%!PS-Adobe-2.0 %%Creator: dvips(k) 5.78 Copyright 1998 Radical Eye Software (www.radicaleye.com) %%Title: dsp_preprint.dvi %%Pages: 31 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSCommandLine: dvips -f dsp_preprint %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2002.02.27:1339 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}B /TR{translate}N /isls false N /vsize 11 72 mul N /hsize 8.5 72 mul N /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[matrix currentmatrix{dup dup round sub abs 0.00001 lt{round}if} forall round exch round exch]setmatrix}N /@landscape{/isls true N}B /@manualfeed{statusdict /manualfeed true put}B /@copies{/#copies X}B /FMat[1 0 0 -1 0 0]N /FBB[0 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y(carrier)f(densit)m(y)i(has)f(to)f(b)s(e)h (rede\034ned.)296 5491 y(The)50 b(situation)d(can)i(b)s(e)g(impro)m(v)m (ed)f(if)g(w)m(e)h(c)m(ho)s(ose)h(dissipativ)m(e)e(b)s(oundary)h (conditions,)178 5612 y(cf.)61 b([7].)g(This)38 b(enables)h(us)g(to)f (use)h(the)g(dilation)c(theory)k(for)f(dissipativ)m(e)g(op)s(erators)g (as)g(the)p eop %%Page: 4 6 4 5 bop 178 46 a FA(4)945 b Ft(M.)36 b(Bar)m(o,)f(H.-Ch.)47 b(Kaiser,)36 b(H.)g(Neidhardt,)f(J.)i(Rehber)m(g)178 295 y FA(tec)m(hnical)43 b(to)s(ol)f(to)g(o)m(v)m(ercome)j (di\036culties)d(arising)f(from)h(the)i(non-selfadjoin)m(tness.)75 b(F)-8 b(rom)178 415 y(the)41 b(ph)m(ysical)f(p)s(oin)m(t)g(of)f(view)i (the)g(minimal)36 b(self-adjoin)m(t)i(dilations)g Fp(K)2916 379 y Fk(\006)2975 415 y Fq(\()p Fp(V)22 b Fq(\))40 b FA(pla)m(y)g(the)h(role)178 535 y(of)36 b(the)h(Hamiltonians)c(of)j(a)g (larger,)g(closed)h(systems)h(whic)m(h)f(con)m(tains)f(the)h(original)c (system)178 656 y(describ)s(ed)j(b)m(y)g Fp(H)839 620 y Fk(\006)897 656 y Fq(\()p 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(tro)s(duce)g(their)g(dilations)e(and)i(generalized)g(eigenfunc-)178 1666 y(tion)d(expansions,)h(cf.)52 b([8],)36 b(and)f(recall)f(the)i (de\034nition)e(of)h(the)g(carrier)g(densit)m(y)h(giv)m(en)f(in)g([9)o (].)178 1787 y(F)-8 b(ollo)m(wing)39 b([5])i(w)m(e)h(in)m(tro)s(duce)g (the)f(\(nonlinear\))f(carrier)h(densit)m(y)h(op)s(erator)f(assigning)f (to)h(a)178 1907 y(Sc)m(hr\366)s(dinger)31 b(p)s(oten)m(tial)e(the)j (corresp)s(onding)e(carrier)g(densit)m(y)-8 b(.)44 b(In)31 b(Section)g(3)g(w)m(e)h(in)m(v)m(estigate)178 2027 y(the)g(con)m(v)m (ergence)h(prop)s(erties)e(of)f(Sc)m(hr\366)s(dinger-t)m(yp)s(e)h(op)s (erators,)g(their)f(dilations)f(and)h(eigen-)178 2148 y(function)37 b(expansions)i(as)f(w)m(ell)f(as)h(of)f(the)h(carrier)f (densit)m(y)h(op)s(erators)g(with)f(resp)s(ect)i(to)e(the)178 2268 y(p)s(oten)m(tials)g(in)g Fp(L)819 2232 y Fk(1)894 2268 y Fq(\([)p Fp(a;)17 b(b)p Fq(]\))p FA(.)60 b(In)38 b(Section)g(4)g(w)m(e)h(rigorously)d(de\034ne)j(dissipativ)m(e)e(Sc)m 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b(functions)e(on)g Fq([)p Fp(a;)17 b(b)p Fq(])34 b FA(will)c(b)s(e)j(denoted)g(b)m(y)g Fp(L)2242 3817 y Fk(1)2350 3853 y FA(in)f(the)h(sequel.)296 4021 y(In)j(order)e(to)h(a)m(v)m(oid)f(confusion)h(w)m(e)h(denote)f(the)g (space)h(of)e(complex)g(v)-5 b(alued,)35 b(square)h(in)m(te-)178 4141 y(grable)k(functions)g(on)h(the)f(in)m(terv)-5 b(al)40 b Fq([)p Fp(a;)17 b(b)p Fq(])41 b FA(b)m(y)g Fi(H)p FA(.)67 b(F)-8 b(urthermore,)42 b(w)m(e)f(denote)g(b)m(y)h Fp(W)3452 4105 y Fl(1)p Fj(;)p Fl(2)3586 4141 y FA(the)178 4262 y(usual)34 b(complex)f(Sob)s(olev)g(space)h Fp(W)1549 4225 y Fl(1)p Fj(;)p Fl(2)1643 4262 y Fq([)p Fp(a;)17 b(b)p Fq(])34 b FA(and)g(b)m(y)g Fp(C)7 b Fq([)p Fp(a;)17 b(b)p Fq(])34 b FA(the)g(space)h(of)e(complex)g(v)-5 b(alued,)178 4382 y(con)m(tin)m(uous)34 b(functions)e(on)g Fq([)p Fp(a;)17 b(b)p Fq(])p FA(.)296 4550 y(If)36 b Fn(H)h FA(is)e(an)m(y)i(Hilb)s(ert)d(space)i(then)h Fi(L)1701 4565 y Fl(1)1741 4550 y Fq(\()p Fn(H)q Fq(\))e FA(denotes)i(the)f 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Fr(Assumptions)k(2.1)p eop %%Page: 5 7 5 6 bop 178 46 a Ft(Dissip)-8 b(a)g(tive)36 b(Schr)m (\366dinger-Poisson)f(systems)1432 b FA(5)178 295 y(\()p Fp(A)289 310 y Fl(1)329 295 y FA(\))48 b(The)d(functions)f Fp(m)1144 310 y Fk(\006)1204 295 y FA(,)j(whic)m(h)e(are)f(called)g (the)g('e\033ectiv)m(e)i(masses',)i(are)d(p)s(ositiv)m(e)f(and)415 415 y(ob)s(ey)33 b Fp(m)733 430 y Fk(\006)792 415 y Fp(;)913 376 y Fl(1)p 873 392 115 4 v 873 449 a Fj(m)935 458 y Fh(\006)1025 415 y Fn(2)c Fp(L)1186 379 y Fk(1)1261 415 y FA(.)43 b(W)-8 b(e)33 b(set)g Fn(k)p Fp(m)1786 430 y Fk(\006)1845 415 y Fn(k)28 b Fq(:=)g Fn(k)p Fp(m)2189 430 y Fk(\006)2248 415 y Fn(k)2298 430 y Fj(L)2346 411 y Fh(1)2415 415 y FA(.)178 640 y(\()p Fp(A)289 655 y Fl(2)329 640 y FA(\))48 b(The)30 b(b)s(oundary)f(co)s(e\036cien)m(ts)h Fp(\024)1593 603 y Fk(\006)1593 664 y Fj(a)1652 640 y Fp(;)17 b(\024)1752 598 y Fk(\006)1752 667 y Fj(b)1840 640 y FA(are)29 b(from)f(the)h(upp)s(er)h(half)e(plane)g Fo(C)3177 655 y Fl(+)3270 640 y Fq(=)g Fn(f)p Fp(z)k Fn(2)c Fo(C)54 b Fq(:)415 760 y Fn(=)p FA(m)o Fq(\()p Fp(z)t Fq(\))28 b Fp(>)g Fq(0)p Fn(g)p FA(.)296 1012 y(The)36 b(distinction)e(b)s(et)m(w)m(een)j(')p Fq(+)p FA(')e(and)g(')p Fn(\000)p FA(')h(is)e(not)h(relev)-5 b(an)m(t)35 b(for)f(the)i(in)m(v)m(estigations)e(in)g(the)178 1132 y(sections)f(2)g(and)f(3,)h(therefore)g(w)m(e)g(will)e(abbreviate) h(the)h(notations)f(to)g Fp(m;)17 b(\024)3039 1147 y Fj(a)3113 1132 y FA(and)33 b Fp(\024)3359 1147 y Fj(b)3393 1132 y FA(.)178 1470 y Fs(2.2)131 b(Sc)l(hr\366)t(dinger-t)l(yp)t(e)45 b(op)t(erators)178 1702 y FA(F)-8 b(ollo)m(wing)27 b(the)j(prop)s(osal) f(of)g([5])h(w)m(e)h(consider)f(the)g(non-selfadjoin)m(t)e(Sc)m(hr\366) s(dinger-t)m(yp)s(e)i(op)s(er-)178 1823 y(ator)i Fp(H)8 b Fq(\()p Fp(V)21 b Fq(\))33 b FA(on)f(the)h(Hilb)s(ert)e(space)j Fi(H)e FA(de\034ned)i(b)m(y)815 2190 y(dom)o Fq(\()p Fp(H)8 b Fq(\()p Fp(V)21 b Fq(\)\))27 b(=)1449 1956 y Fm(8)1449 2046 y(>)1449 2076 y(<)1449 2255 y(>)1449 2285 y(:)1538 2190 y Fp(g)k Fn(2)d Fp(W)1816 2149 y Fl(1)p Fj(;)p Fl(2)1937 2190 y Fq(:)2104 2008 y Fl(1)p 2044 2024 158 4 v 2044 2081 a Fj(m)p Fl(\()p Fj(x)p Fl(\))2211 2047 y Fp(g)2262 2011 y Fk(0)2284 2047 y Fq(\()p Fp(x)p Fq(\))g Fn(2)g Fp(W)2643 2011 y Fl(1)p Fj(;)p Fl(2)2737 2047 y Fp(;)2121 2143 y Fl(1)p 2044 2159 191 4 v 2044 2217 a(2)p Fj(m)p Fl(\()p Fj(a)p Fl(\))2244 2182 y Fp(g)2295 2146 y Fk(0)2317 2182 y Fq(\()p Fp(a)p Fq(\))g(=)f Fn(\000)p Fp(\024)2708 2197 y Fj(a)2751 2182 y Fp(g)t Fq(\()p Fp(a)p Fq(\))p Fp(;)2117 2278 y Fl(1)p 2044 2295 183 4 v 2044 2352 a(2)p Fj(m)p Fl(\()p Fj(b)p Fl(\))2236 2317 y Fp(g)2287 2281 y Fk(0)2310 2317 y Fq(\()p Fp(b)p Fq(\))h(=)f Fp(\024)2614 2332 y Fj(b)2649 2317 y Fp(g)t Fq(\()p Fp(b)p Fq(\))2997 1956 y Fm(9)2997 2046 y(>)2997 2076 y(=)2997 2255 y(>)2997 2285 y(;)3521 2190 y FA(\(2.1\))178 2552 y(and)622 2674 y Fq(\()p Fp(H)8 b Fq(\()p Fp(V)21 b Fq(\))p Fp(g)t Fq(\)\()p Fp(x)p Fq(\))27 b(=)h(\()p Fp(l)r Fq(\()p Fp(g)t Fq(\)\)\()p Fp(x)p Fq(\))p Fp(;)113 b(g)31 b Fn(2)e FA(dom)o Fq(\()p Fp(H)8 b Fq(\()p Fp(V)21 b Fq(\)\))p Fp(;)114 b(g)31 b Fn(2)d FA(dom)o Fq(\()p Fp(H)8 b Fq(\()p Fp(V)21 b Fq(\)\))p Fp(;)243 b FA(\(2.2\))178 2849 y(where)1011 2995 y Fq(\()p Fp(l)r Fq(\()p Fp(g)t Fq(\)\)\()p Fp(x)p Fq(\))27 b(:=)h Fn(\000)1621 2927 y Fq(1)p 1621 2972 49 4 v 1621 3063 a(2)1718 2927 y Fp(d)p 1690 2972 107 4 v 1690 3063 a(dx)1900 2927 y Fq(1)p 1816 2972 217 4 v 1816 3063 a Fp(m)p Fq(\()p Fp(x)p Fq(\))2080 2927 y Fp(d)p 2053 2972 107 4 v 2053 3063 a(dx)2169 2995 y(g)t Fq(\()p Fp(x)p Fq(\))21 b(+)h Fp(V)g Fq(\()p Fp(x)p Fq(\))p Fp(g)t Fq(\()p Fp(x)p Fq(\))p Fp(;)632 b FA(\(2.3\))178 3226 y(cf.)41 b([7,)23 b(8].)40 b(F)-8 b(urthermore,)24 b(it)e(is)h(alw)m(a)m(ys)h(assumed)f(that)g(the)h(o)s(ccuring)e(Sc)m (hr\366)s(dinger)h(p)s(oten)m(tials)178 3347 y(are)i(from)e Fp(L)621 3310 y Fk(1)696 3347 y FA(.)41 b(The)25 b(op)s(erator)f Fp(H)32 b FA(is)24 b(maximal)d(dissipativ)m(e)j(and)h(completely)e (non-selfadjoin)m(t,)178 3467 y(see)k([7].)41 b(The)26 b(sp)s(ectrum)f(of)g Fp(H)8 b Fq(\()p Fp(V)21 b Fq(\))k FA(consists)h(of)f(isolated)f(eigen)m(v)-5 b(alues)25 b(in)g(the)g(lo)m(w)m(er)h(half-plane)178 3587 y(with)i(the)h(only)f (accum)m(ulation)f(p)s(oin)m(t)h(at)g(in\034nit)m(y)-8 b(.)41 b(Since)29 b(the)g(op)s(erator)e Fp(H)8 b Fq(\()p Fp(V)21 b Fq(\))29 b FA(is)f(completely)178 3708 y(non-selfadjoin)m(t)j (there)i(do)g(not)f(exist)h(real)f(eigen)m(v)-5 b(alues.)178 3875 y(Moreo)m(v)m(er,)26 b Fp(H)8 b Fq(\()p Fp(V)21 b Fq(\))h FA(can)g(b)s(e)g(en)m(tirely)f(c)m(haracterized)i(b)m(y)g (its)e(c)m(haracteristic)h(function)f Fq(\002)3390 3891 y Fj(H)5 b Fl(\()p Fj(V)16 b Fl(\))3569 3875 y Fq(\()p Fp(z)t Fq(\))p FA(,)178 3996 y(with)28 b Fp(z)k Fn(2)c Fp(\045)p Fq(\()p Fp(H)8 b Fq(\()p Fp(V)22 b Fq(\)\))12 b Fn(\\)g Fp(\045)p Fq(\()p Fp(H)c Fq(\()p Fp(V)22 b Fq(\))1359 3960 y Fk(\003)1398 3996 y Fq(\))p FA(,)29 b(cf.)42 b([1].)g(In)28 b(our)g(case)g(the)h(de\034nition)d(of)i(the)g (c)m(haracteristic)178 4116 y(function)48 b(relies)g(on)g(the)h(b)s (oundary)g(op)s(erators)f Fp(T)14 b Fq(\()p Fp(V)22 b Fq(\)\()p Fp(z)t Fq(\))55 b(:)g Fi(H)f Fn(\000)-16 b(!)55 b Fo(C)2975 4080 y Fl(2)3020 4116 y FA(,)e Fp(z)59 b Fn(2)c Fp(\045)p Fq(\()p Fp(H)8 b Fq(\()p Fp(V)22 b Fq(\)\))p FA(,)178 4236 y(and)39 b Fp(T)431 4251 y Fk(\003)471 4236 y Fq(\()p Fp(V)21 b Fq(\)\()p Fp(z)t Fq(\))39 b(:)g Fi(H)f Fn(\000)-16 b(!)37 b Fo(C)1229 4200 y Fl(2)1275 4236 y FA(,)j Fp(z)j Fn(2)c Fp(\045)p Fq(\()p Fp(H)8 b Fq(\()p Fp(V)21 b Fq(\))1866 4200 y Fk(\003)1906 4236 y Fq(\))p FA(.)62 b(Let)39 b(us)g(in)m(tro)s(duce)g(the)g(unclosed)g (op)s(erator)178 4357 y Fp(\013)29 b Fq(:)e Fi(H)h Fn(\000)-16 b(!)27 b Fo(C)677 4321 y Fl(2)722 4357 y FA(,)933 4630 y Fp(\013)q(f)38 b Fq(=)1186 4490 y Fm(\022)1348 4569 y Fp(\013)1410 4584 y Fj(b)1444 4569 y Fp(f)11 b Fq(\()p Fp(b)p Fq(\))1301 4690 y Fn(\000)p Fp(\013)1440 4705 y Fj(a)1482 4690 y Fp(f)g Fq(\()p Fp(a)p Fq(\))1709 4490 y Fm(\023)1799 4630 y Fp(;)212 b(f)38 b Fn(2)28 b FA(dom)o 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b Fq(\))j Fn(\000)f Fp(z)t Fq(\))2433 5571 y Fk(\000)p Fl(1)2528 5612 y Fp(f)945 b FA(\(2.6\))p eop %%Page: 6 8 6 7 bop 178 46 a FA(6)945 b Ft(M.)36 b(Bar)m(o,)f(H.-Ch.)47 b(Kaiser,)36 b(H.)g(Neidhardt,)f(J.)i(Rehber)m(g)178 295 y FA(and)1270 415 y Fp(T)1327 430 y Fk(\003)1366 415 y Fq(\()p Fp(V)22 b Fq(\)\()p Fp(z)t Fq(\))p Fp(f)39 b Fq(:=)27 b Fp(\013)q Fq(\()p Fp(H)8 b Fq(\()p Fp(V)21 b Fq(\))2207 374 y Fk(\003)2268 415 y Fn(\000)i Fp(z)t Fq(\))2455 374 y Fk(\000)p Fl(1)2550 415 y Fp(f)5 b(;)891 b FA(\(2.7\))178 588 y Fp(f)54 b Fn(2)44 b Fi(H)p FA(.)71 b(The)43 b(c)m(haracteristic)e(function)g Fq(\002)1845 603 y Fj(H)5 b Fl(\()p Fj(V)17 b Fl(\))2024 588 y Fq(\()p Fn(\001)p Fq(\))41 b FA(of)h(the)g(maximal)c(dissipativ)m(e)j(op)s (erator)178 708 y Fp(H)8 b Fq(\()p Fp(V)21 b Fq(\))33 b FA(is)f(a)g(t)m(w)m(o-b)m(y-t)m(w)m(o)i(matrix-v)-5 b(alued)30 b(function)i(whic)m(h)h(satis\034es)g(the)g(relation)660 925 y Fq(\002)736 940 y Fj(H)5 b Fl(\()p Fj(V)17 b Fl(\))915 925 y Fq(\()p Fp(z)t 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y(The)34 b(c)m(haracteristic)e(function)g(is)g(giv)m(en)h(b)m(y)1368 1910 y Fq(\002)1444 1925 y Fj(H)1511 1910 y Fq(\()p Fp(z)t Fq(\))28 b(=)g Fp(I)1811 1927 y Fg(C)1853 1908 y Ff(2)1919 1910 y Fn(\000)23 b Fp(i\013)q(T)14 b Fq(\()p Fp(V)21 b Fq(\)\()p 2378 1856 50 4 v Fp(z)5 b Fq(\))2466 1869 y Fk(\003)2505 1910 y Fp(;)940 b FA(\(2.10\))178 2127 y(cf.)44 b([8].)296 2294 y(The)28 b(op)s(erator)f Fp(H)8 b Fq(\()p Fp(V)21 b Fq(\))26 b FA(admits)g(a)h(description)f(in)h (terms)f(of)h(quadratic)g(forms.)40 b(T)-8 b(o)27 b(this)g(end)178 2415 y(w)m(e)34 b(in)m(tro)s(duce)e(the)h(sesquilinear)f(form)g Fi(h)1730 2430 y Fl(0)1769 2415 y Fq([)p Fn(\001)p Fp(;)17 b Fn(\001)p Fq(])p FA(,)877 2699 y Fi(h)929 2714 y Fl(0)968 2699 y Fq([)p Fp(g)t(;)g(f)11 b Fq(])27 b(:=)1334 2564 y Fm(Z)1433 2590 y Fj(b)1389 2789 y(a)1484 2699 y Fp(dx)1635 2559 y Fm(\032)1828 2632 y Fq(1)p 1719 2676 266 4 v 1719 2768 a(2)p Fp(m)p Fq(\()p Fp(x)p Fq(\))1995 2699 y Fp(g)2046 2658 y Fk(0)2068 2699 y Fq(\()p Fp(x)p Fq(\))p 2199 2613 214 4 v Fp(f)2258 2671 y Fk(0)2282 2699 y Fq(\()p Fp(x)p Fq(\))22 b(+)g Fp(g)t Fq(\()p Fp(x)p Fq(\))p 2715 2613 191 4 v Fp(f)11 b Fq(\()p Fp(x)p Fq(\))2905 2559 y Fm(\033)2996 2699 y Fp(;)449 b FA(\(2.11\))178 2977 y Fp(f)5 b(;)17 b(g)35 b Fn(2)d FA(dom)p Fq(\()p Fi(h)730 2992 y Fl(0)769 2977 y Fq(\))g(=)f Fp(W)1052 2941 y Fl(1)p Fj(;)p Fl(2)1146 2977 y FA(,)36 b(see)g([7].)50 b(The)36 b(form)e Fi(h)2037 2992 y Fl(0)2111 2977 y FA(is)g(symmetric)g(and)h(non-negativ)m(e.)51 b(Since)178 3098 y Fi(h)230 3113 y Fl(0)302 3098 y FA(is)32 