878 y Ft(j;n)970 2858 y Fv(=)22 b Fw(u)1105 2870 y Ft(\045)1139 2878 y Fm(p)1178 2858 y Fv(\()p Fw(x)1257 2870 y Ft(j)1293 2858 y Fw(;)14 b(\025)p Fv(\))p Fw(;)98 b FO(and)82 b Fw(\013)1800 2824 y Fr(2)1800 2878 y Ft(j;p)1909 2858 y Fv(=)22 b Fw(u)2044 2870 y Ft(\045)2078 2878 y Fm(n)2123 2858 y Fv(\()p Fw(x)2202 2870 y Ft(j)2238 2858 y Fw(;)14 b(\025)p Fv(\))p Fw(;)98 b(j)28 b Fv(=)22 b(1)p Fw(;)14 b Fu(\001)g(\001)g(\001) 27 b Fw(;)14 b(N)t(:)522 b FO(\(9.7\))109 3025 y(The)40 b(equations)g(\(9.4\))o(,\(9.5\))h(can)f(no)n(w)g(b)r(e)h(view)n(ed)f (as)g(a)g(stationary)f(con)n(tin)n(uit)n(y)g(equation)h(with)h(recom-) 109 3125 y(bination-generation)25 b(term)j(giv)n(en)e(b)n(y)532 3217 y Fs(Z)578 3406 y Ft(!)640 3330 y Fw(R)q Fv(\()p Fw(u)784 3342 y Ft(\045)818 3350 y Fm(n)863 3330 y Fv(\()p Fw(x)p Fv(\))p Fw(;)14 b(u)1059 3342 y Ft(\045)1093 3350 y Fm(p)1132 3330 y Fv(\()p Fw(x)p Fv(\)\))g Fw(dx)25 b Fv(=)1561 3251 y Fs(X)1492 3430 y Ft(j)s Fo(2T)1605 3438 y Fm(!)1648 3430 y Fo(\\R)1763 3330 y Fw(u)1811 3342 y Ft(\045)1845 3350 y Fm(n)1890 3330 y Fv(\()p Fw(x)1969 3342 y Ft(j)2005 3330 y Fv(\))p Fw(u)2085 3342 y Ft(\045)2119 3350 y Fm(p)2158 3330 y Fv(\()p Fw(x)2237 3342 y Ft(j)2273 3330 y Fv(\))18 b Fu(\000)2468 3251 y Fs(X)2406 3430 y Ft(j)s Fo(2T)2519 3438 y Fm(!)2562 3430 y Fo(\\G)2664 3330 y Fw(u)2712 3342 y Ft(\045)2746 3350 y Fm(n)2791 3330 y Fv(\()p Fw(x)2870 3342 y Ft(j)2906 3330 y Fv(\))p Fw(u)2986 3342 y Ft(\045)3020 3350 y Fm(p)3059 3330 y Fv(\()p Fw(x)3138 3342 y Ft(j)3174 3330 y Fv(\))p Fw(:)109 3642 y FO(T)-7 b(o)29 b(solv)n(e)g(the)h(nonlinear)f(equation)g(giv)n (en)g(b)n(y)i(\(9.7\))e(w)n(e)h(in)n(tro)r(duce)f(the)h(op)r(erators)e Fu(N)2909 3654 y Ft(k)2977 3642 y Fv(:)f Fq(R)3081 3612 y Ft(N)3081 3663 y Fr(+)3177 3642 y Fu(!)f Fq(R)3340 3612 y Ft(N)3340 3663 y Fr(+)3409 3642 y FO(,)31 b(giv)n(en)109 3742 y(b)n(y)1054 3842 y Fu(N)1122 3854 y Ft(k)1163 3842 y Fv(\()p Fw(\013)1248 3807 y Fr(2)1248 3862 y(1)p Ft(;k)1342 3842 y Fw(;)14 b Fu(\001)g(\001)g(\001)27 b Fw(;)14 b(\013)1593 3807 y Fr(2)1593 3862 y Ft(N)s(;k)1709 3842 y Fv(\))23 b(=)g(\()p Fw(u)1932 3854 y Ft(\045)1966 3863 y