b(closed)h(there)g(is)f(a)h(self-adjoin)m(t)d(op)s(erator)i Fp(H)2092 3113 y Fl(0)2164 3098 y FA(suc)m(h)i(that)e(the)h(represen)m (tation)864 3314 y Fi(h)916 3329 y Fl(0)955 3314 y Fq([)p Fp(g)t(;)17 b(f)11 b Fq(])27 b(=)g(\()p Fp(H)1412 3329 y Fl(0)1452 3314 y Fp(g)t(;)17 b(f)11 b Fq(\))p Fp(;)112 b(g)31 b Fn(2)d FA(dom)p Fq(\()p Fp(H)2259 3329 y Fl(0)2298 3314 y Fq(\))p Fp(;)114 b(f)38 b Fn(2)29 b FA(dom)o Fq(\()p Fi(h)2932 3329 y Fl(0)2971 3314 y Fq(\))p Fp(;)436 b FA(\(2.12\))178 3530 y(holds.)43 b(This)33 b(op)s(erator)f Fp(H)1168 3545 y Fl(0)1240 3530 y FA(can)h(b)s(e)f(explicitly)f (describ)s(ed)i(as)g(follo)m(ws:)770 3828 y(dom)o Fq(\()p Fp(H)1073 3843 y Fl(0)1112 3828 y Fq(\))28 b(:=)1308 3658 y Fm(\()1389 3828 y Fp(g)j Fn(2)d Fp(W)1667 3787 y Fl(1)p Fj(;)p Fl(2)1788 3828 y Fq(:)1973 3713 y Fl(1)p 1895 3729 193 4 v 1895 3787 a(2)p Fj(m)p Fl(\()p Fj(x)p Fl(\))2097 3752 y Fp(g)2148 3716 y Fk(0)2170 3752 y Fq(\()p Fp(x)p Fq(\))g Fn(2)g Fp(W)2529 3716 y Fl(1)p Fj(;)p Fl(2)1968 3848 y(1)p 1895 3865 183 4 v 1895 3922 a(2)p Fj(m)p Fl(\()p Fj(b)p Fl(\))2087 3888 y Fp(g)2138 3851 y Fk(0)2161 3888 y Fq(\()p Fp(b)p Fq(\))g(=)2479 3848 y Fl(1)p 2419 3865 155 4 v 2419 3922 a Fj(m)p Fl(\()p Fj(a)p Fl(\))2584 3888 y Fp(g)2635 3851 y Fk(0)2658 3888 y Fq(\()p Fp(a)p Fq(\))g(=)f(0)3006 3658 y Fm(\))3103 3828 y Fp(;)342 b FA(\(2.13\))178 4126 y(and)786 4271 y Fq(\()p Fp(H)905 4286 y Fl(0)944 4271 y Fp(g)t Fq(\)\()p Fp(x)p Fq(\))27 b(=)h Fn(\000)1410 4203 y Fp(d)p 1382 4248 107 4 v 1382 4339 a(dx)1617 4203 y Fq(1)p 1508 4248 266 4 v 1508 4339 a(2)p Fp(m)p Fq(\()p Fp(x)p Fq(\))1822 4203 y Fp(d)p 1794 4248 107 4 v 1794 4339 a(dx)1910 4271 y(g)t Fq(\()p Fp(x)p Fq(\))22 b(+)g Fp(g)t Fq(\()p Fp(x)p Fq(\))p Fp(;)113 b(g)31 b Fn(2)e FA(dom)o Fq(\()p Fp(H)3010 4286 y Fl(0)3049 4271 y Fq(\))p Fp(:)358 b FA(\(2.14\))178 4493 y(Ob)m(viously)-8 b(,)31 b(the)g(op)s(erator)f Fp(H)1292 4508 y Fl(0)1362 4493 y FA(is)g(non-negativ)m(e.)43 b(In)30 b(order)h(to)f(obtain)g(further)h(prop)s(erties)f(of)178 4614 y(the)37 b(op)s(erators)f Fp(H)8 b Fq(\()p Fp(V)21 b Fq(\))36 b FA(w)m(e)h(in)m(tro)s(duce)f(certain)g(quadratic)g(forms)f (in)g(terms)h(of)g(whic)m(h)h Fp(H)8 b Fq(\()p Fp(V)21 b Fq(\))178 4734 y FA(can)43 b(b)s(e)f(understo)s(o)s(d)g(as)h(a)f (\(form\))e(p)s(erturbation)h(of)h Fp(H)2347 4749 y Fl(0)2386 4734 y FA(.)72 b(W)-8 b(e)43 b(start)f(with)g(the)g(b)s(oundary)178 4854 y(form)32 b Fi(t)442 4869 y Fj(@)t Fl(\012)538 4854 y Fq([)p Fn(\001)p Fp(;)17 b Fn(\001)p Fq(])32 b FA(de\034ned)i(b)m(y) 1142 5071 y Fi(t)1175 5086 y Fj(@)t Fl(\012)1271 5071 y Fq([)p Fp(g)t(;)17 b(f)11 b Fq(])27 b(:=)h Fn(\000)p Fp(\024)1770 5086 y Fj(a)1812 5071 y Fp(g)t Fq(\()p Fp(a)p Fq(\))p 1990 4984 186 4 v Fp(f)11 b Fq(\()p Fp(a)p Fq(\))21 b Fn(\000)i Fp(\024)2353 5086 y Fj(b)2387 5071 y Fp(g)t Fq(\()p Fp(b)p Fq(\))p 2555 4984 177 4 v Fp(f)11 b Fq(\()p Fp(b)p Fq(\))p Fp(;)714 b FA(\(2.15\))178 5287 y Fp(f)5 b(;)17 b(g)31 b Fn(2)d FA(dom)p Fq(\()p Fi(t)703 5302 y Fj(@)t Fl(\012)799 5287 y Fq(\))g(=)f Fp(W)1074 5251 y Fl(1)p Fj(;)p Fl(2)1168 5287 y FA(.)44 b(Next)33 b(w)m(e)g(de\034ne)h (the)f(p)s(oten)m(tial)e(form)g Fi(t)2745 5302 y Fj(V)2806 5287 y Fq([)p Fn(\001)p Fp(;)17 b Fn(\001)p Fq(])p FA(,)1274 5572 y Fi(t)1307 5587 y Fj(V)1368 5572 y Fq([)p Fp(g)t(;)g(f)11 b Fq(])27 b(:=)1733 5436 y Fm(Z)1833 5462 y Fj(b)1789 5662 y(a)1884 5572 y Fp(dx)h(V)21 b Fq(\()p Fp(x)p Fq(\))p Fp(g)t Fq(\()p Fp(x)p Fq(\))p 2409 5485 191 4 v Fp(f)11 b Fq(\()p Fp(x)p Fq(\))p Fp(;)846 b FA(\(2.16\))p eop %%Page: 7 9 7 8 bop 178 46 a Ft(Dissip)-8 b(a)g(tive)36 b(Schr)m (\366dinger-Poisson)f(systems)1432 b FA(7)178 295 y Fp(f)5 b(;)17 b(g)31 b Fn(2)d FA(dom)p Fq(\()p Fi(t)703 310 y Fj(V)764 295 y Fq(\))f(=)h Fp(W)1039 259 y Fl(1)p Fj(;)p Fl(2)1165 295 y FA(and)33 b(the)g(form)e(sum)1583 504 y Fi(t)1616 519 y Fj(@)t Fl(\012)p Fj(;V)1816 504 y Fq(:=)d Fi(t)1980 519 y Fj(@)t Fl(\012)2098 504 y Fq(+)22 b Fi(t)2229 519 y Fj(V)2290 504 y Fp(:)1155 b FA(\(2.17\))178 714 y(As)24 b(usual,)h(w)m(e)f(will)d(denote)j(the)g(corresp)s(onding)f (quadratic)f(forms)h(b)m(y)h(the)g(same)f(sym)m(b)s(ols,)h(but)178 835 y(only)30 b(one)g(argumen)m(t)f(o)s(ccuring.)42 b(In)30 b(the)g(follo)m(wing)d(w)m(e)k(will)c(supply)k(relativ)m(e)e(form)f (estimates)178 955 y(for)k Fi(t)360 970 y Fj(@)t Fl(\012)457 955 y FA(,)g Fi(t)549 970 y Fj(V)643 955 y FA(and)h Fi(t)866 970 y Fj(@)t Fl(\012)p Fj(;V)1071 955 y FA(with)f(resp)s(ect)i(to)e Fi(h)1798 970 y Fl(0)1837 955 y FA(.)44 b(Let)754 1226 y Fi(g)804 1241 y Fl(1)871 1226 y Fq(:=)127 b(sup)1001 1313 y Fl(0)p Fk(6)p Fl(=)p Fj( )r Fk(2)p Fj(W)1263 1294 y Ff(1)p Fe(;)p Ff(2)1544 1159 y Fn(k)p Fp( )t Fn(k)1711 1174 y Fj(L)1759 1155 y Fh(1)p 1374 1203 625 4 v 1374 1316 a Fn(k)p Fp( )t Fn(k)1541 1265 y Fl(1)p Fj(=)p Fl(2)1541 1349 y Fj(W)1618 1330 y Ff(1)p Fe(;)p Ff(2)1722 1316 y Fn(k)p Fp( )t Fn(k)1889 1265 y Fl(1)p Fj(=)p Fl(2)1889 1344 y Fd(H)2036 1226 y Fq(=)g(sup)2140 1313 y Fl(0)p Fk(6)p Fl(=)p Fj( )r Fk(2)p Fj(W)2402 1294 y Ff(1)p Fe(;)p Ff(2)2648 1155 y Fn(k)p Fp( )t Fn(k)2815 1171 y Fj(C)5 b Fl([)p Fj(a;b)p Fl(])p 2512 1203 V 2512 1316 a Fn(k)p Fp( )t Fn(k)2679 1265 y Fl(1)p Fj(=)p Fl(2)2679 1349 y Fj(W)2756 1330 y Ff(1)p Fe(;)p Ff(2)2860 1316 y Fn(k)p Fp( )t Fn(k)3027 1265 y Fl(1)p Fj(=)p Fl(2)3027 1344 y Fd(H)3472 1226 y FA(\(2.18\))178 1515 y(b)s(e)33 b(the)g (Gagliardo-Niren)m(b)s(erg)c(constan)m(t)k(and)1537 1725 y Fq(~)-67 b Fp(m)28 b Fq(:=)f(max)17 b Fn(f)o Fq(1)p Fp(;)g Fn(k)p Fp(m)p Fn(kg)f Fp(:)1091 b FA(\(2.19\))178 1934 y(Applying)22 b(the)h(Gagliardo-Niren)m(b)s(erg)c(inequalit)m(y)i (w)m(e)j(can)e(estimate)g(the)h(form)e Fi(t)3178 1949 y Fj(@)t Fl(\012)3297 1934 y FA(as)i(follo)m(ws:)183 2144 y Fn(j)p Fi(t)244 2159 y Fj(@)t Fl(\012)340 2144 y Fq([)p Fp(g)t(;)17 b(f)11 b Fq(])o Fn(j)27 b(\024)h Fq(\()p Fn(j)p Fp(\024)829 2159 y Fj(a)871 2144 y Fn(j)22 b Fq(+)g Fn(j)p Fp(\024)1103 2159 y Fj(b)1137 2144 y Fn(j)p Fq(\))f Fn(\001)h(k)p Fp(f)11 b Fn(k)1433 2160 y Fj(C)5 b Fl([)p Fj(a;b)p Fl(])1640 2144 y Fn(\001)22 b(k)p Fp(g)t Fn(k)1841 2160 y Fj(C)5 b Fl([)p Fj(a;b)p Fl(])3472 2144 y FA(\(2.20\))469 2352 y Fn(\024)q Fq(\(\()p Fn(j)p Fp(\024)707 2367 y Fj(a)748 2352 y Fn(j)22 b Fq(+)g Fn(j)p Fp(\024)980 2367 y Fj(b)1014 2352 y Fn(j)p Fq(\))p Fi(g)1130 2311 y Fl(2)1130 2376 y(1)1169 2352 y Fn(k)p Fp(f)11 b Fn(k)1328 2368 y Fj(W)1405 2349 y Ff(1)p Fe(;)p Ff(2)1509 2352 y Fn(k)p Fp(f)g Fn(k)1668 2367 y Fd(H)1721 2241 y Fm(\021)1780 2263 y Fl(1)p Fj(=)p Fl(2)1890 2352 y Fq(\(\()p Fn(j)p Fp(\024)2050 2367 y Fj(a)2092 2352 y Fn(j)22 b Fq(+)g Fn(j)p Fp(\024)2324 2367 y Fj(b)2358 2352 y Fn(j)p Fq(\))p Fi(g)2474 2311 y Fl(2)2474 2376 y(1)2513 2352 y Fn(k)p Fp(g)t Fn(k)2664 2368 y Fj(W)2741 2349 y Ff(1)p Fe(;)p Ff(2)2844 2352 y Fn(k)p Fp(g)t Fn(k)2995 2367 y Fd(H)3048 2241 y Fm(\021)3107 2263 y Fl(1)p Fj(=)p Fl(2)469 2588 y Fn(\024)547 2477 y Fm(\020)606 2588 y Fq(\()p Fn(j)p Fp(\024)728 2603 y Fj(a)770 2588 y Fn(j)g Fq(+)g Fn(j)p Fp(\024)1002 2603 y Fj(b)1036 2588 y Fn(j)p Fq(\))p Fi(g)1152 2547 y Fl(2)1152 2612 y(1)1209 2588 y Fq(~)-67 b Fp(m)1276 2547 y Fl(1)p Fj(=)p Fl(2)1386 2588 y Fi(h)1438 2603 y Fl(0)1478 2588 y Fq([)p Fp(f)11 b Fq(])1591 2547 y Fl(1)p Fj(=)p Fl(2)1701 2588 y Fn(k)p Fp(f)g Fn(k)1860 2603 y Fd(H)1913 2477 y Fm(\021)1972 2499 y Fl(1)p Fj(=)p Fl(2)2082 2477 y Fm(\020)2142 2588 y Fq(\()p Fn(j)p Fp(\024)2264 2603 y Fj(a)2305 2588 y Fn(j)22 b Fq(+)g Fn(j)p Fp(\024)2537 2603 y Fj(b)2571 2588 y Fn(j)p Fq(\))p Fi(g)2687 2547 y Fl(2)2687 2612 y(1)2745 2588 y Fq(~)-68 b Fp(m)2811 2547 y Fl(1)p Fj(=)p Fl(2)2922 2588 y Fi(h)2974 2603 y Fl(0)3013 2588 y Fq([)p Fp(g)t Fq(])3118 2547 y Fl(1)p Fj(=)p Fl(2)3228 2588 y Fn(k)p Fp(g)t Fn(k)3379 2603 y Fd(H)3431 2477 y Fm(\021)3491 2499 y Fl(1)p Fj(=)p Fl(2)469 2856 y Fn(\024)563 2715 y Fm(\022)637 2856 y Fp(\016)t Fi(h)736 2871 y Fl(0)775 2856 y Fq([)p Fp(g)t Fq(])22 b(+)1010 2788 y(\()p Fn(j)p Fp(\024)1132 2803 y Fj(a)1173 2788 y Fn(j)g Fq(+)g Fn(j)p Fp(\024)1405 2803 y Fj(b)1439 2788 y Fn(j)p Fq(\))1505 2752 y Fl(2)1544 2788 y Fi(g)1594 2752 y Fl(4)1594 2813 y(1)1652 2788 y Fq(~)-67 b Fp(m)p 1010 2833 710 4 v 1316 2924 a Fq(4)p Fp(\016)1729 2856 y Fn(k)p Fp(g)t Fn(k)1880 2814 y Fl(2)1918 2715 y Fm(\023)1992 2736 y Fl(1)p Fj(=)p Fl(2)2118 2715 y Fm(\022)2192 2856 y Fp(\016)t Fi(h)2291 2871 y Fl(0)2330 2856 y Fq([)p Fp(f)11 b Fq(])22 b(+)2573 2788 y(\()p Fn(j)p Fp(\024)2695 2803 y Fj(a)2736 2788 y Fn(j)g Fq(+)g Fn(j)p Fp(\024)2968 2803 y Fj(b)3002 2788 y Fn(j)p Fq(\))3068 2752 y Fl(2)3107 2788 y Fi(g)3157 2752 y Fl(4)3157 2813 y(1)3215 2788 y Fq(~)-67 b Fp(m)p 2573 2833 V 2880 2924 a Fq(4)p Fp(\016)3292 2856 y Fn(k)p Fp(f)11 b Fn(k)3451 2814 y Fl(2)3490 2715 y Fm(\023)3563 2736 y Fl(1)p Fj(=)p Fl(2)3690 2856 y Fp(:)178 3117 y FA(Setting)1477 3319 y Fp(c)28 b Fq(:=)1688 3165 y Fm(\000)1733 3245 y Fn(j)p Fp(\024)1817 3260 y Fj(a)1859 3245 y Fn(j)22 b Fq(+)g Fn(j)p Fp(\024)2091 3260 y Fj(b)2125 3245 y Fn(j)2153 3165 y Fm(\001)2198 3187 y Fl(2)2238 3245 y Fi(g)2288 3209 y Fl(4)2288 3270 y(1)2346 3245 y Fq(~)-68 b Fp(m)p 1688 3296 725 4 v 2026 3387 a Fq(4)3472 3319 y FA(\(2.21\))178 3514 y(and)33 b(summing)e(up)i(with)f(the)h(ob)m (vious)f(inequalit)m(y)1292 3724 y Fi(t)1325 3739 y Fj(V)1386 3724 y Fq([)p Fp(g)t(;)17 b(f)11 b Fq(])p Fn(j)26 b(\024)i(k)p Fp(V)22 b Fn(k)1932 3739 y Fj(L)1980 3720 y Fh(1)2071 3724 y Fn(\001)f(k)p Fp(f)11 b Fn(k)2279 3739 y Fd(H)2355 3724 y Fn(\001)22 b(k)p Fp(g)t Fn(k)2556 3739 y Fd(H)178 3933 y FA(one)33 b(gets)g(the)g(follo)m(wing)d(estimate)i(for)g(the)h (form)e Fi(t)2109 3948 y Fj(@)t Fl(\012)p Fj(;V)2282 3933 y FA(:)329 4143 y Fn(j)p Fi(t)390 4158 y Fj(@)t Fl(\012)p Fj(;V)562 4143 y Fq([)p Fp(g)t(;)o(f)11 b Fq(])p Fn(j)2692 b FA(\(2.22\))694 4351 y Fn(\024)799 4240 y Fm(\020)859 4351 y Fp(\016)t Fi(h)958 4366 y Fl(0)997 4351 y Fq([)p Fp(g)t Fq(])22 b(+)1234 4283 y Fp(c)p 1232 4328 47 4 v 1232 4419 a(\016)1289 4351 y Fn(k)p Fp(g)t Fn(k)1440 4310 y Fl(2)1440 4375 y Fd(H)1492 4240 y Fm(\021)1552 4262 y Fl(1)p Fj(=)p Fl(2)1678 4240 y Fm(\020)1738 4351 y Fp(\016)t Fi(h)1837 4366 y Fl(0)1876 4351 y Fq([)p Fp(f)11 b Fq(])22 b(+)2122 4283 y Fp(c)p 2119 4328 V 2119 4419 a(\016)2176 4351 y Fn(k)p Fp(f)11 b Fn(k)2335 4310 y Fl(2)2335 4375 y Fd(H)2388 4240 y Fm(\021)2448 4262 y Fl(1)p Fj(=)p Fl(2)2580 4351 y Fq(+)22 b Fn(k)p Fp(V)f Fn(k)2856 4366 y Fj(L)2904 4347 y Fh(1)2973 4351 y Fn(k)p Fp(f)11 b Fn(k)3132 4366 y Fd(H)3185 4351 y Fn(k)p Fp(g)t Fn(k)3336 4366 y Fd(H)694 4590 y Fn(\024)799 4480 y Fm(\020)859 4590 y Fp(\016)t Fi(h)958 4605 y Fl(0)997 4590 y Fq([)p Fp(g)t Fq(])22 b(+)1222 4510 y Fm(\000)1280 4523 y Fp(c)p 1277 4567 V 1277 4659 a(\016)1356 4590 y Fq(+)g Fn(k)p Fp(V)g Fn(k)1633 4605 y Fj(L)1681 4586 y Fh(1)1750 4510 y Fm(\001)1795 4590 y Fn(k)p Fp(g)t Fn(k)1946 4549 y Fl(2)1946 4615 y Fd(H)1999 4480 y Fm(\021)2059 4502 y Fl(1)p Fj(=)p Fl(2)2185 4480 y Fm(\020)2245 4590 y Fp(\016)t Fi(h)2344 4605 y Fl(0)2383 4590 y Fq([)p Fp(f)11 b Fq(])22 b(+)2616 4510 y Fm(\000)2674 4523 y Fp(c)p 2672 4567 V 2672 4659 a(\016)2751 4590 y Fq(+)g Fn(k)p Fp(V)f Fn(k)3027 4605 y Fj(L)3075 4586 y Fh(1)3144 4510 y Fm(\001)3190 4590 y Fn(k)p Fp(f)11 b Fn(k)3349 4549 y Fl(2)3349 4615 y Fd(H)3402 4480 y Fm(\021)3461 4502 y Fl(1)p Fj(=)p Fl(2)178 4831 y Fp(f)5 b(;)17 b(g)32 b Fn(2)c FA(dom)o Fq(\()p Fi(t)703 4846 y Fj(@)t Fl(\012)p Fj(;V)876 4831 y Fq(\))p FA(.)44 b(By)33 b(the)g(last)f(inequalit)m(y)f (it)h(turns)h(out)g(that)f(the)h(quadratic)f(form)g Fi(t)3549 4846 y Fj(@)t Fl(\012)p Fj(;V)178 4951 y FA(is)e(in\034nitesimally)c (small)i(with)i(resp)s(ect)i(to)d Fi(h)1876 4966 y Fl(0)1916 4951 y FA(.)42 b(Hence,)32 b(the)f(quadratic)f(form)f(corresp)s(onding) 178 5072 y(to)k(the)g(sesquilinear)e(form)h Fi(h)1272 5087 y Fj(V)1333 5072 y Fq([)p Fp(g)t(;)17 b(f)11 b Fq(])31 b FA(is)h(giv)m(en)h(b)m(y)439 5282 y Fi(h)491 5297 y Fj(V)552 5282 y Fq([)p Fp(g)t(;)17 b(f)11 b Fq(])27 b(:=)g Fi(h)969 5297 y Fl(0)1009 5282 y Fq([)p Fp(g)t(;)17 b(f)11 b Fq(])21 b(+)h Fi(t)1369 5297 y Fj(@)t Fl(\012)p Fj(;V)1541 5282 y Fq([)p Fp(g)t(;)17 b(f)11 b Fq(])22 b Fn(\000)g Fq(\()p Fp(g)t(;)17 b(f)11 b Fq(\))43 b(=)h Fi(h)2315 5297 y Fl(0)2355 5282 y Fq([)p Fp(g)t(;)17 b(f)11 b Fq(])21 b(+)h Fi(t)2715 5297 y Fj(@)t Fl(\012)p Fj(;V)15 b Fk(\000)p Fl(1)2978 5282 y Fq([)p Fp(g)t(;)i(f)11 b Fq(])p Fp(;)259 b FA(\(2.23\))178 5491 y Fp(f)5 b(;)17 b(g)41 b Fn(2)c FA(dom)o Fq(\()p Fi(h)740 5506 y Fj(V)801 5491 y Fq(\))g(=)g Fp(W)1095 5455 y Fl(1)p Fj(;)p Fl(2)1189 5491 y FA(,)i(is)e(closed)i (and)f(sectorial.)58 b(Consequen)m(tly)-8 b(,)41 b(there)e(is)f(a)f (\(unique\))178 5612 y(maximal)21 b(sectorial)i(op)s(erator)g Fp(H)8 b Fq(\()p Fp(V)21 b Fq(\))j FA(suc)m(h)i(that)e(the)g(represen)m (tation)h Fi(h)2851 5627 y Fj(V)2912 5612 y Fq([)p Fp(g)t(;)17 b(f)11 b Fq(])27 b(=)g(\()p Fp(H)8 b Fq(\()p Fp(V)21 b Fq(\))p Fp(g)t(;)c(f)11 b Fq(\))p eop %%Page: 8 10 8 9 bop 178 46 a FA(8)945 b Ft(M.)36 b(Bar)m(o,)f(H.-Ch.)47 b(Kaiser,)36 b(H.)g(Neidhardt,)f(J.)i(Rehber)m(g)178 295 y FA(is)46 b(v)-5 b(alid)43 b(for)j Fp(g)53 b Fn(2)d FA(dom)p Fq(\()p Fp(H)8 b Fq(\()p Fp(V)21 b Fq(\)\))45 b FA(and)h Fp(f)61 b Fn(2)50 b FA(dom)o Fq(\()p Fi(h)2168 310 y Fj(V)2229 295 y Fq(\))p FA(.)83 b(The)47 b(so)f(de\034ned)h(op)s (erator)e Fp(H)8 b Fq(\()p Fp(V)21 b Fq(\))178 415 y FA(coincides)33 b(with)f(that)g(one)h(giv)m(en)g(b)m(y)g(\(2.1\),)f (\(2.2\))g(and)h(\(2.3\).)296 583 y(Next)38 b(w)m(e)g(in)m(tend)g(for)e Fp(\026)f Fn(\025)h Fq(0)h FA(and)g Fp(V)57 b Fn(2)36 b Fp(L)1915 547 y Fk(1)2027 583 y FA(to)h(de\034ne)h(an)f(op)s(erator)f Fp(B)3048 598 y Fj(\026)3095 583 y Fq(\()p Fp(V)21 b Fq(\))37 b FA(whic)m(h)h(will)178 703 y(allo)m(w)33 b(us)h(a)g(certain) f(factorization)f(of)h(the)h(resolv)m(en)m(t)h(of)f Fp(H)8 b Fq(\()p Fp(V)21 b Fq(\))p FA(.)47 b(F)-8 b(or)33 b(this,)h(let)f(us)i (in)m(tro)s(duce)178 823 y(the)e(sesquilinear)f(form)531 1044 y Fp(b)572 1059 y Fj(\026)619 1044 y Fq(\()p Fp(V)22 b Fq(\)[)p Fp(f)5 b(;)17 b(g)t Fq(])27 b(:=)h Fi(t)1167 1059 y Fj(@)t Fl(\012)p Fj(;V)15 b Fk(\000)p Fl(1)1430 1044 y Fq([\()p Fp(H)1576 1059 y Fl(0)1637 1044 y Fq(+)22 b Fp(\026)p Fq(\))1832 1003 y Fk(\000)p Fl(1)p Fj(=)p Fl(2)1997 1044 y Fp(f)5 b(;)17 b Fq(\()p Fp(H)2213 1059 y Fl(0)2274 1044 y Fq(+)22 b Fp(\026)p Fq(\))2469 1003 y Fk(\000)p Fl(1)p Fj(=)p Fl(2)2633 1044 y Fp(g)t Fq(])p Fp(;)114 b(\026)27 b Fn(\025)i Fq(0)p Fp(;)352 b FA(\(2.24\))178 1264 y Fp(f)5 b(;)17 b(g)37 b Fn(2)c FA(dom)o Fq(\()p Fp(b)721 1279 y Fj(\026)768 1264 y Fq(\()p Fp(V)22 b Fq(\)\))32 b(=)h Fi(H)p FA(.)53 b(The)36 b(form)f Fp(b)1732 1279 y Fj(\026)1779 1264 y Fq(\()p Fp(V)21 b Fq(\))36 b FA(de\034nes)h(a)e(b)s(ounded)i(op)s(erator)e Fp(B)3248 1279 y Fj(\026)3294 1264 y Fq(\()p Fp(V)22 b Fq(\))35 b FA(on)h Fi(H)p FA(.)178 1384 y(F)-8 b(or)32 b(all)f(what)h(follo)m (ws)g(the)h(norm)e(of)h(the)h(op)s(erator)f Fp(B)2221 1399 y Fj(\026)2268 1384 y Fq(\()p Fp(V)21 b Fq(\))33 b FA(is)f(of)g(fundamen)m(tal)f(in)m(terest:)178 1635 y Fr(Lemma)37 b(2.2)49 b Fc(Assume)35 b Fp(V)49 b Fn(2)28 b Fp(L)1397 1599 y Fk(1)1507 1635 y Fc(and)35 b Fp(\016)c Fn(2)p Fq(]0)p Fp(;)17 b Fq(1[)p Fc(.)45 b(If)34 b Fp(\026)27 b Fn(\025)i Fq(0)p Fc(,)34 b(then)990 1900 y Fn(k)p Fp(B)1114 1915 y Fj(\026)1160 1900 y Fq(\()p Fp(V)22 b Fq(\))p Fn(k)1365 1915 y Fk(B)r Fl(\()p Fd(H)p Fl(\))1549 1900 y Fn(\024)28 b Fp(\016)e Fq(+)1821 1819 y Fm(\000)1879 1832 y Fp(c)p 1877 1877 47 4 v 1877 1968 a(\016)1956 1900 y Fq(+)c(1)g(+)g Fn(k)p Fp(V)f Fn(k)2401 1915 y Fj(L)2449 1896 y Fh(1)2518 1819 y Fm(\001)2586 1900 y Fn(\001)2735 1832 y Fq(1)p 2646 1877 228 4 v 2646 1968 a(1)g(+)i Fp(\026)2883 1900 y(:)562 b FA(\(2.25\))178 2164 y Fc(In)35 b(p)-5 b(articular,)34 b(if)1479 2285 y Fp(\026)27 b Fn(\025)h Fq(4)p Fp(c)22 b Fq(+)g(2)g(+)g(2)p Fn(k)p Fp(V)f Fn(k)2277 2300 y Fj(L)2325 2281 y Fh(1)2394 2285 y Fp(;)1051 b FA(\(2.26\))178 2459 y Fc(then)35 b Fn(k)p Fp(B)519 2474 y Fj(\026)566 2459 y Fq(\()p Fp(V)21 b Fq(\))p Fn(k)770 2475 y Fk(B)r Fl(\()p Fd(H)p Fl(\))954 2459 y Fp(<)28 b Fq(1)34 b Fc(and)887 2742 y Fn(k)p Fq(\(1)21 b(+)h Fp(B)1217 2757 y Fj(\026)1264 2742 y Fq(\()p Fp(V)f Fq(\)\))1456 2701 y Fk(\000)p Fl(1)1551 2742 y Fn(k)1601 2757 y Fk(B)r Fl(\()p Fd(H)p Fl(\))1785 2742 y Fn(\024)2262 2674 y Fq(2\()p Fp(\026)h Fq(+)g(1\))p 1900 2719 1077 4 v 1900 2810 a(1)g(+)g Fp(\026)g Fn(\000)g Fq(4)p Fp(c)g Fn(\000)h Fq(2)f Fn(\000)g Fq(2)p Fn(k)p Fp(V)f Fn(k)2859 2825 y Fj(L)2907 2806 y Fh(1)2986 2742 y Fp(:)459 b FA(\(2.27\))178 3065 y Fr(Pro)s(of.)58 b FA(\(2.25\))37 b(follo)m(ws)f(from)g (\(2.22\).)58 b(Setting)36 b Fp(\016)k Fq(=)c(1)p Fp(=)p Fq(2)h FA(w)m(e)i(get)e(from)f(\(2.25\))h(and)g(\(2.26\))178 3185 y(that)c Fn(k)p Fp(B)514 3200 y Fj(\026)561 3185 y Fq(\()p Fp(V)21 b Fq(\))p Fn(k)765 3201 y Fk(B)r Fl(\()p Fd(H)p Fl(\))950 3185 y Fp(<)29 b Fq(1)p FA(.)45 b(The)34 b(last)e(assertion)h(follo)m(ws)f(from)g(\(2.25\),)g(\(2.26\))h(and)g (the)g(repre-)178 3306 y(sen)m(tation)g(of)f(the)h(resolv)m(en)m(t)g(b) m(y)h(Neumann's)f(series.)1459 b Fb(\003)178 3582 y Fr(Lemma)37 b(2.3)49 b Fc(Assume)35 b Fp(V)49 b Fn(2)28 b Fp(L)1397 3546 y Fk(1)1472 3582 y Fc(.)45 b(If)34 b Fp(\026)28 b Fn(\025)g Fq(4)p Fp(c)22 b Fq(+)g(2)g(+)g(2)p Fn(k)p Fp(V)f Fn(k)2448 3597 y Fj(L)2496 3578 y Fh(1)2565 3582 y Fc(,)35 b(then)f(the)h(r)-5 b(epr)g(esentation)740 3802 y Fq(\()p Fp(H)8 b Fq(\()p Fp(V)21 b Fq(\))h(+)g Fp(\026)p Fq(\))1238 3761 y Fk(\000)p Fl(1)1360 3802 y Fq(=)28 b(\()p Fp(H)1583 3817 y Fl(0)1644 3802 y Fq(+)22 b Fp(\026)p Fq(\))1839 3761 y Fk(\000)p Fl(1)p Fj(=)p Fl(2)2003 3802 y Fq(\()p Fp(I)30 b Fq(+)22 b Fp(B)2286 3817 y Fj(\026)2333 3802 y Fq(\()p Fp(V)g Fq(\)\))2526 3761 y Fk(\000)p Fl(1)2620 3802 y Fq(\()p Fp(H)2739 3817 y Fl(0)2800 3802 y Fq(+)g Fp(\026)p Fq(\))2995 3761 y Fk(\000)p Fl(1)p Fj(=)p Fl(2)3472 3802 y FA(\(2.28\))178 4022 y Fc(holds.)178 4298 y Fr(Pro)s(of.)44 b FA(One)32 b(has)h(for)f(an)m(y)i Fp(f)5 b(;)17 b(g)31 b Fn(2)d Fp(W)1613 4262 y Fl(1)p Fj(;)p Fl(2)550 4540 y Fi(h)602 4555 y Fj(V)663 4540 y Fq([)p Fp(g)t(;)17 b(f)11 b Fq(])21 b(+)h Fp(\026)p Fq(\()p Fp(g)t(;)17 b(f)11 b Fq(\))26 b(=)1409 4429 y Fm(\020)1468 4451 y(p)p 1568 4451 300 4 v 89 x Fp(H)1649 4555 y Fl(0)1710 4540 y Fq(+)c Fp(\026g)t(;)1962 4451 y Fm(p)p 2060 4451 V 2060 4540 a Fp(H)2141 4555 y Fl(0)2203 4540 y Fq(+)g Fp(\026)o(f)2418 4429 y Fm(\021)2500 4540 y Fq(+)g Fi(t)2631 4555 y Fj(@)t Fl(\012)p Fj(;V)15 b Fk(\000)p Fl(1)2894 4540 y Fq([)p Fp(g)t(;)i(f)11 b Fq(])370 b FA(\(2.29\))178 4781 y(whic)m(h)33 b(yields)726 5024 y Fi(h)778 5039 y Fj(V)839 5024 y Fq([)p Fp(g)t(;)17 b(f)11 b Fq(])21 b(+)h Fp(\026)p Fq(\()p Fp(g)t(;)17 b(f)11 b Fq(\))26 b(=)1585 4913 y Fm(\020)1644 5024 y Fq(\()p Fp(I)k Fq(+)22 b Fp(B)1927 5039 y Fj(\026)1974 5024 y Fq(\()p Fp(V)f Fq(\)\))2166 4934 y Fm(p)p 2266 4934 V 90 x Fp(H)2347 5039 y Fl(0)2408 5024 y Fq(+)h Fp(\026g)t(;)2660 4931 y Fm(p)p 2758 4931 298 4 v 2758 5024 a Fp(H)2839 5039 y Fl(0)2901 5024 y Fq(+)g Fp(\025f)3115 4913 y Fm(\021)3472 5024 y FA(\(2.30\))178 5271 y(for)44 b Fp(\026)j(>)g Fq(0)p FA(.)77 b(F)-8 b(rom)43 b(the)h(previous)g (lemma)e(w)m(e)j(get)f Fn(k)p Fp(B)2346 5286 y Fj(\026)2393 5271 y Fq(\()p Fp(V)21 b Fq(\))p Fn(k)47 b Fp(<)g Fq(1)p FA(.)78 b(Hence,)48 b(the)c(in)m(v)m(erse)178 5392 y(op)s(erator)32 b(of)g Fp(I)e Fq(+)22 b Fp(B)927 5407 y Fj(\026)974 5392 y Fq(\()p Fp(V)f Fq(\))33 b FA(exists)g(and)g(is)f(b)s(ounded.)44 b(Therefore,)34 b(the)f(de\034nition)885 5612 y Fp(R)959 5627 y Fj(\026)1006 5612 y Fq(\()p Fp(V)22 b Fq(\))27 b(:=)h(\()p Fp(H)1438 5627 y Fl(0)1499 5612 y Fq(+)22 b Fp(\026)p Fq(\))1694 5571 y Fk(\000)p Fl(1)p Fj(=)p Fl(2)1859 5612 y Fq(\()p Fp(I)30 b Fq(+)22 b Fp(B)2142 5627 y Fj(\026)2188 5612 y Fq(\()p Fp(V)g Fq(\)\))2381 5571 y Fk(\000)p Fl(1)2475 5612 y Fq(\()p Fp(H)2594 5627 y Fl(0)2655 5612 y Fq(+)g Fp(\026)p Fq(\))2850 5571 y Fk(\000)p Fl(1)p Fj(=)p Fl(2)3472 5612 y FA(\(2.31\))p eop %%Page: 9 11 9 10 bop 178 46 a Ft(Dissip)-8 b(a)g(tive)36 b(Schr)m (\366dinger-Poisson)f(systems)1432 b FA(9)178 295 y(mak)m(es)33 b(sense.)46 b(Since)32 b Fp(R)1088 310 y Fj(\026)1135 295 y Fq(\()p Fp(V)22 b Fq(\))p Fp(g)31 b Fn(2)d Fp(W)1568 259 y Fl(1)p Fj(;)p Fl(2)1662 295 y Fq(\([)p Fp(a;)17 b(b)p Fq(]\))33 b FA(w)m(e)g(\034nd)444 503 y Fi(h)496 518 y Fj(V)557 503 y Fq([)p Fp(R)658 518 y Fj(\026)705 503 y Fq(\()p Fp(V)21 b Fq(\))p Fp(g)t(;)c(f)11 b Fq(])21 b(+)h Fp(\025)p Fq(\()p Fp(R)1328 518 y Fj(\026)1375 503 y Fq(\()p Fp(V)f Fq(\))p Fp(g)t(;)c(f)11 b Fq(\))1751 b FA(\(2.32\))846 679 y Fq(=)950 568 y Fm(\020)1009 589 y(p)p 1109 589 300 4 v 90 x Fp(H)1190 694 y Fl(0)1251 679 y Fq(+)22 b Fp(\026R)1482 694 y Fj(\026)1529 679 y Fq(\()p Fp(V)f Fq(\))p Fp(g)t(;)1778 589 y Fm(p)p 1877 589 V 1877 679 a Fp(H)1958 694 y Fl(0)2019 679 y Fq(+)h Fp(\026f)2235 568 y Fm(\021)2316 679 y Fq(+)g Fi(t)2447 694 y Fj(@)t Fl(\012)p Fj(;V)16 b Fk(\000)p Fl(1)2710 679 y Fq([)p Fp(R)2811 694 y Fj(\026)2858 679 y Fq(\()p Fp(V)22 b Fq(\))p Fp(g)t(;)17 b(f)11 b Fq(])p Fp(;)178 919 y(g)31 b Fn(2)d Fi(H)33 b FA(and)f Fp(f)39 b Fn(2)28 b Fp(W)931 883 y Fl(1)p Fj(;)p Fl(2)1025 919 y FA(.)43 b(Consequen)m(tly)-8 b(,)35 b(w)m(e)e(obtain)420 1128 y Fi(h)472 1143 y Fj(V)533 1128 y Fq([)p Fp(R)634 1143 y Fj(\026)681 1128 y Fq(\()p Fp(V)22 b Fq(\))p Fp(g)t(;)17 b(f)11 b Fq(])20 b(+)i Fp(\026)p Fq(\()p Fp(R)1306 1143 y Fj(\026)1353 1128 y Fq(\()p Fp(V)f Fq(\))p Fp(g)t(;)c(f)11 b Fq(\))1773 b FA(\(2.33\))822 1273 y Fq(=)926 1192 y Fm(\000)972 1273 y Fq(\()p Fp(I)30 b Fq(+)22 b Fp(B)1255 1288 y Fj(\026)1301 1273 y Fq(\()p Fp(V)g Fq(\)\))1494 1232 y Fk(\000)p Fl(1)1588 1273 y Fq(\()p Fp(H)1707 1288 y Fl(0)1768 1273 y Fq(+)g Fp(\026)p Fq(\))1963 1232 y Fk(\000)p Fl(1)p Fj(=)p Fl(2)2128 1273 y Fp(g)t(;)17 b Fq(\()p Fp(H)2342 1288 y Fl(0)2402 1273 y Fq(+)22 b Fp(\026)p Fq(\))2597 1232 y Fl(1)p Fj(=)p Fl(2)2707 1273 y Fp(f)2766 1192 y Fm(\001)947 1432 y Fq(+)1040 1351 y Fm(\000)1085 1432 y Fp(B)1159 1447 y Fj(\026)1206 1432 y Fq(\()p Fp(V)f Fq(\)\()p Fp(I)30 b Fq(+)22 b Fp(B)1643 1447 y Fj(\026)1690 1432 y Fq(\()p Fp(V)f Fq(\)\))1882 1391 y Fk(\000)p Fl(1)1977 1432 y Fq(\()p Fp(H)2096 1447 y Fl(0)2157 1432 y Fq(+)h Fp(\026)p Fq(\))2352 1391 y Fk(\000)p Fl(1)p Fj(=)p Fl(2)2516 1432 y Fp(g)t(;)17 b Fq(\()p Fp(H)2730 1447 y Fl(0)2791 1432 y Fq(+)22 b Fp(\026)p Fq(\))2986 1391 y Fl(1)p Fj(=)p Fl(2)3095 1432 y Fp(f)3154 1351 y Fm(\001)3216 1432 y Fp(;)178 1641 y FA(whic)m(h)33 b(sho)m(ws)i(that)1118 1761 y Fi(h)1170 1776 y Fj(V)1231 1761 y Fq([)p Fp(R)1332 1776 y Fj(\026)1378 1761 y Fq(\()p Fp(V)22 b Fq(\))p Fp(g)t(;)17 b(f)11 b Fq(])21 b(+)h Fp(\026)p Fq(\()p Fp(R)2004 1776 y Fj(\026)2050 1761 y Fq(\()p Fp(V)g Fq(\))p Fp(g)t(;)17 b(f)11 b Fq(\))26 b(=)i(\()p Fp(g)t(;)17 b(f)11 b Fq(\))p Fp(;)688 b FA(\(2.34\))178 1931 y Fp(g)37 b Fn(2)d Fi(H)p FA(,)i Fp(f)44 b Fn(2)34 b Fp(W)795 1894 y Fl(1)p Fj(;)p Fl(2)889 1931 y FA(.)54 b(Ho)m(w)m(ev)m(er,)39 b(the)d(relation)e(\(2.34\))h(implies)e(that)j Fp(R)2834 1946 y Fj(\026)2881 1931 y Fq(\()p Fp(V)21 b Fq(\))p Fp(g)37 b Fn(2)d FA(dom)o Fq(\()p Fp(H)8 b Fq(\()p Fp(V)21 b Fq(\)\))178 2051 y FA(and)k Fq(\()p Fp(H)8 b Fq(\()p Fp(V)21 b Fq(\))6 b(+)g Fp(\026)p Fq(\))p Fp(R)900 2066 y Fj(\026)947 2051 y Fq(\()p Fp(V)22 b Fq(\))p Fp(g)31 b Fq(=)c Fp(g)h FA(for)d(an)m(y)g Fp(g)31 b Fn(2)d Fi(H)p FA(.)41 b(Similarly)-8 b(,)22 b(one)j(pro)m(v)m(es)i (that)d Fp(R)3157 2066 y Fj(\026)3204 2051 y Fq(\()p Fp(V)e Fq(\)\()p Fp(H)8 b Fq(\()p Fp(V)20 b Fq(\))6 b(+)178 2171 y Fp(\026)p Fq(\))p Fp(g)31 b Fq(=)c Fp(g)32 b FA(for)c(an)m(y)i Fp(g)h Fn(2)d FA(dom)o Fq(\()p Fp(H)8 b Fq(\()p Fp(V)21 b Fq(\)\))p FA(.)42 b(Both)28 b(relations)f(imply)g(that)h Fq(\()p Fp(H)8 b Fq(\()p Fp(V)21 b Fq(\))14 b(+)g Fp(\026)p Fq(\))3194 2135 y Fk(\000)p Fl(1)3316 2171 y Fq(=)27 b Fp(R)3493 2186 y Fj(\026)3540 2171 y Fq(\()p Fp(V)22 b Fq(\))p FA(.)3644 2292 y Fb(\003)178 2626 y Fs(2.3)131 b(Dilations)178 2858 y FA(Since)32 b Fp(H)8 b Fq(\()p Fp(V)21 b Fq(\))32 b FA(is)g(a)f(maximal)e(dissipativ)m(e)i(op)s (erator)h(there)g(is)g(a)f(larger)g(Hilb)s(ert)f(space)j Fi(K)28 b Fn(\023)g Fi(H)178 2978 y FA(and)33 b(a)f(self-adjoin)m(t)f (op)s(erator)h Fp(K)7 b Fq(\()p Fp(V)21 b Fq(\))33 b FA(on)f Fi(K)g FA(suc)m(h)i(that)872 3187 y Fp(P)949 3146 y Fd(K)935 3212 y(H)999 3187 y Fq(\()p Fp(K)7 b Fq(\()p Fp(V)22 b Fq(\))g Fn(\000)g Fp(z)t Fq(\))1490 3146 y Fk(\000)p Fl(1)1585 3187 y Fn(j)p Fi(H)27 b Fq(=)h(\()p Fp(H)8 b Fq(\()p Fp(V)21 b Fq(\))h Fn(\000)h Fp(z)t Fq(\))2306 3146 y Fk(\000)p Fl(1)2401 3187 y Fp(;)114 b Fn(=)p FA(m)o Fq(\()p Fp(z)t Fq(\))28 b Fp(>)g Fq(0)p Fp(;)444 b FA(\(2.35\))178 3396 y(see)26 b([1].)41 b(The)26 b(op)s(erator)e Fp(K)7 b Fq(\()p Fp(V)21 b Fq(\))k FA(is)f(called)g(a)h(self-adjoin)m(t)d (dilation)g(of)j(the)g(maximal)c(dissipativ)m(e)178 3516 y(op)s(erator)32 b Fp(H)8 b Fq(\()p Fp(V)21 b Fq(\))p FA(.)44 b(Ob)m(viously)-8 b(,)32 b(from)f(the)i(condition)e(\(2.35\))h (one)h(gets)853 3725 y Fp(P)930 3684 y Fd(K)916 3749 y(H)979 3725 y Fq(\()p Fp(K)7 b Fq(\()p Fp(V)22 b Fq(\))g Fn(\000)h Fp(z)t Fq(\))1471 3684 y Fk(\000)p Fl(1)1566 3725 y Fn(j)p Fi(H)k Fq(=)g(\()p Fp(H)8 b Fq(\()p Fp(V)21 b Fq(\))2077 3684 y Fk(\003)2139 3725 y Fn(\000)h Fp(z)t Fq(\))2325 3684 y Fk(\000)p Fl(1)2420 3725 y Fp(;)114 b Fn(=)p FA(m)p Fq(\()p Fp(z)t Fq(\))28 b Fp(<)g Fq(0)p Fp(:)424 b FA(\(2.36\))178 3933 y(If)33 