Fm(k)2007 3842 y Fv(\()p Fw(x)2086 3854 y Fr(1)2124 3842 y Fv(\))p Fw(;)14 b Fu(\001)g(\001)g(\001)27 b Fw(;)14 b(u)2402 3854 y Ft(\045)2436 3863 y Fm(k)2477 3842 y Fv(\()p Fw(x)2556 3854 y Ft(N)2619 3842 y Fv(\)\))p Fw(;)109 3982 y FO(where)27 b Fw(k)f Fv(=)c Fw(n;)14 b(p)p FO(.)109 4129 y(A)28 b(solution)f(of)34 b(\(9.7\))27 b(is)g(th)n(us)h(giv)n(en)f(b)n(y)g(a)g(\034x)h(p)r(oin)n (t)f(of)h(the)g(op)r(erator)e Fu(E)35 b FO(de\034ned)27 b(b)n(y)1047 4296 y Fu(E)7 b Fv(\()p Fw(\013)1183 4262 y Fr(2)1183 4317 y(1)p Ft(;n)1281 4296 y Fw(;)14 b Fu(\001)g(\001)g (\001)28 b Fw(;)14 b(\013)1533 4262 y Fr(2)1533 4317 y Ft(N)s(;n)1653 4296 y Fv(\))23 b(:=)g Fu(N)1887 4308 y Ft(p)1926 4296 y Fv(\()p Fu(N)2026 4308 y Ft(n)2071 4296 y Fv(\()p Fw(\013)2156 4262 y Fr(2)2156 4317 y(1)p Ft(;n)2255 4296 y Fw(;)14 b Fu(\001)g(\001)g(\001)27 b Fw(;)14 b(\013)2506 4262 y Fr(2)2506 4317 y Ft(N)s(;n)2626 4296 y Fv(\)\))p Fw(:)109 4511 y FO(As)37 b(in)h([KNR01c)n(,)g(BKNR02a) n(])g(one)e(can)h(set)h(up)f(a)g(dissipativ)n(e)g(Sc)n(hr\366)r (dinger-P)n(oisson)d(system)j(using)g(the)109 4611 y(de\034nition)f(of) f(carrier)f(densities)i(in)n(tro)r(duced)f(in)h(this)g(article.)61 b(In)36 b(addition)f(to)h(the)g(Sc)n(hr\366)r(dinger-P)n(oisson)109 4710 y(system)30 b(considered)g(in)h([BKNR02a)n(])g(the)h(curren)n(t)e (is)g(not)h(constan)n(t.)46 b(This)31 b(system)g(will)g(b)r(e)g (discussed)f(in)h(a)109 4810 y(forthcoming)26 b(pap)r(er.)109 4957 y(In)f(section)g(8)g(w)n(e)h(constructed)e(explicitly)i(a)f (densit)n(y)g(matrix)g Fw(\045)p Fv(\()p Fu(\001)p Fv(\))h FO(suc)n(h)f(that)h(loss)f(and)g(gain)g(e\033ects)g(o)r(ccur)g(at)109 5056 y(prede\034ned)j(p)r(oin)n(ts.)38 b(F)-7 b(urthermore)27 b(w)n(e)h(sho)n(w)n(ed)f(that)h(one)g(has)f(t)n(w)n(o)h(free)g (parameters)e(in)i(the)h(c)n(hoice)e(of)h Fw(\045)p Fv(\()p Fu(\001)p Fv(\))p FO(,)109 5156 y(i.e.)55 b Fw(u)318 5168 y Ft(b)351 5156 y Fv(\()p Fw(\025)p Fv(\))35 b FO(and)e Fw(u)713 5168 y Ft(a)753 5156 y Fv(\()p Fw(\025)p Fv(\))p FO(.)56 b(These)34 b(t)n(w)n(o)e(parameters)g(giv)n(e)h(the)h(p)r (ossibilit)n(y)f(to)h(couple)f(\(for)h(example\))f(a)h(drift)109 5256 