b(the)g(condition)1467 3959 y Fm(_)1415 4175 y Fj(z)s Fk(2)p Fg(C)9 b Fk(n)p Fg(R)1629 4054 y Fq(\()p Fp(K)e Fq(\()p Fp(V)22 b Fq(\))g Fn(\000)g Fp(z)t Fq(\))2120 4013 y Fk(\000)p Fl(1)2215 4054 y Fi(H)28 b Fq(=)f Fi(K)987 b FA(\(2.37\))178 4334 y(is)37 b(satis\034ed,)j(then)e Fp(K)7 b Fq(\()p Fp(V)21 b Fq(\))38 b FA(is)f(called)f(a)h(minimal)c(self-adjoin)m(t)j(dilation) f(of)i Fp(H)8 b Fq(\()p Fp(V)21 b Fq(\))p FA(.)58 b(Minimal)178 4454 y(self-adjoin)m(t)24 b(dilations)f(of)i(maximal)d(dissipativ)m(e)j (op)s(erators)g(are)g(determined)g(up)h(to)f(a)g(certain)178 4574 y(isomorphism,)e(in)g(particular,)h(all)e(minimal)d(self-adjoin)m (t)j(dilations)g(are)h(unitarily)f(equiv)-5 b(alen)m(t.)296 4742 y(In)39 b(the)f(presen)m(t)i(case)f(the)f(minimal)c(self-adjoin)m (t)i(dilation)f(of)j(the)h(maximal)34 b(dissipativ)m(e)178 4862 y(op)s(erator)29 b Fp(H)8 b Fq(\()p Fp(V)21 b Fq(\))29 b FA(can)g(b)s(e)h(constructed)h(in)d(an)h(explicit)f(manner,)h(cf.)43 b([8].)f(The)30 b(dilation)d(space)178 4983 y Fi(K)32 b FA(is)h(giv)m(en)f(b)m(y)1544 5103 y Fi(K)27 b Fq(=)h Fn(D)1819 5118 y Fk(\000)1900 5103 y Fn(\010)22 b Fi(H)g Fn(\010)h(D)2270 5118 y Fl(+)2329 5103 y Fp(;)1116 b FA(\(2.38\))178 5273 y(where)38 b Fn(D)541 5288 y Fk(\006)635 5273 y Fq(:=)d Fp(L)839 5236 y Fl(2)879 5273 y Fq(\()p Fo(R)983 5288 y Fk(\006)1048 5273 y Fp(;)17 b Fo(C)1157 5236 y Fl(2)1203 5273 y Fq(\))p FA(.)56 b(In)m(tro)s(ducing)36 b(the)h(domain)2390 5247 y Fq(^)2379 5273 y(\012)f(:=)f Fo(R)2689 5288 y Fk(\000)2779 5273 y Fn(\002)25 b Fq([)p Fp(a;)17 b(b)p Fq(])26 b Fn(\002)f Fo(R)3265 5288 y Fl(+)3330 5273 y FA(,)38 b(w)m(e)g(ma)m(y)178 5393 y(write)33 b Fi(K)27 b Fq(=)g Fp(L)691 5357 y Fl(2)731 5393 y Fq(\()780 5368 y(^)769 5393 y(\012)q Fp(;)17 b(dx)p Fq(\))p FA(.)43 b(In)33 b(accordance)g(with)f(\(2.38\))g(one)h(writes)1489 5585 y Fp(~)1476 5612 y(f)38 b Fq(:=)28 b Fp(f)1741 5627 y Fk(\000)1822 5612 y Fn(\010)22 b Fp(f)33 b Fn(\010)23 b Fp(f)2150 5627 y Fl(+)2237 5612 y Fn(2)28 b Fi(K)p Fp(:)1047 b FA(\(2.39\))p eop %%Page: 10 12 10 11 bop 178 46 a FA(10)896 b Ft(M.)36 b(Bar)m(o,)f(H.-Ch.)47 b(Kaiser,)36 b(H.)g(Neidhardt,)f(J.)i(Rehber)m(g)178 295 y FA(The)d(resolv)m(en)m(t)f(of)g Fp(K)39 b FA(is)32 b(giv)m(en)h(b)m(y)257 525 y Fq(\()p Fp(K)7 b Fq(\()p Fp(V)21 b Fq(\))i Fn(\000)f Fp(z)t Fq(\))748 484 y Fk(\000)p Fl(1)860 525 y Fq(\()p Fp(f)946 540 y Fk(\000)1027 525 y Fn(\010)h Fp(f)32 b Fn(\010)23 b Fp(f)1355 540 y Fl(+)1414 525 y Fq(\))2020 b FA(\(2.40\))423 735 y Fq(=)28 b Fp(i)577 600 y Fm(Z)676 626 y Fj(x)632 825 y Fk(\0001)778 735 y Fp(dy)i(e)952 694 y Fj(i)p Fl(\()p Fj(x)p Fk(\000)p Fj(y)r Fl(\))p Fj(z)1203 735 y Fp(f)1251 750 y Fk(\000)1310 735 y Fq(\()p Fp(y)t Fq(\))49 b Fn(\010)h Fq(\()p Fp(H)8 b Fq(\()p Fp(V)21 b Fq(\))i Fn(\000)f Fp(z)t Fq(\))2104 694 y Fk(\000)p Fl(1)2199 735 y Fp(f)33 b Fq(+)22 b Fp(iT)2468 750 y Fk(\003)2508 735 y Fq(\()p Fp(V)f Fq(\)\()p 2700 681 50 4 v Fp(z)5 b Fq(\))2788 694 y Fk(\003)2844 600 y Fm(Z)2944 626 y Fl(0)2899 825 y Fk(\0001)3046 735 y Fp(dy)30 b(e)3220 694 y Fk(\000)p Fj(iy)r(z)3376 735 y Fp(f)3424 750 y Fk(\000)3483 735 y Fq(\()p Fp(y)t Fq(\))506 1018 y Fn(\010)e Fp(i)661 882 y Fm(Z)761 908 y Fj(x)716 1108 y Fl(0)821 1018 y Fp(dy)j(e)996 976 y Fj(i)p Fl(\()p Fj(x)p Fk(\000)p Fj(y)r Fl(\))p Fj(z)1246 1018 y Fp(f)1294 1033 y Fl(+)1354 1018 y Fq(\()p Fp(y)t Fq(\))21 b(+)h Fp(ie)1679 976 y Fj(iz)s(x)1783 1018 y Fp(T)14 b Fq(\()p Fp(V)21 b Fq(\)\()p Fp(z)t Fq(\))p Fp(f)33 b Fq(+)22 b Fp(i)p Fq(\002)2421 1033 y Fj(H)5 b Fl(\()p Fj(V)17 b Fl(\))2600 1018 y Fq(\()p 2638 963 V Fp(z)5 b Fq(\))2726 976 y Fk(\003)2782 882 y Fm(Z)2882 908 y Fl(0)2837 1108 y Fk(\0001)2984 1018 y Fp(dy)30 b(e)3158 976 y Fj(i)p Fl(\()p Fj(x)p Fk(\000)p Fj(y)r Fl(\))p Fj(z)3409 1018 y Fp(f)3457 1033 y Fk(\000)3516 1018 y Fq(\()p Fp(y)t Fq(\))178 1308 y FA(for)i Fn(=)p FA(m)p Fq(\()p Fp(z)t Fq(\))c Fp(>)f Fq(0)33 b FA(and)195 1538 y Fq(\()p Fp(K)7 b Fq(\()p Fp(V)22 b Fq(\))g Fn(\000)h Fp(z)t Fq(\))687 1497 y Fk(\000)p Fl(1)798 1538 y Fq(\()p Fp(f)884 1553 y Fk(\000)965 1538 y Fn(\010)g Fp(f)33 b Fn(\010)22 b Fp(f)1293 1553 y Fl(+)1353 1538 y Fq(\))2081 b FA(\(2.41\))361 1748 y Fq(=)28 b Fn(\000)p Fp(i)592 1613 y Fm(Z)692 1639 y Fl(0)648 1838 y Fj(x)748 1748 y Fp(dy)i(e)922 1707 y Fj(i)p Fl(\()p Fj(x)p Fk(\000)p Fj(y)r Fl(\))p Fj(z)1173 1748 y Fp(f)1221 1763 y Fk(\000)1280 1748 y Fq(\()p Fp(y)t Fq(\))21 b Fn(\000)i Fp(ie)1607 1707 y Fj(iz)s(x)1711 1748 y Fp(T)1768 1763 y Fk(\003)1808 1748 y Fq(\()p Fp(V)e Fq(\)\()p Fp(z)t Fq(\))p Fp(f)34 b Fn(\000)22 b Fp(i)p Fq(\002)2377 1764 y Fj(H)5 b Fl(\()p Fj(V)17 b Fl(\))2556 1748 y Fq(\()p Fp(z)t Fq(\))2698 1613 y Fm(Z)2798 1639 y Fk(1)2754 1838 y Fl(0)2889 1748 y Fp(dy)31 b(e)3064 1707 y Fj(i)p Fl(\()p Fj(x)p Fk(\000)p Fj(y)r Fl(\))p Fj(z)3315 1748 y Fp(f)3363 1763 y Fl(+)3422 1748 y Fq(\()p Fp(y)t Fq(\))444 2009 y Fn(\010)d Fq(\()p Fp(H)8 b Fq(\()p Fp(V)21 b Fq(\))830 1968 y Fk(\003)892 2009 y Fn(\000)i Fp(z)t Fq(\))1079 1968 y Fk(\000)p Fl(1)1174 2009 y Fp(f)32 b Fn(\000)23 b Fp(iT)14 b Fq(\()p Fp(V)22 b Fq(\)\()p 1651 1954 V Fp(z)t Fq(\))1738 1968 y Fk(\003)1794 1874 y Fm(Z)1894 1900 y Fk(1)1849 2099 y Fl(0)1985 2009 y Fp(dy)31 b(e)2160 1968 y Fk(\000)p Fj(iy)r(z)2316 2009 y Fp(f)2364 2024 y Fl(+)2423 2009 y Fq(\()p Fp(y)t Fq(\))49 b Fn(\010)h(\000)p Fp(i)2854 1874 y Fm(Z)2954 1900 y Fk(1)2910 2099 y Fj(x)3045 2009 y Fp(dy)31 b(e)3220 1968 y Fj(i)p Fl(\()p Fj(x)p Fk(\000)p Fj(y)r Fl(\))p Fj(z)3471 2009 y Fp(f)3519 2024 y Fl(+)3578 2009 y Fq(\()p Fp(y)t Fq(\))178 2292 y FA(for)36 b Fn(=)p FA(m)o Fq(\()p Fp(z)t Fq(\))e Fp(<)g Fq(0)p FA(.)53 b(The)37 b(self-adjoin)m(t)e(op)s(erator) g Fp(K)43 b FA(is)36 b(absolutely)f(con)m(tin)m(uous)h(and)h(its)e(sp)s (ec-)178 2413 y(trum)g(coincides)g(with)g(the)g(real)f(axis,)i(i.e.)51 b Fp(\033)t Fq(\()p Fp(K)7 b Fq(\))32 b(=)g Fo(R)5 b FA(.)57 b(The)36 b(m)m(ultiplicit)m(y)c(of)j(its)f(sp)s(ectrum)178 2533 y(is)e(t)m(w)m(o.)45 b(F)-8 b(or)31 b(more)h(details)f(the)i (reader)g(is)g(referred)g(to)f([8].)178 2887 y Fs(2.4)131 b(Eigenfunction)45 b(expansions)178 3131 y FA(The)30 b(generalized)d(eigenfunctions)1538 3105 y Fp(~)1507 3131 y( )1574 3102 y Fk(\000)1634 3131 y Fq(\()p Fp(V)21 b Fq(\)\()p Fn(\001)p Fp(;)c(\025;)g(\034)11 b Fq(\))p FA(,)29 b Fp(\025)e Fn(2)h Fo(R)5 b FA(,)35 b Fp(\034)40 b Fq(=)27 b Fp(a;)17 b(b)p FA(,)30 b(of)e Fp(K)7 b Fq(\()p Fp(V)21 b Fq(\))29 b FA(are)f(giv)m(en)h(b)m(y)312 3345 y Fp(~)281 3372 y( )348 3343 y Fk(\000)407 3372 y Fq(\()p Fp(V)21 b Fq(\)\()p Fp(x;)c(\025;)g(\034)11 b Fq(\))28 b(:=)g Fp( )1116 3331 y Fk(\000)1112 3396 y(\000)1175 3372 y Fq(\()p Fp(V)21 b Fq(\)\()p Fp(x;)c(\025;)g(\034)11 b Fq(\))22 b Fn(\010)h Fp( )1847 3331 y Fk(\000)1906 3372 y Fq(\()p Fp(V)e Fq(\)\()p Fp(x;)c(\025;)g(\034)11 b Fq(\))23 b Fn(\010)f Fp( )2578 3331 y Fk(\000)2574 3396 y Fl(+)2637 3372 y Fq(\()p Fp(V)g Fq(\)\()p Fp(x;)17 b(\025;)g(\034)11 b Fq(\))351 b FA(\(2.42\))918 3573 y Fq(=)1102 3505 y(1)p 1031 3550 191 4 v 1031 3570 a Fn(p)p 1114 3570 108 4 v 82 x Fq(2)p Fp(\031)1232 3573 y(e)1277 3532 y Fj(ix\025)1386 3573 y Fp(e)1431 3588 y Fj(\034)1497 3573 y Fn(\010)1677 3505 y Fq(1)p 1606 3550 191 4 v 1606 3570 a Fn(p)p 1690 3570 108 4 v 1690 3652 a Fq(2)p Fp(\031)1807 3492 y Fm(\000)1853 3573 y Fq(\()p Fp(T)1948 3588 y Fk(\003)1987 3573 y Fq(\()p Fp(V)22 b Fq(\)\()p Fp(\025)p Fq(\))2275 3492 y Fm(\001)2320 3514 y Fk(\003)2360 3573 y Fp(e)2405 3588 y Fj(\034)2448 3573 y Fq(\)\()p Fp(x)p Fq(\))g Fn(\010)2820 3505 y Fq(1)p 2749 3550 191 4 v 2749 3570 a Fn(p)p 2832 3570 108 4 v 82 x Fq(2)p Fp(\031)2950 3573 y(e)2995 3532 y Fj(ix\025)3104 3573 y Fq(\002)3180 3588 y Fj(H)5 b Fl(\()p Fj(V)16 b Fl(\))3359 3573 y Fq(\()p Fp(\025)p Fq(\))3492 3532 y Fk(\003)3531 3573 y Fp(e)3576 3588 y Fj(\034)178 3852 y FA(where)1224 4016 y Fp(e)1269 4031 y Fj(b)1331 4016 y Fq(:=)1462 3876 y Fm(\022)1576 3955 y Fq(1)1576 4076 y(0)1667 3876 y Fm(\023)1854 4016 y FA(and)98 b Fp(e)2154 4031 y Fj(a)2223 4016 y Fq(:=)2354 3876 y Fm(\022)2469 3955 y Fq(0)2469 4076 y(1)2559 3876 y Fm(\023)2649 4016 y Fp(:)796 b FA(\(2.43\))178 4255 y(The)34 b(eigenfunctions)e(are)h (orthogonal,)d(i.e.)823 4397 y Fm(\020)914 4481 y Fp(~)883 4507 y( )950 4478 y Fk(\000)1009 4507 y Fq(\()p Fp(V)21 b Fq(\)\()p Fn(\001)p Fp(;)c(\025;)g(\034)11 b Fq(\))p Fp(;)1539 4481 y(~)1509 4507 y( )1576 4478 y Fk(\000)1634 4507 y Fq(\()p Fp(V)22 b Fq(\)\()p Fn(\001)p Fp(;)17 b(\025)1956 4466 y Fk(0)1978 4507 y Fp(;)g(\034)2075 4466 y Fk(0)2099 4507 y Fq(\))2137 4397 y Fm(\021)2196 4576 y Fj(L)2244 4557 y Ff(2)2279 4576 y Fl(\()2314 4559 y(^)2306 4576 y(\012\))2417 4507 y Fq(=)27 b Fp(\016)t Fq(\()p Fp(\025)22 b Fn(\000)h Fp(\025)2841 4466 y Fk(0)2864 4507 y Fq(\))p Fp(\016)2945 4522 y Fj(\034)8 b(\034)3023 4503 y Fh(0)3049 4507 y Fp(;)396 b FA(\(2.44\))178 4786 y Fp(\025;)17 b(\025)336 4750 y Fk(0)403 4786 y Fn(2)44 b Fo(R)5 b FA(,)51 b Fp(\034)6 b(;)17 b(\034)802 4750 y Fk(0)869 4786 y Fq(=)44 b Fp(a;)17 b(b)42 b FA(in)g(the)g(sense)i(of) d(distributions,)i(cf.)73 b([8)o(],)45 b(and)d(their)g(linear)e(span) 178 4906 y(\(mo)s(dulo)33 b(the)j(scalar,)f(con)m(tin)m(uous,)h (compactly)e(supp)s(orted)i(functions\))f(is)f(dense)j(in)d Fi(K)p FA(.)50 b(W)-8 b(e)178 5037 y(note)29 b(that)g(the)h (generalized)e(eigenfunctions)1928 5010 y Fp(~)1898 5037 y( )1965 5008 y Fk(\000)2024 5037 y Fq(\()p Fp(V)21 b Fq(\)\()p Fn(\001)p Fp(;)c(\025;)g(\034)11 b Fq(\))28 b FA(are)h(usually)f(called)g(the)h(incom-)178 5157 y(ing)j (eigenfunctions.)43 b(Using)31 b(the)i(incoming)d(eigenfunctions)i(one) g(de\034nes)i(a)e(transformation)178 5277 y Fq(\010)248 5292 y Fk(\000)308 5277 y Fq(\()p Fp(V)21 b Fq(\))28 b(:)g Fi(K)f Fn(\000)-16 b(!)836 5251 y Fq(^)827 5277 y Fi(K)27 b Fq(=)h Fp(L)1091 5241 y Fl(2)1131 5277 y Fq(\()p Fo(R)5 b Fp(;)17 b Fo(C)1350 5241 y Fl(2)1396 5277 y Fq(\))1174 5566 y(\(\010)1282 5581 y Fk(\000)1342 5566 y Fq(\()p Fp(V)k Fq(\))l Fp(~)-45 b(g)s Fq(\)\)\()p Fp(\025)p Fq(\))28 b(=:)j(^)-53 b Fp(g)t Fq(\()p Fp(\025)p Fq(\))27 b(=)2228 5426 y Fm(\022)2350 5505 y Fq(^)-53 b Fp(g)2397 5469 y Fj(b)2431 5505 y Fq(\()p Fp(\025)p Fq(\))2346 5625 y(^)g Fp(g)2393 5589 y Fj(a)2434 5625 y Fq(\()p Fp(\025)p Fq(\))2609 5426 y Fm(\023)2699 5566 y Fp(;)746 b FA(\(2.45\))p eop %%Page: 11 13 11 12 bop 178 46 a Ft(Dissip)-8 b(a)g(tive)36 b(Schr)m (\366dinger-Poisson)f(systems)1383 b FA(11)178 295 y(where)893 442 y Fq(^)-53 b Fp(g)940 401 y Fj(\034)982 442 y Fq(\()p Fp(\025)p Fq(\))28 b(:=)1274 306 y Fm(Z)1337 515 y Fl(^)1329 532 y(\012)1401 442 y Fp(dx)1551 331 y Fm(\020)1606 442 y Fp(~)-44 b(g)s Fq(\()p Fp(x)p Fq(\))p Fp(;)1867 416 y(~)1836 442 y( )1903 413 y Fk(\000)1962 442 y Fq(\()p Fp(V)21 b Fq(\)\()p Fp(x;)c(\025;)g(\034)11 b Fq(\))2445 331 y Fm(\021)2522 442 y Fp(;)114 b(\034)39 b Fq(=)27 b Fp(a;)17 b(b:)462 b FA(\(2.46\))178 673 y Fq(\010)248 688 y Fk(\000)308 673 y Fq(\()p Fp(V)21 b Fq(\))37 b FA(is)g(unitary)f(and)h(called)f(the)h(incoming)e(F)-8 b(ourier)35 b(transformation.)55 b(The)38 b(in)m(v)m(erse)g(in-)178 793 y(coming)31 b(F)-8 b(ourier)32 b(transformation)e Fq(\010)1586 808 y Fk(\000)1645 793 y Fq(\()p Fp(V)22 b Fq(\))1800 757 y Fk(\000)p Fl(1)1927 793 y FA(is)32 b(giv)m(en)g(b)m(y)413 1065 y Fq(\(\010)521 1080 y Fk(\000)580 1065 y Fq(\()p Fp(V)22 b Fq(\))735 1024 y Fk(\000)p Fl(1)832 1065 y Fq(^)-52 b Fp(g)s Fq(\)\()p Fp(x)p Fq(\))28 b(=)1180 929 y Fm(Z)1235 1155 y Fg(R)1304 1065 y Fp(d\025)1474 970 y Fm(X)1456 1183 y Fj(\034)8 b Fl(=)p Fj(a;b)1684 1039 y Fp(~)1653 1065 y( )1720 1036 y Fk(\000)1779 1065 y Fq(\()p Fp(V)22 b Fq(\)\()p Fp(x;)17 b(\025;)g(\034)11 b Fq(\))s(^)-52 b Fp(g)2314 1024 y Fj(\034)2356 1065 y Fq(\()p Fp(\025)p Fq(\))p Fp(;)118 b Fq(^)-53 b Fp(g)31 b Fn(2)d Fp(L)2868 1024 y Fl(2)2908 1065 y Fq(\()p Fo(R)5 b Fp(;)17 b Fo(C)3128 1024 y Fl(2)3173 1065 y Fq(\))p Fp(:)234 b FA(\(2.47\))178 1385 y(W)-8 b(e)33 b(note)g(that)1368 1509 y Fq(\010)1438 1524 y Fk(\000)1497 1509 y Fq(\()p Fp(V)22 b Fq(\))p Fp(K)7 b Fq(\()p Fp(V)21 b Fq(\)\010)1966 1524 y Fk(\000)2026 1509 y Fq(\()p Fp(V)g Fq(\))2180 1468 y Fk(\000)p Fl(1)2302 1509 y Fq(=)28 b Fp(M)5 b(;)940 b FA(\(2.48\))178 1687 y(where)34 b Fp(M)43 b FA(is)32 b(the)h(m)m(ultiplication)28 b(op)s(erator)k(b)m(y)h(the)g(indep)s (enden)m(t)h(v)-5 b(ariable)31 b Fp(\025)h FA(on)3338 1661 y Fq(^)3329 1687 y Fi(K)o FA(,)h(i.e.)845 1902 y(dom)o Fq(\()p Fp(M)10 b Fq(\))83 b(:=)g Fn(f)t Fq(^)-53 b Fp(g)31 b Fn(2)d Fp(L)1766 1866 y Fl(2)1806 1902 y Fq(\()p Fo(R)5 b Fp(;)17 b Fo(C)2026 1866 y Fl(2)2071 1902 y Fq(\))28 b(:)f Fp(\025)s Fq(^)-52 b Fp(g)t Fq(\()p Fp(\025)p Fq(\))27 b Fn(2)h Fp(L)2619 1866 y Fl(2)2659 1902 y Fq(\()p Fo(R)5 b Fp(;)17 b Fo(C)2879 1866 y Fl(2)2924 1902 y Fq(\))p Fn(g)f Fp(;)845 2070 y Fq(\()p Fp(M)e Fq(^)-53 b Fp(g)t Fq(\)\()p Fp(\025)p Fq(\))83 b(:=)g Fp(\025)s Fq(^)-52 b Fp(g)t Fq(\()p Fp(\025)p Fq(\))p Fp(;)117 b Fq(^)-52 b Fp(g)31 b Fn(2)d FA(dom)o Fq(\()p Fp(M)10 b Fq(\))p Fp(:)3472 1987 y FA(\(2.49\))178 2292 y(The)37 b(represen)m(tation)f (\(2.49\))f(induced)h(b)m(y)h Fq(\010)1882 2307 y Fk(\000)1941 2292 y Fq(\()p Fp(V)22 b Fq(\))35 b FA(is)g(called)g(the)h(incoming)d (sp)s(ectral)i(repre-)178 2413 y(sen)m(tation)e(of)f Fp(K)7 b FA(.)296 2580 y(Finally)-8 b(,)24 b(w)m(e)j(note)e(that)g(eac) m(h)h(b)s(ounded)g(self-adjoin)m(t)d(op)s(erator)i Fp(G)g FA(on)g Fi(K)p FA(,)h(whic)m(h)g(comm)m(utes)178 2701 y(with)d Fp(K)7 b FA(,)25 b(corresp)s(onds)f(to)e(a)g(measurable)g (family)f Fn(f)p Fp(G)p Fq(\()p Fp(\025)p Fq(\))p Fn(g)2337 2716 y Fj(\025)p Fk(2)p Fg(R)2499 2701 y FA(of)h(t)m(w)m(o-b)m(y-t)m(w) m(o)i(matrices)e(whic)m(h)178 2821 y(is)33 b(uniformly)e(b)s(ounded,)k (i.e.)45 b Fp(G)p Fq(\()p Fn(\001)p Fq(\))29 b Fn(2)h Fp(L)1691 2785 y Fk(1)1766 2821 y Fq(\()p Fo(R)5 b Fp(;)17 b Fn(B)s Fq(\()p Fo(C)2092 2785 y Fl(2)2137 2821 y Fq(\)\))p FA(,)34 b(suc)m(h)h(that)e(the)h(m)m(ultiplication)28 b(op)s(er-)178 2941 y(ator)406 2916 y Fq(^)384 2941 y Fp(G)33 b FA(on)f Fp(L)695 2905 y Fl(2)735 2941 y Fq(\()p Fo(R)5 b Fp(;)17 b Fo(C)954 2905 y Fl(2)1000 2941 y Fq(\))32 b FA(de\034ned)i(b)m(y)782 3172 y(dom)o Fq(\()1026 3147 y(^)1004 3172 y Fp(G)p Fq(\))83 b(:=)g Fn(f)s Fq(^)-52 b Fp(g)31 b Fn(2)d Fp(L)1676 3136 y Fl(2)1716 3172 y Fq(\()p Fo(R)5 b Fp(;)17 b Fo(C)1935 3136 y Fl(2)1981 3172 y Fq(\))27 b(:)h Fp(G)p Fq(\()p Fp(\025)p Fq(\))s(^)-52 b Fp(g)s Fq(\()p Fp(\025)p Fq(\))28 b Fn(2)g Fp(L)2682 3136 y Fl(2)2722 3172 y Fq(\()p Fo(R)t Fp(;)17 b Fo(C)2941 3136 y Fl(2)2986 3172 y Fq(\))p Fn(g)g Fp(;)782 3350 y Fq(\()842 3324 y(^)820 3350 y Fp(G)s Fq(^)-52 b Fp(g)s Fq(\)\()p Fp(\025)p Fq(\))84 b(:=)f Fp(G)p Fq(\()p Fp(\025)p Fq(\))s(^)-52 b Fp(g)s Fq(\()p Fp(\025)p Fq(\))p Fp(;)118 b Fq(^)-53 b Fp(g)31 b Fn(2)d FA(dom)o Fq(\()2339 3324 y(^)2316 3350 y Fp(G)p Fq(\))3472 3257 y FA(\(2.50\))178 3566 y(is)k(unitarily)f(equiv)-5 b(alen)m(t)32 b(to)g Fp(G)p FA(,)h(i.e.)1463 3789 y Fq(\010)1533 3804 y Fk(\000)1592 3789 y Fq(\()p Fp(V)22 b Fq(\))p Fp(G)p Fq(\010)1894 3804 y Fk(\000)1953 3789 y Fq(\()p Fp(V)g Fq(\))2108 3748 y Fk(\000)p Fl(1)2230 3789 y Fq(=)2355 3764 y(^)2333 3789 y Fp(G:)1035 b FA(\(2.51\))178 4012 y(The)34 b(represen)m(tation)f (\(2.50\))f(is)g(called)f(in)h(incoming)e(sp)s(ectral)i(represen)m (tation)i(of)e Fp(G)p FA(.)178 4354 y Fs(2.5)131 b(Carrier)45 b(densities)178 4587 y FA(In)33 b(the)g(follo)m(wing)c(w)m(e)34 b(call)d(an)h(op)s(erator)g Fp(\045)c Fq(:)f Fi(K)h Fn(\000)-17 b(!)28 b Fi(K)k FA(a)g(densit)m(y)h(matrix)e(if)g Fp(\045)i FA(is)f(a)g(b)s(ounded,)178 4708 y(non-negativ)m(e,)47 b(self-adjoin)m(t)42 b(op)s(erator.)77 b(The)45 b(op)s(erator)e Fp(\045)h FA(is)g(called)f(a)g(steady)i(state,)j(if)42 b Fp(\045)178 4828 y FA(comm)m(utes)35 b(with)g Fp(K)7 b Fq(\()p Fp(V)21 b Fq(\))p FA(,)36 b(see)g([9].)51 b(Th)m(us)37 b(an)m(y)e(steady)h(state)g Fp(\045)f FA(is)g(unitarily)d(equiv)-5 b(alen)m(t)35 b(to)g(a)178 4949 y(m)m(ultiplication)d(op)s(erator)45 b Fq(^)-58 b Fp(\045)37 b FA(on)f(the)h(Hilb)s(ert)e(space)i Fp(L)2273 4912 y Fl(2)2313 4949 y Fq(\()p Fo(R)5 b Fp(;)17 b Fo(C)2533 4912 y Fl(2)2578 4949 y Fq(\))37 b FA(induced)g(b)m(y)g(a)f (measurable)178 5069 y(function)e Fp(\045)p Fq(\()p Fn(\001)p Fq(\))d Fn(2)g Fp(L)910 5033 y Fk(1)986 5069 y Fq(\()p Fo(R)t Fp(;)17 b Fn(B)t Fq(\()p Fo(C)1311 5033 y Fl(2)1357 5069 y Fq(\)\))p FA(.)49 b(In)35 b(the)g(follo)m(wing)c(w)m(e)36 b(assume)f(that)f(the)h(function)f Fp(\045)p Fq(\()p Fn(\001)p Fq(\))g FA(is)178 5189 y(\034xed,)44 b(i.e.)65 b(the)41 b(function)e Fp(\045)p Fq(\()p Fn(\001)p Fq(\))h FA(do)s(es)h(not)e(dep)s(end)j(on)e(the)g(p)s(oten)m(tial)e Fp(V)22 b FA(.)66 b(This)40 b(leads)g(to)g(a)178 5310 y(steady)34 b(state)f(of)f(the)h(form)1412 5433 y Fp(\045)p Fq(\()p Fp(V)22 b Fq(\))28 b(=)f(\010)1818 5448 y Fk(\000)1878 5433 y Fq(\()p Fp(V)21 b Fq(\))2032 5392 y Fk(\000)p Fl(1)2135 5433 y Fq(^)-58 b Fp(\045)q Fq(\010)2247 5448 y Fk(\000)2306 5433 y Fq(\()p Fp(V)22 b Fq(\))p Fp(;)984 b FA(\(2.52\))178 5612 y(whic)m(h)33 b(dep)s(ends)h(on)f Fp(V)21 b FA(.)p eop %%Page: 12 14 12 13 bop 178 46 a FA(12)896 b Ft(M.)36 b(Bar)m(o,)f(H.-Ch.)47 b(Kaiser,)36 b(H.)g(Neidhardt,)f(J.)i(Rehber)m(g)296 295 y FA(In)31 b(order)f(to)f(de\034ne)i(the)g(carrier)e(densit)m(y)i Fp(u)1931 310 y Fl(^)-41 b Fj(\045)1965 295 y Fq(\()p Fp(V)21 b Fq(\)\()p Fn(\001)p Fq(\))29 b FA(one)i(has)f(\(in)f (accordance)i(with)e([9]\))h(to)178 415 y(in)m(tro)s(duce)j(the)g (carrier)f(densit)m(y)h(observ)-5 b(able)366 629 y Fp(D)s Fq(\()p Fp(V)21 b Fq(\)\()p Fp(x;)c(\025)p Fq(\))28 b(:=)2505 b FA(\(2.53\))650 703 y Fm(\022)1021 782 y Fn(j)p Fp( )1116 746 y Fk(\000)1175 782 y Fq(\()p Fp(V)21 b Fq(\)\()p Fp(x;)c(\025;)g(b)p Fq(\))p Fn(j)1674 746 y Fl(2)2053 782 y Fp( )2120 746 y Fk(\000)2179 782 y Fq(\()p Fp(V)k Fq(\)\()p Fp(x;)c(\025;)g(a)p Fq(\))p 2660 696 598 4 v Fp( )2727 753 y Fk(\000)2786 782 y Fq(\()p Fp(V)k Fq(\)\()p Fp(x;)c(\025;)g(b)p Fq(\))765 913 y Fp( )832 877 y Fk(\000)891 913 y Fq(\()p Fp(V)k Fq(\)\()p Fp(x;)c(\025;)g(b)p Fq(\))p 1362 826 608 4 v Fp( )1429 884 y Fk(\000)1488 913 y Fq(\()p Fp(V)22 b Fq(\)\()p Fp(x;)17 b(\025;)g(a)p Fq(\))348 b Fp( )2385 877 y Fk(\000)2444 913 y Fq(\()p Fp(V)21 b Fq(\)\()p Fp(x;)c(\025;)g(a)p Fq(\))p Fn(j)2953 877 y Fl(2)3299 703 y Fm(\023)3389 843 y Fp(:)178 1126 y FA(With)42 b(resp)s(ect)h(to)f(the)g(carrier)g(densit)m(y)h(observ)-5 b(able)42 b Fp(D)s Fq(\()p Fp(V)21 b Fq(\)\()p Fp(x;)c(\025)p Fq(\))42 b FA(one)g(de\034nes)i(the)e(carrier)178 1246 y(densit)m(y)34 b Fp(u)577 1261 y Fl(^)-41 b Fj(\045)611 1246 y Fq(\()p Fp(V)21 b Fq(\)\()p Fp(x;)c(\025)p Fq(\))32 b FA(at)h Fp(x)28 b Fn(2)g Fq([)p Fp(a;)17 b(b)p Fq(])33 b FA(and)g(at)f(energy)h Fp(\025)28 b Fn(2)g Fo(R)43 b FA(b)m(y)1123 1461 y Fp(u)1185 1476 y Fl(^)-41 b Fj(\045)1219 1461 y Fq(\()p Fp(V)21 b Fq(\)\()p Fp(x;)c(\025)p Fq(\))28 b(:=)f FA(tr)p Fq(\()p Fp(\045)p Fq(\()p Fp(\025)p Fq(\))p Fp(D)s Fq(\()p Fp(V)22 b Fq(\)\()p Fp(x;)17 b(\025)p Fq(\)\))27 b Fn(\025)h Fq(0)p Fp(:)695 b FA(\(2.54\))178 1675 y(The)34 b(carrier)e(densit)m(y)h Fp(u)1136 1687 y Fl(^)1080 1708 y Fj(\045)p Fl(\()p Fj(V)16 b Fl(\))1232 1675 y Fq(\()p Fn(\001)p Fq(\))32 b FA(is)g(giv)m(en)g(b)m(y)1309 1963 y Fp(u)1371 1978 y Fl(^)-41 b Fj(\045)1405 1963 y Fq(\()p Fp(V)22 b Fq(\)\()p Fp(x)p Fq(\))28 b(=)1822 1827 y Fm(Z)1877 2053 y Fg(R)1946 1963 y Fp(d\025)f(u)2143 1978 y Fl(^)-41 b Fj(\045)2177 1963 y Fq(\()p Fp(V)22 b Fq(\)\()p Fp(x;)17 b(\025)p Fq(\))p Fp(:)881 b FA(\(2.55\))178 2230 y(If)33 b(the)g(function)f Fp(\045)p Fq(\()p Fn(\001)p Fq(\))g FA(satis\034es)h(the)g(condition)1195 2444 y Fp(C)1271 2459 y Fl(^)-41 b Fj(\045)1333 2444 y Fq(:=)27 b(sup)1469 2526 y Fj(\025)p Fk(2)p Fg(R)1627 2357 y Fn(p)p 1710 2357 266 4 v 87 x Fp(\025)1767 2415 y Fl(2)1828 2444 y Fq(+)22 b(1)p Fn(k)p Fp(\045)p Fq(\()p Fp(\025)p Fq(\))p Fn(k)2258 2461 y Fk(B)r Fl(\()p Fg(C)2375 2442 y Ff(2)2416 2461 y Fl(\))2475 2444 y Fp(<)27 b Fn(1)p Fp(;)767 b FA(\(2.56\))178 2717 y(then)43 b(the)f(de\034nition)f (\(2.55\))g(mak)m(es)i(sense)g(for)f(a.e.)72 b Fp(x)44 b Fn(2)g Fq([)p Fp(a;)17 b(b)p Fq(])p FA(.)72 b(Moreo)m(v)m(er,)46 b(in)41 b(this)h(case)178 2838 y Fp(u)240 2853 y Fl(^)-41 b Fj(\045)274 2838 y Fq(\()p Fp(V)22 b Fq(\)\()p Fn(\001)p Fq(\))g FA(is)h(a)g(p)s(ositiv)m(e)p 716 2873 329 4 v 23 w(and)g(in)m(tegrable)p 1248 2873 420 4 v 23 w(function.)39 b(F)-8 b(urthermore,)25 b Fp(P)2770 2801 y Fd(K)2756 2863 y(H)2820 2838 y Fq(\()p Fp(K)7 b Fq(\()p Fp(V)21 b Fq(\))s Fn(\000)s Fp(i)p Fq(\))3256 2801 y Fk(\000)p Fl(1)3379 2838 y Fn(2)28 b Fi(L)3539 2853 y Fl(1)3579 2838 y Fq(\()p Fi(K)p Fq(\))178 2958 y FA(and)33 b(the)g(estimate)812 3172 y Fn(k)p Fp(u)924 3187 y Fl(^)-41 b Fj(\045)958 3172 y Fq(\()p Fp(V)21 b Fq(\))p Fn(k)1162 3189 y Fj(L)1210 3170 y Ff(1)1276 3172 y Fq(=)28 b FA(tr)p Fq(\()p Fp(\045)p Fq(\()p Fp(V)22 b Fq(\))p Fp(P)1776 3131 y Fd(K)1762 3197 y(H)1825 3172 y Fq(\))28 b Fn(\024)g Fp(C)2072 3187 y Fl(^)-41 b Fj(\045)2106 3172 y Fn(k)p Fq(\()p Fp(K)7 b Fq(\()p Fp(V)21 b Fq(\))i Fn(\000)f Fp(i)p Fq(\))2631 3131 y Fk(\000)p Fl(1)2726 3172 y Fp(P)2803 3131 y Fd(K)2789 3197 y(H)2852 3172 y Fn(k)2902 3188 y Fd(L)2948 3197 y Ff(1)2983 3188 y Fl(\()p Fd(K)p Fl(\))3472 3172 y FA(\(2.57\))178 3387 y(is)32 b(v)-5 b(alid,)31 b(cf.)44 b([9].)f(Let)33 b(us)g(in)m(tro)s(duce)f(the)h(op)s(erator)738 3611 y Fq(\()p Fp(M)10 b Fq(\()p Fp(h)p Fq(\))1026 3585 y Fp(~)1012 3611 y(f)i 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b(2.4)49 b Fc(Supp)-5 b(ose)37 b Fp(V)55 b Fn(2)33 b Fp(L)1627 259 y Fk(1)1702 295 y Fc(.)54 b(If)37 b(the)h(density)g(matrix)g Fp(\045)g Fc(satis\034es)f(the)h(c) -5 b(ondition)178 415 y(\(2.56\),)34 b(then)545 669 y Fn(kN)683 684 y Fl(^)-41 b Fj(\045)717 669 y Fq(\()p Fp(V)21 b Fq(\))p Fn(k)921 686 y Fj(L)969 667 y Ff(1)1036 669 y Fn(\024)28 b Fp(C)1217 684 y Fl(^)-41 b Fj(\045)1273 669 y Fn(\001)1323 558 y Fm(h)1370 669 y Fq(8)22 b(+)g(4)1588 582 y Fn(p)p 1671 582 49 4 v 87 x Fq(2)27 b(\()p Fp(b)c Fn(\000)f Fp(a)p Fq(\))2037 579 y Fm(p)p 2137 579 185 4 v 90 x Fn(k)p Fp(m)p Fn(k)g(\001)2394 579 y Fm(p)p 2494 579 724 4 v 90 x Fq(8)p Fp(c)f Fq(+)h(5)g(+)g(4)p Fn(k)p Fp(V)g Fn(k)3101 684 y Fj(L)3149 665 y Fh(1)3472 669 y FA(\(2.61\))829 881 y Fq(+8)p Fi(g)1004 896 y Fl(1)1044 881 y Fq(\()p Fp(\013)1145 840 y Fl(2)1144 906 y Fj(a)1207 881 y Fq(+)g Fp(\013)1368 840 y Fl(2)1367 906 y Fj(b)1407 801 y Fm(\001)1453 823 y Fl(1)p Fj(=)p Fl(2)1563 881 y Fn(k)p Fp(H)1702 831 y Fk(\000)p Fl(1)p Fj(=)p Fl(2)1694 906 y(0)1866 881 y Fn(k)1916 831 y Fl(1)p Fj(=)p Fl(2)1916 914 y Fk(B)r Fl(\()p Fd(H)p Fl(;)p Fj(W)2138 895 y Ff(1)p Fe(;)p Ff(2)2221 914 y Fl(\))2253 881 y Fq(\(8)p Fp(c)g Fq(+)g(5)f(+)i(4)p Fn(k)p Fp(V)e Fn(k)2898 896 y Fj(L)2946 878 y Fh(1)3015 881 y Fq(\))3053 840 y Fl(1)p Fj(=)p Fl(4)3163 771 y Fm(i)3210 881 y Fp(;)178 1135 y Fc(wher)-5 b(e)34 b Fp(c)h Fc(is)g(de\034ne)-5 b(d)34 b(by)h(\(2.21\).)178 1395 y Fr(Pro)s(of.)41 b FA(In)24 b(view)g(of)f(\(2.57\))g(it)g (su\036ces)i(to)f(estimate)e Fn(k)p Fq(\()p Fp(K)7 b Fq(\()p Fp(V)22 b Fq(\))g Fn(\000)h Fp(i)p Fq(\))2655 1358 y Fk(\000)p Fl(1)2749 1395 y Fp(P)2826 1358 y Fd(K)2812 1420 y(H)2876 1395 y Fn(k)2926 1410 y Fd(L)2972 1419 y Ff(1)3006 1410 y Fl(\()p Fd(K)p Fl(\))3111 1395 y FA(.)41 b(Using)24 b(\(2.40\),)178 1515 y(w)m(e)34 b(obtain)d(the)i(equation) 783 1669 y Fm(\000)829 1750 y Fp(K)7 b Fq(\()p Fp(V)22 b Fq(\))g Fn(\000)g Fp(i)1228 1669 y Fm(\001)1274 1691 y Fk(\000)p Fl(1)1369 1750 y Fp(P)1446 1709 y Fd(K)1432 1774 y(H)1509 1724 y Fp(~)1495 1750 y(f)39 b Fq(=)1685 1669 y Fm(\000)1731 1750 y Fq(0)p Fp(;)17 b Fq(\()p Fp(H)8 b Fq(\()p Fp(V)20 b Fq(\))i Fn(\000)h Fp(i)p Fq(\))2297 1709 y Fk(\000)p Fl(1)2392 1750 y Fp(f)5 b(;)17 b(e)2534 1709 y Fk(\000\017)2645 1750 y Fp(iT)d Fq(\()p Fp(V)21 b Fq(\)\()p Fp(i)p Fq(\))p Fp(f)3071 1669 y Fm(\001)178 1996 y FA(where)474 1970 y Fp(~)460 1996 y(f)38 b Fq(=)28 b Fp(f)698 2011 y Fk(\000)779 1996 y Fn(\010)23 b Fp(f)33 b Fn(\010)22 b Fp(f)1107 2011 y Fl(+)1167 1996 y FA(.)43 b(Th)m(us,)34 b(one)f(can)g(estimate)286 2234 y Fn(k)336 2153 y Fm(\000)381 2234 y Fp(K)7 b Fq(\()p Fp(V)22 b Fq(\))g Fn(\000)g Fp(i)780 2153 y Fm(\001)826 2175 y Fk(\000)p Fl(1)921 2234 y Fp(P)998 2193 y Fd(K)984 2258 y(H)1047 2234 y Fn(k)1097 2249 y Fd(L)1143 2258 y Ff(1)1178 2249 y Fl(\()p Fd(K)p Fl(\))1310 2234 y Fn(\024)29 b(k)1466 2153 y Fm(\000)1511 2234 y Fp(H)8 b Fq(\()p Fp(V)21 b Fq(\))h Fn(\000)h Fp(i)p Fq(\))1947 2193 y Fk(\000)p Fl(1)2041 2234 y Fn(k)2091 2249 y Fd(L)2137 2258 y Ff(1)2172 2249 y Fl(\()p Fd(H)p Fl(\))2303 2234 y Fq(+)f Fn(k)p Fp(ie)2529 2193 y Fk(\000\017)2640 2234 y Fp(T)14 b Fq(\()p Fp(V)21 b Fq(\)\()p Fp(i)p Fq(\))p Fn(k)3024 2249 y Fd(L)3070 2258 y Ff(1)3104 2249 y Fl(\()p Fd(H)p Fl(;)p Fk(D)3256 2258 y Ff(+)3306 2249 y Fl(\))3338 2234 y Fp(:)107 b FA(\(2.62\))178 2460 y(W)-8 b(e)38 b(estimate)f(the)h(\034rst)g(addend) g(on)f(the)h(righ)m(t)f(hand)h(side.)58 b(Let)38 b Fp(\026)f FA(b)s(e)g(a)h(su\036cien)m(tly)g(large)178 2581 y(p)s(ositiv)m(e)32 b(n)m(um)m(b)s(er)h(\(to)f(b)s(e)h(sp)s(eci\034ed)g(later\).)43 b(W)-8 b(e)33 b(write)719 2835 y Fq(\()p Fp(H)8 b Fq(\()p Fp(V)21 b Fq(\))h Fn(\000)g Fp(i)p Fq(\))1192 2793 y Fk(\000)p Fl(1)1315 2835 y Fq(=)27 b(\()p Fp(H)8 b Fq(\()p Fp(V)21 b Fq(\))h(+)g Fp(\026)p Fq(\))1916 2793 y Fk(\000)p Fl(1)2010 2724 y Fm(\020)2070 2835 y Fq(1)g(+)g(\()p Fp(\026)f Fq(+)h Fp(i)p Fq(\)\()p