y(di\033usion)d(mo)r(del)h(to)f(the)h(op)r(en)f(quan)n(tum)h (system.)48 b(W)-7 b(e)32 b(in)n(tend)f(to)h(in)n(v)n(estigate)e(this)h (coupled)h(system)f(in)g(a)109 5355 y(forthcoming)26 b(pap)r(er.)109 5511 y(F)-7 b(or)35 b(simplicit)n(y)h(w)n(e)f (considered)g(imaginary)f(p)r(oten)n(tials)h(of)h(the)g(form)g Fu(\000)2521 5479 y Ft(\013)2564 5454 y Fl(2)p 2520 5493 76 4 v 2542 5540 a Fr(2)2606 5511 y Fw(\016)s Fv(\()p Fu(\001)24 b(\000)g Fw(x)2861 5523 y Ft(j)2896 5511 y Fv(\))p FO(.)62 b(A)36 b(further)g(step)g(in)109 5611 y(order)28 b(to)i(include)g(recom)n(bination)f(and)h(generation)e(in)n (to)i(the)g(mo)r(del)g(is)g(to)g(allo)n(w)f(complex)h(p)r(oten)n (tials,)g(i.e.)109 5720 y Fv(Im\()p Fw(V)19 b Fv(\))33 b(=)f Fu(\000)544 5687 y Ft(\013)587 5662 y Fl(2)p 544 5701 V 565 5749 a Fr(2)629 5720 y FO(,)j(where)e Fw(\013)g Fv(=)f Fw(\013)p Fv(\()p Fw(x)p Fv(\))j Fu(2)e Fp(L)1448 5690 y Ft(p)1519 5720 y FO(for)g(some)g Fw(p)f(>)h Fv(1)p FO(.)53 b(This)34 b(w)n(ould)f(giv)n(e)f(the)i(p)r(ossibilit)n(y)f(to)g (mo)r(del)109 5820 y(recom)n(bination-generation)24 b(rates)i(on)i(the) g(whole)f(in)n(terv)-5 b(al)27 b Fv(\012)p FO(.)p eop %%Page: 26 28 26 27 bop 109 254 a FO(26)2477 b Fx(M.)31 b(Bar)n(o,)f(H.)h(Neidhardt) 109 503 y Fz(A)l(c)l(kno)l(wledgmen)l(ts)109 732 y FO(The)25 b(authors)f(lik)n(e)h(to)g(thank)g(Prof.)36 b(P)-7 b(.)26 b(Degond,)f(H.-Ch.)37 b(Kaiser,)24 b(and)h(J.)g(Reh)n(b)r(erg)f(for)h (fruitful)h(discussions.)109 831 y(The)h(\034nancial)g(supp)r(ort)h(of) f(DF)n(G)h(\(Gran)n(t)f(RE)i(1480/1-2\))24 b(is)j(also)g(gratefully)f (ac)n(kno)n(wledged.)109 1150 y Fz(References)109 1360 y FO([A)n(GHH88])98 b(S.)34 b(Alb)r(ev)n(erio,)g(F.)g(Gesztesy)-7 b(,)35 b(R.)f(H\370gh-Krohn,)f(H.)h(Holden,)h FA(Solvable)i(mo)l(dels)f (in)f(quantum)586 1460 y(me)l(chanics)p FO(,)29 b(T)-7 b(ext)27 b(and)h(Monographs)d(in)j(Ph)n(ysics,)f(Springer,)g(New)h(Y)-7 b(ork)26 b(1988.)109 1618 y([AK00])220 b(S.)40 b(Alb)r(ev)n(erio,)j(P) -7 b(.)41 b(Kuraso)n(v,)g FA(Singular)g(p)l(erturb)l(ations)g(of)i (di\033er)l(ential)f(op)l(er)l(ators)p FO(,)j(Lecture)586 1717 y(Note)28 b(Series)e(271,)h(Cam)n(bridge)f(Univ)n(ersit)n(y)g (Press,)h(Cam)n(bridge)g(2000.)109 1875 y([A87])285 b(B.)25 b(P)-7 b(.)26 b(Allakh)n(v)n(erdiev,)e FA(On)i(dissip)l(ative)k (extensions)d(of)h(the)g(symmetric)g(Schr)l(\366)l(dinger)g(op)l(er)l (ator)586 1975 y(in)i(W)-6 b(eyl's)30 b(limit-cir)l(cle)h(c)l(ase)p FO(,)d(Dokl.)g(Ak)-5 b(ad.)27 b(Nauk)h(SSSR)g Fy(V)-8 b(ol)31 b(293)p FO(,)c(777-781)d(1987.)109 2133 y([A88])285 b(B.)35 b(P)-7 b(.)36 b(Allakh)n(v)n(erdiev,)f(F.)g(G.)g(Maksudo)n(v,)h FA(On)g(the)g(the)l(ory)i(of)f(the)g(char)l(acteristic)h(function)586 2232 y(and)c(sp)l(e)l(ctr)l(al)f(analysis)h(of)g(a)g(dissip)l(ative)h (Schr)l(\366)l(dinger)f(op)l(er)l(ator)p FO(,)g(Dokl.)d(Ak)-5 b(ad.)31 b(Nauk)h(SSSR)586 2332 y Fy(303)p FO(,)27 b(No.6,)g (1307-1309,)c(1988.)109 2490 y([A89])285 b(B.)28 b(P)-7 b(.)29 b(Allakh)n(v)n(erdiev,)d FA(Schr)l(\366)l(dinger)32 b(typ)l(e)e(dissip)l(ative)i(op)l(er)l(ator)f(with)g(a)g(matrix)f(p)l (otential)p FO(,)f(in)586 2589 y(Sp)r(ectral)e(theory)g(of)h(op)r (erators)d(and)i(its)h(applications)f(No.9,)g(11-41,)e(\020\311lm\021,) j(Baku,)f(1989.)109 2747 y([A90])285 b(B.)19 b(P)-7 b(.)21 b(Allakh)n(v)n(erdiev,)e FA(On)j(dilation)i(the)l(ory)f(and)g(sp)l(e)l (ctr)l(al)f(analysis)i(of)f(dissip)l(ative)i(Schr)l(\366)l(dinger)586 2847 y(op)l(er)l(ators)j(in)f(W)-6 b(eyl's)27 b(limit-cir)l(cle)h(c)l (ase)p FO(,)e(Izv.)f(Ak)-5 b(ad.)24 b(Nauk)h(SSSR,)g(Ser.)f(Mat.)g Fy(54)p FO(,)h(No.2,)g(242-)586 2946 y(257,)h(1990.)109 3104 y([A93])285 b(B.)33 b(P)-7 b(.)34 b(Allakh)n(v)n(erdiev,)f FA(On)h(the)h(the)l(ory)h(of)g(non-selfadjoint)g(op)l(er)l(ators)g(of)g (Schr)l(\366)l(dinger)g(typ)l(e)586 3204 y(with)30 b(a)g(matrix)g(p)l (otential)p FO(,)e(Izv.)g(Ross.)e(Ak)-5 b(ad.)28 b(Nauk,)g(Ser.)f(Mat.) g Fy(56)p FO(,)h(No.)f(2,)g(193-205)e(1993.)109 3362 y([A97])285 b(B.)32 b(P)-7 b(.)33 b(Allakh)n(v)n(erdiev,)f(A.)h (Canoglu,)f FA(Sp)l(e)l(ctr)l(al)i(analysis)h(of)g(dissip)l(ative)h (Schr)l(\366)l(dinger)f(op)l(er)l(a-)586 3461 y(tors)p FO(,)28 b(Pro)r(c.)f(R.)h(So)r(c.)f(Edin)n(b.,)i(Sect.)f(A,)g(Math)f Fy(127)p FO(,)g(No.)h(6,)f(1113-1121,)c(1997.)109 3619 y([BKNR02a])58 b(M.)31 b(Baro,)f(H.