Fp(H)8 b Fq(\()p Fp(V)21 b Fq(\))i Fn(\000)f Fp(i)p Fq(\))3000 2793 y Fk(\000)p Fl(1)3095 2724 y Fm(\021)3154 2835 y Fp(:)291 b FA(\(2.63\))178 3093 y(Since)25 b Fp(H)8 b Fq(\()p Fp(V)21 b Fq(\))k FA(is)f(a)g(maximal)e(dissipativ)m(e)i(op)s(erator)f(one)i(has)g Fn(k)p Fq(\()p Fp(H)8 b Fq(\()p Fp(V)21 b Fq(\))6 b Fn(\000)g Fp(i)p Fq(\))2929 3056 y Fk(\000)p Fl(1)3024 3093 y Fn(k)3074 3108 y Fk(B)r Fl(\()p Fd(H)p Fl(\))3258 3093 y Fn(\024)28 b Fq(1)p FA(.)41 b(Th)m(us,)178 3213 y(\(2.63\))32 b(implies)849 3439 y Fn(k)p Fq(\()p Fp(H)8 b Fq(\()p Fp(V)21 b Fq(\))h Fn(\000)h Fp(i)p Fq(\))1373 3398 y Fk(\000)p Fl(1)1467 3439 y Fn(k)1517 3455 y Fd(L)1563 3464 y Ff(1)1598 3455 y Fl(\()p Fd(H)p Fl(\))1734 3439 y Fn(\024)28 b Fq(\(2)22 b(+)g Fp(\026)p Fq(\))p Fn(k)p Fq(\()p Fp(H)8 b Fq(\()p Fp(V)20 b Fq(\))j(+)f Fp(\026)p Fq(\))2691 3398 y Fk(\000)p Fl(1)2785 3439 y Fn(k)2835 3455 y Fd(L)2881 3464 y Ff(1)2915 3455 y Fl(\()p Fd(H)p Fl(\))3024 3439 y Fp(:)421 b FA(\(2.64\))178 3666 y(Applying)32 b(the)h(factorization)d(form)m(ula)h(\(2.28\))g(one) i(gets)580 3892 y Fn(k)p Fq(\()p Fp(H)8 b Fq(\()p Fp(V)21 b Fq(\))h(+)g Fp(\026)p Fq(\))1128 3851 y Fk(\000)p Fl(1)1222 3892 y Fn(k)1272 3908 y Fd(L)1318 3917 y Ff(1)1352 3908 y Fl(\()p Fd(H)p Fl(\))1489 3892 y Fn(\024)28 b(k)p Fq(\()p Fp(H)1763 3907 y Fl(0)1824 3892 y Fq(+)22 b Fp(\026)p Fq(\))2019 3851 y Fk(\000)p Fl(1)p Fj(=)p Fl(2)2183 3892 y Fn(k)2233 3851 y Fl(2)2233 3917 y Fd(L)2279 3926 y Ff(2)2314 3917 y Fl(\()p Fd(H)p Fl(\))2422 3892 y Fn(k)p Fq(\(1)g(+)g Fp(B)2753 3907 y Fj(\026)2800 3892 y Fq(\()p Fp(V)f Fq(\)\))2992 3851 y Fk(\000)p Fl(1)3086 3892 y Fn(k)3136 3908 y Fk(B)r Fl(\()p Fd(H)p Fl(\))3293 3892 y Fp(:)178 4130 y FA(The)j(\034rst)f(factor)e(of)h(the)h(righ)m(t)f (hand)g(side)h(is)f(calculated)f(to)2379 4055 y Fm(P)2484 4082 y Fk(1)2484 4159 y Fj(l)q Fl(=0)2731 4091 y(1)p 2627 4107 243 4 v 2627 4164 a Fj(\020)2658 4176 y Fe(l)2682 4164 y Fl(+1+)p Fj(\026)2880 4130 y FA(,)j(where)g(the)f(n)m(um)m(b)s (ers)178 4280 y Fp(\020)221 4295 y Fj(l)280 4280 y FA(are,)34 b(of)f(course,)h(the)g(eigen)m(v)-5 b(alues)33 b(of)g(the)g(op)s (erator)g Fp(H)2338 4295 y Fl(0)2400 4280 y Fn(\000)23 b Fq(1)p FA(.)45 b(Let)2822 4255 y Fq(^)2797 4280 y Fp(H)2878 4295 y Fl(0)2950 4280 y FA(b)s(e)34 b(the)f(self-adjoin)m(t)178 4400 y(op)s(erator)f(de\034ned)i(b)m(y)g(\(2.13\))e(and)h(\(2.14\))e (where)j Fp(m)f FA(is)f(sp)s(eci\034ed)i(to)e Fp(m)p Fq(\()p Fp(x)p Fq(\))d Fn(\021)f Fq(1)p FA(.)44 b(Ob)m(viously)-8 b(,)178 4521 y(one)43 b(has)609 4481 y Fl(1)p 561 4498 133 4 v 561 4555 a Fk(k)p Fj(m)p Fk(k)704 4521 y Fq(\()767 4495 y(^)742 4521 y Fp(H)823 4536 y Fl(0)891 4521 y Fn(\000)29 b Fq(1\))44 b Fn(\024)h Fp(H)1331 4536 y Fl(0)1399 4521 y Fn(\000)29 b Fq(1)p FA(.)73 b(The)43 b(eigen)m(v)-5 b(alues)42 b(of)2524 4495 y Fq(^)2499 4521 y Fp(H)2580 4536 y Fl(0)2648 4521 y Fn(\000)29 b Fq(1)42 b FA(are)h(giv)m(en)f(b)m (y)3494 4481 y Fj(\031)3537 4458 y Ff(2)3571 4481 y Fj(l)3593 4458 y Ff(2)p 3437 4498 248 4 v 3437 4555 a Fl(2\()p Fj(b)p Fk(\000)p Fj(a)p Fl(\))3648 4536 y Ff(2)3695 4521 y FA(,)178 4675 y Fp(l)30 b Fq(=)e(0)p Fp(;)17 b Fq(1)p Fp(;)g(:)g(:)g(:)n FA(.)43 b(Th)m(us)33 b(the)e(minimax)d(principle)h (implies)2311 4635 y Fl(1)p 2263 4652 133 4 v 2263 4709 a Fk(k)p Fj(m)p Fk(k)2472 4635 y Fj(\031)2515 4612 y Ff(2)2550 4635 y Fj(l)2572 4612 y Ff(2)p 2416 4652 248 4 v 2416 4709 a Fl(2\()p Fj(b)p Fk(\000)p Fj(a)p Fl(\))2627 4690 y Ff(2)2700 4675 y Fn(\024)g Fp(\020)2849 4690 y Fj(l)2874 4675 y FA(,)j Fp(l)d Fq(=)f(0)p Fp(;)17 b Fq(1)p Fp(;)g(:)g(:)g(:)n FA(.)43 b(Hence)178 4795 y(w)m(e)34 b(obtain)946 4973 y Fn(k)p Fq(\()p Fp(H)1115 4988 y Fl(0)1176 4973 y Fq(+)22 b Fp(\026)p Fq(\))1371 4932 y Fk(\000)p Fl(1)p Fj(=)p Fl(2)1536 4973 y Fn(k)1586 4932 y Fl(2)1586 4998 y Fd(L)1632 5007 y Ff(2)1666 4998 y Fl(\()p Fd(H)p Fl(\))1803 4973 y Fn(\024)1944 4848 y Fk(1)1908 4878 y Fm(X)1924 5090 y Fj(l)q Fl(=0)2473 4906 y Fq(1)p 2078 4950 839 4 v 2137 5013 a Fl(1)p 2088 5029 133 4 v 2088 5087 a Fk(k)p Fj(m)p Fk(k)2326 5013 y Fj(\031)2369 4994 y Ff(2)p 2241 5029 248 4 v 2241 5087 a Fl(2\()p Fj(b)p Fk(\000)p Fj(a)p Fl(\))2452 5068 y Ff(2)2498 5052 y Fp(l)2529 5023 y Fl(2)2591 5052 y Fq(+)g(1)g(+)g Fp(\026)2927 4973 y(:)178 5242 y FA(F)-8 b(or)32 b(an)m(y)h Fp(l)d Fn(\025)e Fq(1)33 b FA(w)m(e)g(ha)m(v)m(e)1333 5464 y Fq(1)p 955 5508 804 4 v 1014 5571 a Fl(1)p 965 5588 133 4 v 965 5645 a Fk(k)p Fj(m)p Fk(k)1168 5571 y Fl(4)p Fj(\031)1246 5552 y Ff(2)p 1118 5588 213 4 v 1118 5645 a Fl(\()p Fj(b)p Fk(\000)p Fj(a)p Fl(\))1294 5626 y Ff(2)1340 5611 y Fp(l)1371 5582 y Fl(2)1433 5611 y Fq(+)22 b(1)g(+)g Fp(\026)1796 5531 y Fn(\024)1901 5396 y Fm(Z)2001 5422 y Fj(l)1957 5621 y(l)q Fk(\000)p Fl(1)2460 5464 y Fp(ds)p 2100 5508 819 4 v 2158 5571 a Fl(1)p 2110 5588 133 4 v 2110 5645 a Fk(k)p Fj(m)p Fk(k)2312 5571 y Fl(4)p Fj(\031)2390 5552 y Ff(2)p 2262 5588 213 4 v 2262 5645 a Fl(\()p Fj(b)p Fk(\000)p Fj(a)p Fl(\))2438 5626 y Ff(2)2484 5611 y Fp(s)2530 5582 y Fl(2)2592 5611 y Fq(+)g(1)g(+)g Fp(\026)2928 5531 y(:)p eop %%Page: 14 16 14 15 bop 178 46 a FA(14)896 b Ft(M.)36 b(Bar)m(o,)f(H.-Ch.)47 b(Kaiser,)36 b(H.)g(Neidhardt,)f(J.)i(Rehber)m(g)178 295 y FA(Th)m(us,)e(w)m(e)e(get)661 584 y Fn(k)p Fq(\()p Fp(H)830 599 y Fl(0)891 584 y Fq(+)22 b Fp(\026)p Fq(\))1086 543 y Fk(\000)p Fl(1)p Fj(=)p Fl(2)1251 584 y Fn(k)1301 543 y Fl(2)1301 609 y Fd(L)1347 618 y Ff(2)1381 609 y Fl(\()p Fd(H)p Fl(\))1517 584 y Fn(\024)1722 517 y Fq(1)p 1632 561 228 4 v 1632 652 a(1)g(+)g Fp(\026)1892 584 y Fq(+)2027 459 y Fk(1)1990 489 y Fm(X)2006 701 y Fj(l)q Fl(=1)2151 448 y Fm(Z)2250 475 y Fj(l)2206 674 y(l)q Fk(\000)p Fl(1)2727 517 y Fp(ds)p 2349 561 854 4 v 2408 624 a Fl(1)p 2359 640 133 4 v 2359 698 a Fk(k)p Fj(m)p Fk(k)2597 624 y Fj(\031)2640 605 y Ff(2)p 2512 640 248 4 v 2512 698 a Fl(2\()p Fj(b)p Fk(\000)p Fj(a)p Fl(\))2723 679 y Ff(2)2769 663 y Fp(s)2815 634 y Fl(2)2876 663 y Fq(+)g(1)g(+)g Fp(\026)3212 584 y(;)178 889 y FA(whic)m(h)33 b(yields)797 1152 y Fn(k)p Fq(\()p Fp(H)966 1167 y Fl(0)1027 1152 y Fq(+)22 b Fp(\026)p Fq(\))1222 1111 y Fk(\000)p Fl(1)p Fj(=)p 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Fl(2)1744 3693 y Fj(a)1808 3669 y Fq(+)g Fp(\013)1969 3627 y Fl(2)1968 3693 y Fj(b)2008 3669 y Fq(\))2046 3627 y Fl(1)p Fj(=)p Fl(2)2156 3669 y Fn(k)p Fq(\()p Fp(H)8 b Fq(\()p Fp(V)21 b Fq(\))h Fn(\000)g Fp(i)p Fq(\))2679 3627 y Fk(\000)p Fl(1)2774 3669 y Fn(k)2824 3684 y Fk(B)r Fl(\()p Fd(H)p Fl(;)p Fj(C)5 b Fl([)p Fj(a;b)p Fl(]\))3182 3669 y Fp(:)263 b FA(\(2.66\))178 3887 y(It)46 b(remains)e(to)h(estimate)f Fn(k)p Fq(\()p Fp(H)8 b Fq(\()p Fp(V)21 b Fq(\))31 b Fn(\000)g Fp(i)p Fq(\))1749 3851 y Fk(\000)p Fl(1)1844 3887 y Fn(k)1894 3903 y Fk(B)r Fl(\()p Fd(H)p Fl(;)p Fj(C)5 b Fl([)p Fj(a;b)p Fl(]\))2251 3887 y FA(.)82 b(T)-8 b(aking)44 b(in)m(to)h(accoun)m(t)h(\(2.63\))e (one)178 4008 y(obtains)32 b(\(analogous)f(to)i(\(2.64\)\))681 4227 y Fn(k)p Fq(\()p Fp(H)8 b Fq(\()p Fp(V)21 b Fq(\))h Fn(\000)g Fp(i)p Fq(\))1204 4185 y Fk(\000)p Fl(1)1299 4227 y Fn(k)1349 4242 y Fk(B)r Fl(\()p Fd(H)p Fl(;)p Fj(C)5 b Fl([)p Fj(a;b)p Fl(]\))1734 4227 y Fn(\024)28 b Fq(\(2)22 b(+)g Fp(\026)p Fq(\))p 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1361 y Fn(\024)28 b Fi(g)2152 1376 y Fl(1)2201 1273 y Fn(k)p Fp(H)2340 1222 y Fk(\000)p Fl(1)p Fj(=)p Fl(2)2332 1297 y(0)2505 1273 y Fn(k)2555 1222 y Fl(1)p Fj(=)p Fl(2)2555 1306 y Fk(B)r Fl(\()p Fd(H)p Fj(;W)2777 1287 y Ff(1)p Fe(;)p Ff(2)2859 1306 y Fl(\))p 2201 1338 V 2340 1429 a Fq(\(1)21 b(+)h Fp(\026)p Fq(\))2643 1401 y Fl(1)p Fj(=)p Fl(4)2901 1361 y Fp(:)178 1599 y FA(T)-8 b(ogether)33 b(with)g(\(2.69\))e(this)h(giv)m(es)730 1794 y Fn(k)p Fq(\()p Fp(H)8 b Fq(\()p Fp(V)21 b Fq(\))h Fn(\000)g Fp(i)p Fq(\))1253 1753 y Fk(\000)p Fl(1)1348 1794 y Fn(k)1398 1809 y Fd(B)p Fl(\()p Fd(H)p Fj(;C)5 b Fl([)p Fj(a;b)p Fl(]\))1796 1794 y Fn(\024)28 b Fq(8)p Fi(g)2000 1809 y Fl(1)2040 1794 y Fn(k)p Fp(H)2179 1743 y Fk(\000)p Fl(1)p Fj(=)p Fl(2)2171 1818 y(0)2343 1794 y Fn(k)2393 1743 y Fl(1)p Fj(=)p Fl(2)2393 1827 y Fk(B)r Fl(\()p Fd(H)p Fj(;W)2615 1808 y Ff(1)p Fe(;)p Ff(2)2698 1827 y Fl(\))2729 1794 y Fq(\(1)22 b(+)g Fp(\026)p Fq(\))3033 1753 y Fl(1)p Fj(=)p Fl(4)3143 1794 y Fp(:)178 1988 y FA(Inserting)36 b(the)h(c)m(hosen)g Fp(\026)c Fq(=)g(2\(4)p Fp(c)24 b Fq(+)g(2)g(+)h(2)p Fn(k)p Fp(V)c Fn(k)1982 2003 y Fj(L)2030 1984 y Fh(1)2099 1988 y Fq(\))p FA(,)36 b(w)m(e)h(obtain)e(for)g(the)i(second)g(addend)f(in)178 2108 y(the)d(righ)m(t)f(hand)h(side)f(of)g(\(2.62\):)370 2290 y Fn(k)p Fp(ie)498 2249 y Fk(\000\017)609 2290 y Fp(T)14 b Fq(\()p Fp(V)21 b Fq(\)\()p Fp(i)p Fq(\))p Fn(k)993 2305 y Fd(L)1039 2314 y Ff(1)1074 2305 y Fl(\()p Fd(H)p Fj(;)p Fk(D)1226 2314 y Ff(+)1276 2305 y Fl(\))772 2463 y Fn(\024)28 b Fq(8)926 2376 y Fn(p)p 1009 2376 49 4 v 87 x Fq(2)p Fi(g)1108 2478 y Fl(1)1148 2463 y Fq(\()p Fp(\013)1249 2422 y Fl(2)1248 2488 y Fj(a)1311 2463 y Fq(+)22 b Fp(\013)1472 2422 y Fl(2)1471 2488 y Fj(b)1512 2383 y Fm(\001)1557 2405 y Fl(1)p Fj(=)p Fl(2)1667 2463 y Fn(k)p Fp(H)1806 2412 y Fk(\000)p Fl(1)p Fj(=)p Fl(2)1798 2488 y(0)1970 2463 y Fn(k)2020 2412 y Fl(1)p Fj(=)p Fl(2)2020 2496 y Fk(B)r Fl(\()p Fd(H)p Fj(;W)2242 2477 y Ff(1)p Fe(;)p Ff(2)2325 2496 y Fl(\))2357 2463 y Fq(\(8)p Fp(c)g Fq(+)g(5)g(+)g(4)p Fn(k)p Fp(V)f Fn(k)3002 2478 y Fj(L)3050 2459 y Fh(1)3119 2463 y Fq(\))3157 2422 y Fl(1)p Fj(=)p Fl(4)3267 2463 y Fp(:)178 2657 y FA(Fitting)30 b(together)j(\(2.57\),)f(\(2.62\),)g(\(2.65\))g(and)g(this)g(last)g (estimate)g(w)m(e)h(pro)m(v)m(e)h(\(2.61\).)214 b Fb(\003)178 2889 y Fr(Remark)37 b(2.5)49 b FA(One)33 b(can)f(ev)m(en)i(pro)m(v)m(e) f(that)f(the)h(carrier)e(densit)m(y)i(op)s(erator)f(tak)m(es)h(its)f(v) -5 b(alues)178 3009 y(not)41 b(only)f(in)g Fp(L)770 2973 y Fl(1)851 3009 y FA(but)h(in)f(\(real\))f Fp(L)1499 2973 y Fl(2)1539 3009 y FA(.)68 b(A)m(dditionally)-8 b(,)40 b(one)h(can)g(pro)m(v)m(e)h(estimates)f(similar)c(to)178 3129 y(\(2.61\))45 b(but)h(more)f(in)m(v)m(olv)m(ed.)83 b(W)-8 b(e)46 b(do)f(not)h(need)g(these)h(things)e(in)g(this)g(pap)s (er,)k(ho)m(w)m(ev)m(er,)178 3250 y(this)29 b(fact)g(b)s(ecomes)g (essen)m(tial)g(in)f(the)h(momen)m(t)f(when)i(one)f(w)m(an)m(ts)i(to)d (include)h(recom)m(bination)178 3370 y(e\033ects)34 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Fp(\045)p Fq(\()p Fp(H)8 b Fq(\()p Fp(V)21 b Fq(\)\))33 b Fc(is)g(a)f(c)-5 b(omp)g(act)32 b(subset,)i(then)e(for)h(su\036ciently)g(lar)-5 b(ge)33 b Fp(n)28 b Fn(2)g Fo(N)9 b Fc(,)39 b(one)32 b(has)422 4980 y Fn(C)i(\032)28 b Fp(\045)p Fq(\()p Fp(H)8 b Fq(\()p Fp(V)885 4995 y Fj(n)932 4980 y Fq(\))p Fc(,)1016 5162 y Fq(lim)992 5222 y Fj(n)p Fk(!1)1192 5162 y Fq(sup)1204 5243 y Fj(z)s Fk(2C)1356 5077 y Fm(\015)1356 5137 y(\015)1411 5162 y Fq(\()p Fp(H)g Fq(\()p Fp(V)1633 5177 y Fj(n)1679 5162 y Fq(\))22 b Fn(\000)h Fp(z)t Fq(\))1926 5121 y Fk(\000)p Fl(1)2043 5162 y Fn(\000)g Fq(\()p Fp(H)8 b Fq(\()p Fp(V)21 b Fq(\))h Fn(\000)h Fp(z)t Fq(\))2633 5121 y Fk(\000)p Fl(1)2728 5077 y Fm(\015)2728 5137 y(\015)2783 5201 y Fd(L)2829 5210 y Ff(1)2863 5201 y Fl(\()p Fd(H)p Fl(\))3000 5162 y Fq(=)k(0)369 b FA(\(3.1\))422 5396 y Fc(and)1191 5516 y Fq(lim)1167 5576 y Fj(n)p Fk(!1)1367 5516 y Fq(sup)1379 5598 y Fj(z)s Fk(2C)1531 5516 y Fn(k)p Fp(T)14 b Fq(\()p Fp(V)1747 5531 y Fj(n)1793 5516 y Fq(\)\()p Fp(z)t Fq(\))23 b Fn(\000)f Fp(T)14 b Fq(\()p Fp(V)21 b Fq(\)\()p Fp(z)t Fq(\))p Fn(k)2478 5533 y Fd(L)2524 5542 y Ff(1)2559 5533 y Fl(\()p Fd(H)p Fj(;)p Fg(C)2698 5514 y Ff(2)2739 5533 y Fl(\))2798 5516 y Fq(=)27 b(0)p Fp(:)544 b FA(\(3.2\))p eop %%Page: 16 18 16 17 bop 178 46 a FA(16)896 b Ft(M.)36 b(Bar)m(o,)f(H.-Ch.)47 b(Kaiser,)36 b(H.)g(Neidhardt,)f(J.)i(Rehber)m(g)234 295 y Fc(\(ii\))48 b(If)32 b Fn(C)i(\032)28 b Fp(\045)p Fq(\()p Fp(H)8 b Fq(\()p Fp(V)21 b Fq(\))1044 259 y Fk(\003)1084 295 y Fq(\))32 b Fc(is)g(a)g(c)-5 b(omp)g(act)31 b(subset,)i(then)f (for)g(su\036ciently)g(lar)-5 b(ge)31 b Fp(n)d Fn(2)g Fo(N)48 b Fc(one)31 b(has)422 415 y Fn(C)j(\032)28 b Fp(\045)p Fq(\()p Fp(H)8 b Fq(\()p Fp(V)885 430 y Fj(n)932 415 y Fq(\))970 379 y Fk(\003)1009 415 y Fq(\))p Fc(,)977 622 y Fq(lim)952 682 y Fj(n)p Fk(!1)1153 622 y Fq(sup)1165 704 y Fj(z)s Fk(2C)1316 538 y Fm(\015)1316 597 y(\015)1372 622 y Fq(\()p Fp(H)g Fq(\()p Fp(V)1594 637 y Fj(n)1640 622 y Fq(\))1678 581 y Fk(\003)1739 622 y Fn(\000)23 b Fp(z)t Fq(\))1926 581 y Fk(\000)p Fl(1)2043 622 y Fn(\000)g Fq(\()p Fp(H)8 b Fq(\()p Fp(V)21 b Fq(\))2424 581 y Fk(\003)2485 622 y Fn(\000)i Fp(z)t Fq(\))2672 581 y Fk(\000)p Fl(1)2767 538 y Fm(\015)2767 597 y(\015)2822 661 y Fd(L)2868 670 y Ff(1)2903 661 y Fl(\()p Fd(H)p Fl(\))3039 622 y Fq(=)28 b(0)329 b FA(\(3.3\))422 882 y Fc(and)1165 1002 y Fq(lim)1141 1062 y Fj(n)p Fk(!1)1341 1002 y Fq(sup)1353 1083 y Fj(z)s Fk(2C)1505 1002 y Fn(k)p Fp(T)1612 1017 y Fk(\003)1651 1002 y Fq(\()p Fp(V)1746 1017 y Fj(n)1793 1002 y Fq(\)\()p Fp(z)t Fq(\))23 b Fn(\000)f Fp(T)2135 1017 y Fk(\003)2175 1002 y Fq(\()p Fp(V)f Fq(\)\()p Fp(z)t Fq(\))p Fn(k)2504 1019 y Fd(L)2550 1028 y Ff(1)2585 1019 y Fl(\()p Fd(H)p Fj(;)p Fg(C)2724 1000 y Ff(2)2765 1019 y Fl(\))2824 1002 y Fq(=)27 b(0)p Fp(:)518 b FA(\(3.4\))204 1283 y Fc(\(iii\))48 b(If)34 b Fn(C)g(\032)p 716 1202 131 4 v 29 w Fo(C)782 1298 y Fk(\000)881 1283 y Fc(is)h(a)g(c)-5 b(omp)g(act)34 b(set,)g(then)1195 1490 y Fq(lim)1171 1550 y Fj(n)p Fk(!1)1371 1490 y Fq(sup)1383 1571 y Fj(z)s Fk(2C)1535 1490 y Fn(k)p Fq(\002)1661 1505 y Fj(H)5 b Fl(\()p Fj(V)1792 1513 y Fe(n)1835 1505 y Fl(\))1866 1490 y Fq(\()p Fp(z)t Fq(\))23 b Fn(\000)g Fq(\002)2190 1505 y Fj(H)5 b Fl(\()p Fj(V)16 b Fl(\))2368 1490 y Fq(\()p Fp(z)t Fq(\))p Fn(k)2543 1507 y Fd(L)2589 1516 y Ff(1)2624 1507 y Fl(\()p Fg(C)2694 1488 y Ff(2)2734 1507 y Fl(\))2794 1490 y Fq(=)27 b(0)p Fp(:)548 b FA(\(3.5\))422 1769 y Fc(Similarly,)34 b(if)h Fn(C)f(\032)p 1155 1689 V 28 w Fo(C)1221 1784 y Fl(+)1321 1769 y Fc(is)g(a)h(c)-5 b(omp)g(act)34 b(set,)h(then)1159 1976 y Fq(lim)1135 2036 y Fj(n)p Fk(!1)1335 1976 y Fq(sup)1347 2058 y Fj(z)s Fk(2C)1499 1976 y Fn(k)p Fq(\002)1625 1992 y Fj(H)5 b Fl(\()p Fj(V)1756 2000 y Fe(n)1799 1992 y Fl(\))1826 1973 y Fh(\003)1866 1976 y Fq(\()p Fp(z)t Fq(\))23 b Fn(\000)g Fq(\002)2190 1992 y Fj(H)5 b Fl(\()p Fj(V)16 b Fl(\))2364 1973 y Fh(\003)2404 1976 y Fq(\()p Fp(z)t Fq(\))p Fn(k)2579 1993 y Fd(L)2625 2002 y Ff(1)2660 1993 y Fl(\()p Fg(C)2730 1974 y Ff(2)2770 1993 y Fl(\))2830 1976 y Fq(=)27 b(0)p Fp(:)512 b FA(\(3.6\))178 2282 y Fr(Pro)s(of.)41 b FA(\(i\))26 b(By)g(Theorem)h(IV)f(1.16)g(in)f([10])h (w)m(e)h(get)f(that)g Fp(H)8 b Fq(\()p Fp(V)2505 2297 y Fj(n)2552 2282 y Fq(\))h Fn(\000)g Fp(z)31 b FA(is)26 b(b)s(oundedly)g(in)m(v)m(ertible)178 2402 y(if)1028 2609 y Fn(k)p Fp(V)43 b Fn(\000)23 b Fp(V)1335 2624 y Fj(n)1382 2609 y Fn(k)1432 2624 y Fj(L)1480 2605 y Fh(1)1576 2609 y Fp(<)1680 2529 y Fm(\000)1725 2609 y Fn(k)p Fq(\()p Fp(H)8 b Fq(\()p Fp(V)21 b Fq(\))h Fn(\000)h Fp(z)t Fq(\))2265 2568 y Fk(\000)p Fl(1)2360 2609 y Fn(k)2410 2625 y Fk(B)r Fl(\()p Fd(H)p Fl(\))2566 2529 y Fm(\001)2612 2551 y Fk(\000)p Fl(1)3521 2609 y FA(\(3.7\))178 2817 y(\(what)33 b(is)f(true)h(for)f Fp(n)c Fn(2)g Fo(N)48 b FA(large)31 b(enough\).)44 b(F)-8 b(urthermore,)32 b(w)m(e)h(get)g(in)f(this)g (case)661 3087 y Fn(k)p Fq(\()p Fp(H)8 b Fq(\()p Fp(V)933 3102 y Fj(n)979 3087 y Fq(\))22 b Fn(\000)h Fp(z)t 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FA(equation)h(\(3.10\))g(exten)m(ts)i(to)d(all)g Fp(z)43 b Fn(2)p 2731 3334 V 39 w Fo(C)2797 3429 y Fk(\000)2862 3414 y FA(.)64 b(Uniformit)m(y)37 b(o)m(v)m(er)j(a)178 3535 y(compact)e(subset)j(of)d Fp(z)1046 3498 y Fk(0)1070 3535 y Fp(s)g FA(is)g(deriv)m(ed)i(b)m(y)f(the)g(con)m(tin)m(uit)m(y)g (of)f(the)h(map)p 2902 3454 72 4 v 38 w Fo(C)64 b Fn(3)38 b Fp(z)43 b Fn(7!)37 b Fq(\002)3417 3550 y Fj(H)5 b Fl(\()p Fj(V)16 b Fl(\))3596 3535 y Fq(\()p Fp(z)t Fq(\))178 3655 y FA(and)31 b(a)f(simple)f(compactness)j(argumen)m(t.)42 b(Since)31 b Fq(\002)2112 3670 y Fj(H)5 b Fl(\()p Fj(V)16 b Fl(\))2291 3655 y Fq(\()p 2329 3600 50 4 v Fp(z)t Fq(\))2416 3619 y Fk(\003)2484 3655 y Fq(=)27 b(\002)2663 3670 y Fj(H)5 b Fl(\()p Fj(V)16 b Fl(\))2837 3652 y Fh(\003)2878 3655 y Fq(\()p Fp(z)t Fq(\))31 b FA(for)f Fp(z)i Fn(2)d Fp(\045)p Fq(\()p Fp(H)8 b Fq(\()p Fp(V)21 b Fq(\)\))178 3775 y FA(w)m(e)34 b(obtain)d(\(3.6\))h(from)g(\(3.5\).)2327 b Fb(\003)178 4105 y Fs(3.2)131 b(Dilations)178 4337 y Fr(Prop)s(osition)36 b(3.2)49 b Fc(Assume)31 b Fp(V)49 b Fn(2)28 b Fp(L)1602 4301 y Fk(1)1677 4337 y Fc(,)k Fp(V)1796 4352 y Fj(n)1871 4337 y Fn(2)c Fp(L)2031 4301 y Fk(1)2106 4337 y Fc(,)k Fp(n)27 b Fq(=)h(1)p Fp(;)17 b Fq(2)p Fp(;)g(:)g(:)g(:)n Fc(.)44 b(If)31 b Fp(V)2886 4352 y Fj(n)2984 4281 y(L)3032 4257 y Fh(1)2960 4337 y Fn(\000)-16 b(!)27 b Fp(V)53 b Fc(as)31 b Fp(n)d Fn(!)f(1)p Fc(,)178 4458 y(then)900 4578 y Fq(lim)876 4638 y Fj(n)p Fk(!1)1076 4578 y Fq(sup)1088 4659 y Fj(z)s Fk(2C)1240 4578 y Fn(k)p Fq(\()p Fp(K)7 b Fq(\()p Fp(V)1513 4593 y Fj(n)1559 4578 y Fq(\))22 b Fn(\000)h Fp(z)t Fq(\))1806 4537 y Fk(\000)p Fl(1)1923 4578 y Fn(\000)g Fq(\()p Fp(K)7 b Fq(\()p Fp(V)21 b Fq(\))h Fn(\000)h Fp(z)t Fq(\))2514 4537 y Fk(\000)p Fl(1)2609 4578 y Fn(k)2659 4594 y Fd(L)2705 4603 y Ff(1)2739 4594 y Fl(\()p Fd(K)p Fl(\))2872 4578 y Fq(=)28 b(0)447 b FA(\(3.11\))178 4793 y Fc(for)35 b(any)g(c)-5 b(omp)g(act)34 b(set)h Fn(C)f(\022)28 b Fo(C)48 b Fn(n)22 b Fo(R)5 b Fc(.)178 5022 y Fr(Pro)s(of.)43 b FA(T)-8 b(o)29 b(pro)m(v)m(e)i(\(3.11\))e(it)g(is)g(enough)h(to)f(v)m (erify)i(it)d(for)h(a)h(single)e(p)s(oin)m(t)h Fp(z)k Fn(2)28 b Fo(C)3188 5037 y Fl(+)3253 5022 y FA(.)42 b(By)30 b(\(2.40\))178 5142 y(w)m(e)k(get)e(that)339 5321 y Fq(\()p Fp(K)7 b Fq(\()p Fp(V)562 5336 y Fj(n)609 5321 y Fq(\))22 b Fn(\000)h Fp(z)t Fq(\))856 5280 y Fk(\000)p Fl(1)964 5295 y Fp(~)951 5321 y(f)33 b Fn(\000)22 b Fq(\()p Fp(K)7 b Fq(\()p Fp(V)22 b Fq(\))g Fn(\000)h Fp(z)t Fq(\))1623 5280 y Fk(\000)p Fl(1)1731 5295 y Fp(~)1718 5321 y(f)38 b Fq(=)1592 b FA(\(3.12\))529 5466 y Fq(0)49 b Fn(\010)727 5386 y Fm(\000)773 5466 y Fq(\()p Fp(H)8 b Fq(\()p Fp(V)995 5481 y Fj(n)1041 5466 y Fq(\))22 b Fn(\000)h Fp(z)t Fq(\))1288 5425 y Fk(\000)p Fl(1)1405 5466 y Fn(\000)f Fq(\()p Fp(H)8 b Fq(\()p Fp(V)21 b Fq(\))h Fn(\000)h Fp(z)t Fq(\))1994 5425 y Fk(\000)p Fl(1)2089 5386 y Fm(\001)2151 5466 y Fp(f)33 b Fq(+)22 b Fp(i)17 b Fq(\()p Fp(T)2475 5481 y Fk(\003)2515 5466 y Fq(\()p Fp(V)2610 5481 y Fj(n)2656 5466 y Fq(\)\()p 2732 5412 V Fp(z)5 b Fq(\))2820 5425 y Fk(\003)2882 5466 y Fn(\000)22 b Fq(\()p Fp(T)3076 5481 y Fk(\003)3116 5466 y Fq(\()p Fp(V)f Fq(\)\()p 3308 5412 V Fp(z)5 b Fq(\))3396 5425 y Fk(\003)3435 5466 y Fq(\))17 b Fp(\030)529 5612 y Fn(\010)p Fp(ie)684 5571 y Fj(ixz)805 5612 y Fq(\()p Fp(T)d Fq(\()p Fp(V)1009 5627 y Fj(n)1055 5612 y Fq(\)\()p Fp(z)t Fq(\))23 b Fn(\000)g Fp(T)14 b Fq(\()p Fp(V)21 b Fq(\)\()p Fp(z)t Fq(\)\))c Fp(f)33 b Fq(+)22 b Fp(ie)2003 5571 y Fj(ixz)2124 5531 y Fm(\000)2169 5612 y Fq(\002)2245 5627 y Fj(H)5 b Fl(\()p Fj(V)2376 5635 y Fe(n)2419 5627 y Fl(\))2451 5612 y Fq(\()p 2489 5557 V Fp(z)t Fq(\))2576 5571 y Fk(\003)2638 5612 y Fn(\000)23 b Fq(\002)2814 5627 y Fj(H)5 b Fl(\()p Fj(V)16 b Fl(\))2992 5612 y Fq(\()p 3030 5557 V Fp(z)5 b Fq(\))3118 5571 y Fk(\003)3157 5531 y Fm(\001)3220 5612 y Fp(\030)p eop %%Page: 18 20 18 19 bop 178 46 a FA(18)896 b Ft(M.)36 b(Bar)m(o,)f(H.-Ch.)47 b(Kaiser,)36 b(H.)g(Neidhardt,)f(J.)i(Rehber)m(g)178 295 y FA(where)1302 457 y Fp(\030)32 b Fq(:=)1508 321 y Fm(Z)1608 348 y Fl(0)1563 547 y Fk(\0001)1709 457 y Fp(dy)f(e)1884 416 y Fk(\000)p Fj(iy)r(z)2040 457 y Fp(f)2088 472 y Fk(\000)2147 457 y Fq(\()p Fp(y)t Fq(\))c Fn(2)h Fo(C)2462 416 y Fl(2)2507 457 y Fp(ee)875 b FA(\(3.13\))178 687 y(Since)1268 828 y Fn(k)p Fp(\030)5 b Fn(k)1416 845 y Fg(C)1457 826 y Ff(2)1529 828 y Fn(\024)1834 761 y Fq(1)p 1644 805 428 4 v 1644 825 a Fm(p)p 1744 825 328 4 v 85 x Fq(2)p Fn(=)p FA(m)o Fq(\()p Fp(z)t Fq(\))2082 828 y Fn(k)p Fp(f)2180 843 y Fk(\000)2239 828 y Fn(k)2289 845 y Fj(L)2337 826 y Ff(2)2371 845 y Fl(\()p Fg(R)2446 854 y Fh(\000)2498 845 y Fj(;)p Fg(C)2560 826 y Ff(2)2601 845 y Fl(\))3472 828 y FA(\(3.14\))178 1074 y(w)m(e)34 b(obtain)745 1295 y Fn(k)p Fq(\()p Fp(K)7 b Fq(\()p Fp(V)1018 1310 y Fj(n)1064 1295 y Fq(\))22 b Fn(\000)h Fp(z)t Fq(\))1311 1254 y Fk(\000)p Fl(1)1428 1295 y Fn(\000)g Fq(\()p Fp(K)7 b Fq(\()p Fp(V)21 b Fq(\))h Fn(\000)h Fp(z)t Fq(\))2019 1254 y Fk(\000)p Fl(1)2114 1295 y Fn(k)2164 1311 y Fd(L)2210 1320 y Ff(1)2244 1311 y Fl(\()p Fd(K)p Fl(\))3472 1295 y FA(\(3.15\))946 1440 y Fn(\024)83 b(k)p Fq(\()p Fp(H)8 b Fq(\()p Fp(V)1378 1455 y Fj(n)1424 1440 y Fq(\))22 b Fn(\000)h Fp(z)t Fq(\))1671 1399 y Fk(\000)p Fl(1)1788 1440 y Fn(\000)g Fq(\()p Fp(H)8 b Fq(\()p Fp(V)21 b Fq(\))h Fn(\000)g Fp(z)t Fq(\))2377 1399 y Fk(\000)p Fl(1)2472 1440 y Fn(k)2522 1456 y Fd(L)2568 1465 y Ff(1)2603 1456 y Fl(\()p Fd(H)p Fl(\))1106 1637 y Fq(+)1381 1570 y(1)p 1192 1614 428 4 v 1192 1634 a Fm(p)p 1292 1634 328 4 v 86 x Fq(2)p Fn(=)p FA(m)o Fq(\()p Fp(z)t Fq(\))1629 1637 y Fn(k)p Fp(T)1736 1652 y Fk(\003)1776 1637 y Fq(\()p Fp(V)1871 1652 y Fj(n)1917 1637 y Fq(\)\()p 1993 1583 50 4 v Fp(z)5 b Fq(\))2081 1596 y Fk(\003)2143 1637 y Fn(\000)22 b Fq(\()p Fp(T)2337 1652 y Fk(\003)2377 1637 y Fq(\()p Fp(V)f Fq(\)\()p 2569 1583 V Fp(z)5 b Fq(\))2657 1596 y Fk(\003)2696 1637 y Fn(k)2746 1654 y Fd(L)2792 1663 y Ff(1)2827 1654 y Fl(\()p Fg(C)2896 1635 y Ff(2)2937 1654 y Fl(;)p Fd(H)p Fl(\))1106 1915 y Fq(+)1381 1847 y(1)p 1192 1892 428 4 v 1192 1912 a Fm(p)p 1292 1912 328 4 v 85 x Fq(2)p Fn(=)p FA(m)o Fq(\()p Fp(z)t Fq(\))1629 1915 y Fn(k)p Fp(T)14 b Fq(\()p Fp(V)1845 1930 y Fj(n)1892 1915 y Fq(\)\()p Fp(z)t Fq(\))22 b Fn(\000)h Fp(T)14 b Fq(\()p Fp(V)21 b Fq(\)\()p Fp(z)t Fq(\))p Fn(k)2577 1931 y Fd(L)2623 1940 y Ff(1)2658 1931 y Fl(\()p Fg(C)2727 1913 y Ff(2)2768 1931 y Fl(;)p Fd(H)p Fl(\))1106 2192 y Fq(+)1331 2125 y(1)p 1192 2169 V 1192 2261 a(2)p Fn(=)p FA(m)o Fq(\()p Fp(z)t Fq(\))1530 2192 y Fn(k)p Fq(\002)1656 2208 y Fj(H)5 b Fl(\()p Fj(V)1787 2216 y Fe(n)1829 2208 y Fl(\))1861 2192 y Fq(\()p 1899 2137 50 4 v Fp(z)g Fq(\))1987 2151 y Fk(\003)2048 2192 y Fn(\000)23 b Fq(\002)2224 2208 y Fj(H)5 b Fl(\()p Fj(V)16 b Fl(\))2403 2192 y Fq(\()p 2441 2137 V Fp(z)t Fq(\))2528 2151 y Fk(\003)2568 2192 y Fn(k)2618 2209 y Fd(L)2664 2218 y Ff(1)2698 2209 y Fl(\()p Fg(C)2767 2190 y Ff(2)2808 2209 y Fl(\))2840 2192 y Fp(:)178 2470 y FA(Applying)32 b(\(3.1\),)g(\(3.2\),)g(\(3.4\))g (and)h(\(3.6\))f(w)m(e)h(obtain)f(\(3.11\).)1143 b Fb(\003)178 2809 y Fs(3.3)131 b(Eigenfunction)45 b(expansions)178 3064 y Fr(Lemma)37 b(3.3)49 b Fc(Assume)31 b Fp(V)49 b Fn(2)28 b Fp(L)1393 3028 y Fk(1)1468 3064 y Fc(,)j Fp(V)1586 3079 y Fj(n)1661 3064 y Fn(2)d Fp(L)1821 3028 y Fk(1)1896 3064 y Fc(,)j Fp(n)d Fq(=)g(1)p Fp(;)17 b Fq(2)p Fp(;)g(:)g(:)g(:)n Fc(.)43 b(If)30 b Fp(V)2674 3079 y Fj(n)2772 3008 y(L)2820 2984 y Fh(1)2749 3064 y Fn(\000)-17 b(!)28 b Fp(V)52 b Fc(as)30 b Fp(n)e Fn(!)f(1)p Fc(,)k(then)810 3286 y Fq(lim)786 3346 y Fj(n)p Fk(!1)986 3286 y Fq(sup)995 3368 y Fj(\025)p Fk(2C)1150 3286 y Fn(k)p Fp( )1267 3245 y Fk(\000)1263 3310 y(\006)1326 3286 y Fq(\()p Fp(V)1421 3301 y Fj(n)1467 3286 y Fq(\)\()p Fn(\001)p Fp(;)17 b(\025;)g(\034)11 b Fq(\))22 b Fn(\000)g Fp( )1995 3245 y Fk(\000)1991 3310 y(\006)2054 3286 y Fq(\()p Fp(V)g Fq(\)\()p Fn(\001)p Fp(;)17 b(\025;)g(\034)11 b Fq(\))p Fn(k)2561 3303 y Fj(L)2609 3284 y Fh(1)2673 3303 y Fl(\()p Fg(R)2748 3312 y Fh(\006)2801 3303 y Fj(;)p Fg(C)2862 3284 y Ff(2)2903 3303 y Fl(\))2962 3286 y Fq(=)28 b(0)357 b FA(\(3.16\))178 3560 y Fc(and)957 3682 y Fq(lim)932 3742 y Fj(n)p Fk(!1)1133 3682 y Fq(sup)1142 3764 y Fj(\025)p Fk(2C)1296 3682 y Fn(k)p Fp( )1413 3641 y Fk(\000)1472 3682 y Fq(\()p Fp(V)1567 3697 y Fj(n)1614 3682 y Fq(\)\()p Fn(\001)p Fp(;)17 b(\025;)g(\034)11 b Fq(\))21 b Fn(\000)i Fp( )2142 3641 y Fk(\000)2201 3682 y Fq(\()p Fp(V)f Fq(\)\()p Fn(\001)p Fp(;)17 b(\025;)g(\034)11 b Fq(\))p Fn(k)2708 3697 y Fd(H)2788 3682 y Fq(=)28 b(0)p Fp(;)504 b FA(\(3.17\))178 3912 y Fp(\034)40 b Fq(=)27 b Fp(a;)17 b(b)p Fc(,)35 b(for)g(any)g(c)-5 b(omp)g(act)34 b(set)g Fn(C)g(\022)29 b Fo(R)5 b Fc(.)178 4191 y Fr(Pro)s(of.)42 b FA(By)29 b(\(2.42\))f(one)h(gets)g(that)f Fp( )1596 4150 y Fk(\000)1592 4213 y(\000)1655 4191 y Fq(\()p Fp(V)21 b Fq(\)\()p Fp(x;)c(\025;)g (\034)11 b Fq(\))28 b Fn(\021)g Fp( )2338 4150 y Fk(\000)2334 4213 y(\000)2397 4191 y Fq(\()p Fp(V)2492 4206 y Fj(n)2539 4191 y Fq(\)\()p Fp(x;)17 b(\025;)g(\034)11 b Fq(\))28 b FA(for)g Fp(x)h Fn(2)f Fo(R)3323 4206 y Fk(\000)3388 4191 y FA(,)h Fp(\025)f Fn(2)g Fo(R)5 b FA(,)178 4311 y Fp(n)28 b Fq(=)g(1)p Fp(;)17 b Fq(2)p Fp(;)g(:)g(:)g(:)28 b FA(and)h Fp(\034)39 b Fq(=)28 b Fp(a;)17 b(b)p FA(.)43 b(So)29 b(the)h(assertion)f(\(3.16\))g(is)g(ob)m(vious)g(for)g(the)h (sign)f(\020-\021.)41 b(F)-8 b(urther,)178 4432 y(w)m(e)34 b(\034nd)803 4653 y Fp( )870 4612 y Fk(\000)866 4678 y Fl(+)929 4653 y Fq(\()p Fp(V)1024 4668 y Fj(n)1070 4653 y Fq(\)\()p Fp(x;)17 b(\025;)g(\034)11 b Fq(\))23 b Fn(\000)f Fp( )1626 4612 y Fk(\000)1622 4678 y Fl(+)1685 4653 y Fq(\()p Fp(V)g Fq(\)\()p Fp(x;)17 b(\025;)g(\034)11 b Fq(\))1303 b FA(\(3.18\))1122 4850 y Fq(=)1362 4782 y(1)p 1291 4827 191 4 v 1291 4847 a Fn(p)p 1374 4847 108 4 v 82 x Fq(2)p Fp(\031)1491 4850 y(e)1536 4809 y Fj(ix\025)1662 4769 y Fm(\000)1708 