-Ch.)h(Kaiser,)f(H.)i(Neidhardt,)f (J.)g(Reh)n(b)r(erg,)g FA(Dissip)l(ative)j(Schr)l(\366)l (dinger-Poisson)586 3719 y(systems)p FO(,)d(submitted,)i(WIAS-Preprin)n (t)e Fy(719)p FO(,)g(W)-7 b(eierstrass-Institute)29 b(for)h(Applied)i (Analysis)586 3818 y(and)27 b(Sto)r(c)n(hastics,)g(Berlin)g(2002,)586 3918 y(h)n(ttp://www.wias-b)r(erlin.de/publications/preprin)n(ts/719.h) n(tml)17 b(.)109 4076 y([BA98])226 b(N.)33 b(Ben)f(Ab)r(dallah,)i FA(A)g(hybrid)i(kinetic-quantum)e(mo)l(del)h(for)g(stationary)g(ele)l (ctr)l(on)f(tr)l(ansp)l(ort)586 4175 y(in)c(a)g(r)l(esonant)f (tunneling)g(dio)l(de)p FO(,)g(J.Stat.)f(Ph)n(ys.)g Fy(90)p FO(,)f(no.)g(3-4,)f(627-662,)f(1998.)109 4333 y([BADM97])87 b(N.)34 b(Ben)f(Ab)r(dallah,)i(P)-7 b(.)34 b(Degond,)g(P)-7 b(.A.)34 b(Mark)n(o)n(wic)n(h,)f FA(On)h(a)i(one-dimensional)g(Schr)l (\366)l(dinger-)586 4433 y(Poisson)31 b(sc)l(attering)e(mo)l(del)p FO(,)g(Z.)e(angew.)g(Math.)h(Ph)n(ys.)f Fy(48)p FO(,)h(135-155,)c (1997.)109 4591 y([D80])284 b(E.)28 b(B.)g(Da)n(vies,)e FA(One-p)l(ar)l(ameter)k(semigr)l(oups)p FO(.)e(A)n(cademic)f(Press,)h (London)e(1980.)109 4748 y([FN70])231 b(C.)36 b(F)-7 b(oias,)36 b(B.)g(Sz.-Nagy)-7 b(,)37 b FA(Harmonic)g(analysis)i(of)f (op)l(er)l(ators)g(on)f(Hilb)l(ert)g(sp)l(ac)l(e)p FO(,)h(Ak)-5 b(ad\351mia)586 4848 y(Kiad\363,)26 b(Budap)r(est,)i(North-Holland)f (Publishing)h(Compan)n(y)-7 b(,)26 b(Amsterdam-London,)g(1970.)109 5006 y([F)-7 b(re90])230 b(W.)25 b(R.)g(F)-7 b(rensley)g(,)24 b FA(Boundary)k(c)l(onditions)g(for)g(op)l(en)f(quantum)f(systems)g (driven)i(far)g(fr)l(om)f(e)l(qui-)586 5105 y(librium)p FO(,)h(Reviews)f(of)h(Mo)r(dern)f(Ph)n(ysics)g Fy(62)p FO(,)h(745,)e(1990.)109 5263 y([J01])304 b(A.)22 b(J\374ngel,)g FA(Quasi-hydr)l(o)l(dynamic)k(semic)l(onductor)f(e)l(quations)p FO(,)e(Progress)c(in)j(nonlinear)e(di\033er-)586 5363 y(en)n(tial)25 b(equations)f(and)g(their)h(applications,)g (Birkh\344user)e(V)-7 b(erlag,)24 b(Basel-Boston-Berlin)e(2001.)109 5521 y([KNR01a])117 b(H.-Ch.)24 b(Kaiser,)f(H.)h(Neidhardt,)g(J.)g(Reh) n(b)r(erg,)f FA(Macr)l(osc)l(opic)29 b(curr)l(ent)c(induc)l(e)l(d)h(b)l (oundary)h(c)l(on-)586 5620 y(ditions)j(for)g(Schr)l(\366)l(dinger-typ) l(e)g(op)l(er)l(ators)p FO(,)e(ersc)n(hein)n(t)d(in)i(In)n(tgr.)f(Equ.) h(Op)r(er.)g(Theory)-7 b(,)26 b(WIAS-)586 5720 y(Preprin)n(t)20 b Fy(650)p FO(,)h(W)-7 b(eierstrass-Institute)20 b(for)g(Applied)h (Analysis)e(and)i(Sto)r(c)n(hastics,)g(Berlin)f(2001,)586 5820 y(h)n(ttp://www.wias-b)r(erlin.de/publications/preprin)n(ts/650.h) n(tml)d(.)p eop %%Page: 27 29 27 28 bop 109 254 a FH(Dissipativ)n(e)27 b(Sc)n(hr\366)r(dinger-t)n(yp) r(e)e(op)r(erator)2084 b FO(27)109 503 y([KNR01b])113 b(H.-Ch.)20 b(Kaiser,)f(H.)h(Neidhardt,)h(J.)e(Reh)n(b)r(erg,)i FA(On)g(one)i(dimensional)h(dissip)l(ative)g(Schr)l(\366)l(dinger-)586 602 y(typ)l(e)f(op)l(er)l(ators)h(their)g(dilation)g(and)g (eigenfunction)f(exp)l(ansions)p FO(,)f(submitted,)h(WIAS-Preprin)n(t) 586 702 y Fy(664)p FO(,)k(W)-7 b(eierstrass-Institute)26 b(for)h(Applied)h(Analysis)f(and)h(Sto)r(c)n(hastics,)f(Berlin)g(2001,) 586 801 y(h)n(ttp://www.wias-b)r(erlin.de/publications/preprin)n (ts/664.h)n(tml)17 b(.)109 968 y([KNR01c])122 b(H.-Ch.)49 b(Kaiser,)k(H.)c(Neidhardt,)54 b(J.)49 b(Reh)n(b)r(erg,)k FA(Density)c(and)h(curr)l(ent)e(of)i(a)g(dissip)l(ative)586 1067 y(Schr)l(\366)l(dinger)35 b(op)l(er)l(ator)p FO(,)f(submitted,)f (WIAS-Preprin)n(t)f Fy(728)p FO(,)g(W)-7 b(eierstrass-Institute)31 b(for)g(Ap-)586 1167 y(plied)d(Analysis)f(and)g(Sto)r(c)n(hastics,)g (Berlin)g(2002,)586 1266 y(h)n(ttp://www.wias-b)r (erlin.de/publications/preprin)n(ts/728.h)n(tml)17 b(.)109 1432 y([K80])282 b(T.)32 b(Kato,)f FA(Perturb)l(ation)j(The)l(ory)h (for)f(Line)l(ar)g(Op)l(er)l(ators)p FO(,)f(Springer)d(V)-7 b(erlag,)32 b(Berlin-Heidel-)586 1532 y(b)r(erg-New)27 b(Y)-7 b(ork,)27 b(second)f(edition,)i(1980.)109 1698 y([LL71])243 b(L.)28 b(A.)g(Landau,)e(E.)j(M.)e(Lifsc)n(hitz,)h FA(Quantenme)l(chanik)p FO(,)g(Ak)-5 b(ademie-V)e(erlag,)26 b(Berlin,)h(1971.)109 1864 y([M86])271 b(P)-7 b(.)38 b(A.)f(Mark)n(o)n(wic)n(h,)g FA(The)j(stationary)f(semic)l(onductor)g (devic)l(e)h(e)l(quations)p FO(,)g(Springer-V)-7 b(erlag,)586 1964 y(Wien-New)28 b(Y)-7 b(ork)27 b(1986.)109 2130 y([MRS90])164 b(P)-7 b(.)52 b(A.)f(Mark)n(o)n(wic)n(h,)j(Ch.)d(A.)h(Ringhofer,)k(Ch.) 51 b(Sc)n(hmeiser,)56 b FA(Semic)l(onductor)51 b(Equations)p FO(,)586 2229 y(Springer-V)-7 b(erlag,)25 b(Wien-New)j(Y)-7 b(ork)27 b(1990.)109 2396 y([P77])291 b(B.)27 b(S.)g(P)n(a)n(vlo)n(v,)f FA(Self-adjoint)k(dilation)h(of)f(the)g(dissip)l(ative)h(Schr)l(\366)l (dinger)f(op)l(er)l(ator)h(and)e(its)g(r)l(es-)586 2495 y(olution)h(in)g(terms)f(of)i(eigenfunctions)p FO(,)d(Mat.)f(Sb)r (ornik)h Fy(102\(144\))p FO(,)e(511-536,)e(1977.)109 2661 y([P76])291 b(B.)29 b(S.)h(P)n(a)n(vlo)n(v,)e FA(Dilation)33 b(the)l(ory)f(and)g(sp)l(e)l(ctr)l(al)g(analysis)g(of)h (non-selfadjoint)g(di\033er)l(ential)f(op-)586 2761 y(er)l(ators)p FO(,)d(T)-7 b(ransl.,)27 b(I)r(I.)h(Ser.,)g(Am.)h(Math.)f(So)r(c.)g Fy(115)p FO(,)f(103-142,)e(1981;)i(translation)g(from)g(Pro)r(c.)586 2860 y(7th.)h(Win)n(ter)f(Sc)n(ho)r(ol,)g(Drogob)n(yc)n(h)e(1974,)h (3-69,)g(1976.)109 3026 y([P84])291 b(B.)25 b(S.)g(P)n(a)n(vlo)n(v,)f FA(Sp)l(e)l(ctr)l(al)k(the)l(ory)g(of)g(non-selfadjoint)h(op)l(er)l (ators)p FO(,)e(in)e(Pro)r(c.)g(In)n(t.)g(Congr.)f(Math.,)586 3126 y(W)-7 b(arsza)n(w)n(a)25 b(1983,)h(V)-7 b(ol.2,)27 b(1011-1025,)c(1984.)109 3292 y([P96])291 b(B.)25 b(S.)g(P)n(a)n(vlo)n (v,)f FA(Sp)l(e)l(ctr)l(al)j(analysis)i(of)f(a)f(dissip)l(ative)j (singular)d(Schr)l(\366)l(dinger)i(op)l(er)l(ator)f(in)f(terms)586 3392 y(of)32 b(a)g(functional)g(mo)l(del)p FO(,)g(P)n(artial)c (di\033eren)n(tial)i(equations)e Fy(8)p FO(,)i(95-163,)e(in)h(P)n (artial)g(Di\033eren)n(tial)586 3491 y(Equations)e(VI)r(I)r(I.)i (Encycl.)f(Math.)f(Sci.)h Fy(65)p FO(,)f(87-153,)e(1996.)109 3657 y([P99])291 b(B.)19 b(S.)h(P)n(a)n(vlo)n(v,)g FA(Irr)l (eversibility,)26 b(L)l(ax-Phil)t(lips)e(appr)l(o)l(ach)h(to)d(r)l (esonanc)l(e)h(sc)l(attering)f(and)h(sp)l(e)l(ctr)l(al)586 3757 y(analysis)33 b(of)g(non-selfadjoint)h(op)l(er)l(ators)f(in)f (Hilb)l(ert)g(sp)l(ac)l(e)p FO(,)g(In)n(t.)e(J.)g(Theor.)f(Ph)n(ys.)h Fy(38)p FO(,)h(No.1,)586 3857 y(21-45,)25 b(1999.)109 4023 y([S84])301 b(S.)35 b(Selb)r(erherr,)h FA(A)n(nalysis)h(and)f (Simulation)h(of)g(Semic)l(onductor)g(Devic)l(es)p FO(,)g(Springer)d(V) -7 b(erlag,)586 4122 y(Wien,)28 b(1984.)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0204120957420--