4850 y Fq(\002)1784 4865 y Fj(H)5 b Fl(\()p Fj(V)1915 4873 y Fe(n)1958 4865 y Fl(\))1990 4850 y Fq(\()p Fp(\025)p Fq(\))2123 4809 y Fk(\003)2184 4850 y Fn(\000)23 b Fq(\002)2360 4865 y Fj(H)5 b Fl(\()p Fj(V)16 b Fl(\))2538 4850 y Fq(\()p Fp(\025)p Fq(\))2671 4809 y Fk(\003)2711 4769 y Fm(\001)2773 4850 y Fp(e)2818 4865 y Fj(\034)178 5127 y FA(for)32 b Fp(x)c Fn(2)h Fq([)p Fp(a;)17 b(b)p Fq(])p FA(,)33 b Fp(\025)27 b Fn(2)h Fo(R)5 b FA(,)39 b Fp(n)28 b Fq(=)f(1)p Fp(;)17 b Fq(2)p Fp(;)g(:)g(:)g(:)31 b FA(and)i Fp(\034)39 b Fq(=)28 b Fp(a;)17 b(b)33 b FA(whic)m(h)g(yields)368 5264 y Fm(\015)368 5324 y(\015)424 5348 y Fp( )491 5307 y Fk(\000)487 5373 y Fl(+)550 5348 y Fq(\()p Fp(V)645 5363 y Fj(n)692 5348 y Fq(\)\()p Fn(\001)p Fp(;)17 b(\025;)g(\034)11 b Fq(\))21 b Fn(\000)i Fp( )1220 5307 y Fk(\000)1216 5373 y Fl(+)1279 5348 y Fq(\()p Fp(V)e Fq(\)\()p Fn(\001)p Fp(;)c(\025;)g(\034)11 b Fq(\))1735 5264 y Fm(\015)1735 5324 y(\015)1790 5387 y Fj(L)1838 5369 y Fh(1)1903 5387 y Fl(\()p Fg(R)1978 5396 y Ff(+)2029 5387 y Fj(;)p Fg(C)2090 5369 y Ff(2)2131 5387 y Fl(\))3472 5348 y FA(\(3.19\))1160 5569 y Fn(\024)1402 5501 y Fq(1)p 1331 5546 191 4 v 1331 5566 a Fn(p)p 1414 5566 108 4 v 82 x Fq(2)p Fp(\031)1531 5569 y Fn(k)p Fq(\002)1657 5584 y Fj(H)5 b Fl(\()p Fj(V)1788 5592 y Fe(n)1831 5584 y Fl(\))1863 5569 y Fq(\()p Fp(\025)p Fq(\))1996 5528 y Fk(\003)2057 5569 y Fn(\000)23 b Fq(\002)2233 5584 y Fj(H)5 b Fl(\()p Fj(V)16 b Fl(\))2411 5569 y Fq(\()p Fp(\025)p Fq(\))2544 5528 y Fk(\003)2584 5569 y Fn(k)2634 5586 y Fk(B)r Fl(\()p Fg(C)2751 5567 y Ff(2)2791 5586 y Fl(\))p eop %%Page: 19 21 19 20 bop 178 46 a Ft(Dissip)-8 b(a)g(tive)36 b(Schr)m (\366dinger-Poisson)f(systems)1383 b FA(19)178 295 y(for)34 b Fp(\025)c Fn(2)h Fo(R)5 b FA(,)41 b Fp(n)31 b Fq(=)f(1)p Fp(;)17 b Fq(2)p Fp(;)g(:)g(:)g(:)33 b FA(and)h Fp(\034)42 b Fq(=)30 b Fp(a;)17 b(b)p FA(.)49 b(Applying)33 b(\(3.6\))h(w)m(e)h (pro)m(v)m(e)h(\(3.16\).)47 b(F)-8 b(urthermore,)178 415 y(b)m(y)34 b(\(2.42\))e(w)m(e)h(get)420 665 y Fp( )487 624 y Fk(\000)546 665 y Fq(\()p Fp(V)641 680 y Fj(n)687 665 y Fq(\)\()p Fp(x;)17 b(\025;)g(\034)11 b Fq(\))23 b Fn(\000)f Fp( )1243 624 y Fk(\000)1302 665 y Fq(\()p Fp(V)g Fq(\)\()p Fp(x;)17 b(\025;)g(\034)11 b Fq(\))1686 b FA(\(3.20\))1211 855 y Fq(=)1451 788 y(1)p 1380 832 191 4 v 1380 852 a Fn(p)p 1463 852 108 4 v 83 x Fq(2)p Fp(\031)1597 855 y Fq(\()p Fp(T)1692 870 y Fk(\003)1732 855 y Fq(\()p Fp(V)1827 870 y Fj(n)1874 855 y Fq(\)\()p Fp(\025)p Fq(\))2045 814 y Fk(\003)2106 855 y Fn(\000)23 b Fp(T)2263 870 y Fk(\003)2302 855 y Fq(\()p Fp(V)f Fq(\)\()p Fp(\025)p Fq(\))2590 814 y Fk(\003)2629 855 y Fq(\))16 b Fp(e)2728 870 y Fj(\034)178 1161 y FA(for)32 b Fp(x)c Fn(2)h Fq([)p Fp(a;)17 b(b)p Fq(])p FA(,)33 b Fp(\025)27 b Fn(2)h Fo(R)5 b FA(,)39 b Fp(n)28 b Fq(=)f(1)p Fp(;)17 b Fq(2)p Fp(;)g(:)g(:)g(:)31 b FA(and)i Fp(\034)39 b Fq(=)28 b Fp(a;)17 b(b)p FA(.)44 b(Hence)811 1411 y Fn(k)p Fp( )928 1370 y Fk(\000)987 1411 y Fq(\()p Fp(V)1082 1426 y Fj(n)1128 1411 y Fq(\)\()p Fn(\001)p Fp(;)17 b(\025;)g(\034)11 b Fq(\))22 b Fn(\000)g Fp( )1656 1370 y Fk(\000)1715 1411 y Fq(\()p Fp(V)g Fq(\)\()p Fn(\001)p Fp(;)17 b(\025;)g(\034)11 b Fq(\))p Fn(k)2222 1426 y Fd(H)3472 1411 y FA(\(3.21\))1130 1601 y Fn(\024)1371 1534 y Fq(1)p 1300 1578 191 4 v 1300 1598 a Fn(p)p 1383 1598 108 4 v 82 x Fq(2)p Fp(\031)1501 1601 y Fn(k)p Fp(T)1608 1616 y Fk(\003)1648 1601 y Fq(\()p Fp(V)1743 1616 y Fj(n)1789 1601 y Fq(\)\()p Fp(\025)p Fq(\))1960 1560 y Fk(\003)2022 1601 y Fn(\000)22 b Fp(T)2178 1616 y Fk(\003)2218 1601 y Fq(\()p Fp(V)f Fq(\)\()p Fp(\025)p Fq(\))2505 1560 y Fk(\003)2544 1601 y Fn(k)2594 1618 y Fk(B)r Fl(\()p Fg(C)2712 1599 y Ff(2)2752 1618 y Fl(;)p Fd(H)p Fl(\))178 1907 y FA(for)32 b Fp(\025)c Fn(2)g Fo(R)5 b FA(,)39 b Fp(n)27 b Fq(=)h(1)p Fp(;)17 b Fq(2)p Fp(;)g(:)g(:)g(:)31 b FA(and)i Fp(\034)39 b Fq(=)27 b Fp(a;)17 b(b)p FA(.)44 b(Applying)32 b(\(3.4\))g(w)m(e)h(obtain)f (\(3.17\).)524 b Fb(\003)178 2267 y Fr(Prop)s(osition)36 b(3.4)49 b Fc(L)-5 b(et)41 b Fp(V)61 b Fn(2)39 b Fp(L)1439 2231 y Fk(1)1514 2267 y Fc(,)k Fp(V)1644 2282 y Fj(n)1730 2267 y Fn(2)d Fp(L)1902 2231 y Fk(1)1977 2267 y Fc(,)i Fp(n)e Fq(=)f(1)p Fp(;)17 b Fq(2)p Fp(;)g(:)g(:)g(:)n Fc(.)63 b(If)41 b Fp(V)2820 2282 y Fj(n)2930 2211 y(L)2978 2188 y Fh(1)2906 2267 y Fn(\000)-16 b(!)39 b Fp(V)62 b Fc(as)41 b Fp(n)e Fn(!)g(1)p Fc(,)178 2388 y(then)1390 2539 y Fp(s)22 b Fn(\000)47 b Fq(lim)1558 2598 y Fj(n)p Fk(!1)1758 2539 y Fq(\010)1828 2554 y Fk(\000)1888 2539 y Fq(\()p Fp(V)1983 2554 y Fj(n)2030 2539 y Fq(\))27 b(=)h(\010)2269 2554 y Fk(\000)2328 2539 y Fq(\()p Fp(V)22 b Fq(\))p Fp(:)962 b FA(\(3.22\))178 2907 y Fr(Pro)s(of.)43 b FA(Because)34 b(the)f(op)s(erators)f Fq(\010)1566 2922 y Fk(\000)1625 2907 y Fq(\()p Fp(V)1720 2922 y Fj(n)1767 2907 y Fq(\))c(\()p Fp(n)g Fn(2)g Fo(N)9 b Fq(\))38 b FA(and)32 b Fq(\010)2452 2922 y Fk(\000)2512 2907 y Fq(\()p Fp(V)21 b Fq(\))32 b FA(are)g(unitary)-8 b(,)32 b(it)f(is)h(enough)178 3028 y(to)f(v)m(erify)h Fp(w)24 b Fn(\000)f Fq(lim)895 3043 y Fj(n)p Fk(!1)1100 3028 y Fq(\010)1170 3043 y Fk(\000)1230 3028 y Fq(\()p Fp(V)1325 3043 y Fj(n)1371 3028 y Fq(\))28 b(=)f(\010)1610 3043 y Fk(\000)1670 3028 y Fq(\()p Fp(V)21 b Fq(\))p FA(.)43 b(Let)27 b Fp(~)-44 b(g)31 b Fn(2)d Fi(K)i FA(and)2527 3001 y Fq(^)2526 3028 y Fp(h)d Fn(2)i Fp(L)2770 2991 y Fl(2)2809 3028 y Fq(\()p Fo(R)5 b Fp(;)17 b Fo(C)3029 2991 y Fl(2)3074 3028 y Fq(\))31 b FA(with)g(compact)178 3148 y(supp)s(orts.)45 b(By)33 b(\(2.46\))e(w)m(e)j(\034nd)520 3315 y Fm(\020)579 3426 y Fq(\010)649 3441 y Fk(\000)709 3426 y Fq(\()p Fp(V)21 b Fq(\))l Fp(~)-45 b(g)t(;)958 3399 y Fq(^)958 3426 y Fp(h)1013 3315 y Fm(\021)1073 3495 y Fj(L)1121 3476 y Ff(2)1156 3495 y Fl(\()p Fg(R)p Fj(;)p Fg(C)1293 3476 y Ff(2)1333 3495 y Fl(\))3472 3426 y FA(\(3.23\))839 3680 y Fq(=)1017 3585 y Fm(X)998 3798 y Fj(\034)8 b Fl(=)p Fj(a;b)1196 3544 y Fm(Z)1251 3770 y Fg(R)1320 3680 y Fp(d\025)1472 3544 y Fm(Z)1527 3770 y Fg(R)1575 3779 y Fh(\000)1648 3680 y Fp(dx)1798 3599 y Fm(\012)1845 3680 y Fp(g)1892 3695 y Fk(\000)1951 3680 y Fq(\()p Fp(x)p Fq(\))p Fp(;)17 b( )2193 3639 y Fk(\000)2189 3705 y(\000)2252 3680 y Fq(\()p Fp(V)22 b Fq(\)\()p Fp(x;)17 b(\025;)g(\034)11 b Fq(\))2736 3599 y Fm(\013)2783 3719 y Fg(C)2825 3700 y Ff(2)p 2886 3573 233 4 v 2887 3654 a Fq(^)2886 3680 y Fp(h)2942 3651 y Fj(\034)2985 3680 y Fq(\()p Fp(\025)p Fq(\))839 4009 y(+)1017 3914 y Fm(X)998 4127 y Fj(\034)d Fl(=)p Fj(a;b)1196 3873 y Fm(Z)1251 4099 y Fg(R)1320 4009 y Fp(d\025)1472 3873 y Fm(Z)1571 3900 y Fj(b)1527 4099 y(a)1622 4009 y Fp(dx)1773 3928 y Fm(\000)1818 4009 y Fp(g)t Fq(\()p Fp(x)p Fq(\))p Fp(;)17 b( )2111 3968 y Fk(\000)2170 4009 y Fq(\()p Fp(V)k Fq(\)\()p Fp(x;)c(\025;)g(\034)11 b Fq(\))2653 3928 y Fm(\001)2699 4048 y Fg(C)p 2768 3902 V 2769 3983 a Fq(^)2768 4009 y Fp(h)2824 3980 y Fj(\034)2867 4009 y Fq(\()p Fp(\025)p Fq(\))839 4318 y(+)1017 4224 y Fm(X)998 4436 y Fj(\034)d Fl(=)p Fj(a;b)1196 4183 y Fm(Z)1251 4408 y Fg(R)1320 4318 y Fp(d\025)1472 4183 y Fm(Z)1527 4408 y Fg(R)1575 4417 y Ff(+)1647 4318 y Fp(dx)1797 4238 y Fm(\012)1844 4318 y Fp(g)1891 4333 y Fl(+)1950 4318 y Fq(\()p Fp(x)p Fq(\))p Fp(;)17 b( )2192 4277 y Fk(\000)2188 4343 y Fl(+)2251 4318 y Fq(\()p Fp(V)k Fq(\)\()p Fp(x;)c(\025;)g(\034)11 b Fq(\))2734 4238 y Fm(\013)2781 4357 y Fg(C)2823 4338 y Ff(2)p 2885 4211 V 2886 4292 a Fq(^)2885 4318 y Fp(h)2941 4289 y Fj(\034)2984 4318 y Fq(\()p Fp(\025)p Fq(\))p Fp(:)178 4666 y FA(Using)32 b(this)h(form)m(ula)d(one)j(gets)270 4827 y Fm(\020)330 4938 y Fq(\(\010)438 4953 y Fk(\000)497 4938 y Fq(\()p Fp(V)592 4953 y Fj(n)639 4938 y Fq(\))22 b Fn(\000)g Fq(\010)868 4953 y Fk(\000)928 4938 y Fq(\()p Fp(V)f Fq(\)\))l Fp(~)-45 b(g)t(;)1215 4911 y Fq(^)1215 4938 y Fp(h)1270 4827 y Fm(\021)1330 5007 y Fj(L)1378 4988 y Ff(2)1413 5007 y Fl(\()p Fg(R)p Fj(;)p Fg(C)1550 4988 y Ff(2)1590 5007 y Fl(\))3472 4938 y FA(\(3.24\))377 5212 y Fq(=)554 5117 y Fm(X)536 5329 y Fj(\034)8 b Fl(=)p Fj(a;b)733 5076 y Fm(Z)789 5302 y Fg(R)857 5212 y Fp(d\025)1009 5076 y Fm(Z)1109 5103 y Fj(b)1065 5302 y(a)1160 5212 y Fp(dx)1310 5131 y Fm(\000)1356 5212 y Fp(g)t Fq(\()p Fp(x)p Fq(\))p Fp(;)17 b Fq(\()p Fp( )1687 5171 y Fk(\000)1745 5212 y Fq(\()p Fp(V)1840 5227 y Fj(n)1887 5212 y Fq(\)\()p Fp(x;)g(\025;)g(\034)11 b Fq(\))22 b Fn(\000)h Fp( )2443 5171 y Fk(\000)2502 5212 y Fq(\()p Fp(V)e Fq(\)\()p Fp(x;)c(\025;)g (\034)11 b Fq(\)\))3023 5131 y Fm(\001)3069 5251 y Fg(C)p 3138 5104 V 3139 5186 a Fq(^)3138 5212 y Fp(h)3194 5183 y Fj(\034)3237 5212 y Fq(\()p Fp(\025)p Fq(\))536 5521 y(+)647 5426 y Fm(X)629 5639 y Fj(\034)d Fl(=)p Fj(a;b)826 5386 y Fm(Z)881 5611 y Fg(R)950 5521 y Fp(d\025)1102 5386 y Fm(Z)1157 5611 y Fg(R)1205 5620 y Ff(+)1277 5521 y Fp(dx)1427 5440 y Fm(\012)1474 5521 y Fp(g)1521 5536 y Fl(+)1580 5521 y Fq(\()p Fp(x)p Fq(\))p Fp(;)17 b Fq(\()p Fp( )1860 5480 y Fk(\000)1856 5546 y Fl(+)1919 5521 y Fq(\()p Fp(V)2014 5536 y Fj(n)2061 5521 y Fq(\)\()p Fp(x;)g(\025;)g (\034)11 b Fq(\))22 b Fn(\000)g Fp( )2616 5480 y Fk(\000)2612 5546 y Fl(+)2675 5521 y Fq(\()p Fp(V)g Fq(\)\()p Fp(x;)17 b(\025;)g(\034)11 b Fq(\)\))3197 5440 y Fm(\013)3244 5560 y Fg(C)3286 5541 y Ff(2)p 3347 5414 V 3348 5495 a Fq(^)3347 5521 y Fp(h)3403 5492 y Fj(\034)3446 5521 y Fq(\()p Fp(\025)p Fq(\))p Fp(:)p eop %%Page: 20 22 20 21 bop 178 46 a FA(20)896 b Ft(M.)36 b(Bar)m(o,)f(H.-Ch.)47 b(Kaiser,)36 b(H.)g(Neidhardt,)f(J.)i(Rehber)m(g)178 295 y FA(F)-8 b(rom)31 b(\(3.24\))h(w)m(e)i(get)e(the)h(estimate)188 425 y Fm(\014)188 485 y(\014)188 545 y(\014)188 604 y(\014)221 459 y(\020)281 569 y Fq(\(\010)389 584 y Fk(\000)448 569 y Fq(\()p Fp(V)543 584 y Fj(n)590 569 y Fq(\))22 b Fn(\000)h Fq(\010)820 584 y Fk(\000)879 569 y Fq(\()p Fp(V)f Fq(\)\))l Fp(~)-45 b(g)s(;)1166 543 y Fq(^)1166 569 y Fp(h)1222 459 y Fm(\021)1281 638 y Fj(L)1329 619 y Ff(2)1364 638 y Fl(\()p Fg(R)p Fj(;)p Fg(C)1501 619 y Ff(2)1541 638 y Fl(\))1573 425 y Fm(\014)1573 485 y(\014)1573 545 y(\014)1573 604 y(\014)3472 569 y FA(\(3.25\))295 833 y Fn(\024)83 b(k)p Fp(g)t Fn(k)606 848 y Fd(H)694 738 y Fm(X)676 950 y Fj(\034)8 b Fl(=)p Fj(a;b)873 697 y Fm(Z)929 923 y Fg(R)997 833 y Fp(d\025)28 b Fn(k)p Fp( )1250 792 y Fk(\000)1308 833 y Fq(\()p Fp(V)1403 848 y Fj(n)1450 833 y Fq(\)\()p Fn(\001)p Fp(;)17 b(\025;)g(\034)11 b Fq(\))22 b Fn(\000)g Fp( )1978 792 y Fk(\000)2037 833 y Fq(\()p Fp(V)g Fq(\)\()p Fn(\001)p Fp(;)17 b(\025;)g(\034)11 b Fq(\)\))p Fn(k)2582 848 y Fd(H)2635 833 y Fn(j)2664 807 y Fq(^)2663 833 y Fp(h)2719 792 y Fj(\034)2762 833 y Fq(\()p Fp(\025)p Fq(\))p Fn(j)455 1142 y Fq(+)p Fn(k)p Fp(g)628 1157 y Fl(+)687 1142 y Fn(k)737 1159 y Fj(L)785 1140 y Ff(1)819 1159 y Fl(\()p Fg(R)894 1168 y Ff(+)945 1159 y Fj(;)p Fg(C)1007 1140 y Ff(2)1047 1159 y Fl(\))1114 1047 y Fm(X)1096 1260 y Fj(\034)d Fl(=)p Fj(a;b)1293 1006 y Fm(Z)1349 1232 y Fg(R)1417 1142 y Fp(d\025)28 b Fn(k)p Fp( )1670 1101 y Fk(\000)1666 1167 y Fl(+)1728 1142 y Fq(\()p Fp(V)1823 1157 y Fj(n)1870 1142 y Fq(\)\()p Fn(\001)p Fp(;)17 b(\025;)g(\034)11 b Fq(\))22 b Fn(\000)g Fp( )2398 1101 y Fk(\000)2394 1167 y Fl(+)2457 1142 y Fq(\()p Fp(V)g Fq(\)\()p Fn(\001)p Fp(;)17 b(\025;)g(\034)11 b Fq(\)\))p Fn(k)3002 1159 y Fj(L)3050 1140 y Fh(1)3114 1159 y Fl(\()p Fg(R)3189 1168 y Ff(+)3240 1159 y Fj(;)p Fg(C)3302 1140 y Ff(2)3342 1159 y Fl(\))3374 1142 y Fn(j)3403 1116 y Fq(^)3402 1142 y Fp(h)3458 1101 y Fj(\034)3501 1142 y Fq(\()p Fp(\025)p Fq(\))p Fn(j)p Fp(:)178 1456 y FA(Applying)32 b(Lemma)f(3.3)h(w)m(e)i(pro)m(v)m(e)1108 1696 y Fq(lim)1084 1755 y Fj(n)p Fk(!1)1284 1585 y Fm(\020)1344 1696 y Fq(\(\010)1452 1711 y Fk(\000)1511 1696 y Fq(\()p Fp(V)1606 1711 y Fj(n)1653 1696 y Fq(\))22 b Fn(\000)h Fq(\010)1883 1711 y Fk(\000)1942 1696 y Fq(\()p Fp(V)f Fq(\)\))l Fp(~)-45 b(g)s(;)2230 1669 y Fq(^)2229 1696 y Fp(h)2285 1585 y Fm(\021)2344 1765 y Fj(L)2392 1746 y Ff(2)2427 1765 y Fl(\()p Fg(R)p Fj(;)p Fg(C)2564 1746 y Ff(2)2605 1765 y Fl(\))2664 1696 y Fq(=)27 b(0)656 b FA(\(3.26\))178 1982 y(for)32 b Fp(~)-45 b(g)38 b Fn(2)c Fi(K)i FA(and)814 1956 y Fq(^)813 1982 y Fp(h)e Fn(2)g Fp(L)1069 1946 y Fl(2)1109 1982 y Fq(\()p Fo(R)5 b Fp(;)17 b Fo(C)1329 1946 y Fl(2)1374 1982 y Fq(\))37 b FA(with)e(compact)h (supp)s(orts.)56 b(Since)37 b(b)s(oth)f(sets)h(are)g(dense)g(in)178 2102 y Fi(K)30 b FA(and)h Fp(L)529 2066 y Fl(2)569 2102 y Fq(\()p Fo(R)5 b Fp(;)17 b Fo(C)788 2066 y Fl(2)834 2102 y Fq(\))p FA(,)31 b(resp)s(ectiv)m(ely)-8 b(,)32 b(and)e Fq(\010)1736 2117 y Fk(\000)1796 2102 y Fq(\()p Fp(V)1891 2117 y Fj(n)1938 2102 y Fq(\))p FA(,)h Fp(n)d Fq(=)f(1)p Fp(;)17 b Fq(2)p Fp(;)g(:)g(:)g(:)o FA(,)31 b(are)f(isometric)f(op)s(erators)h(the)178 2223 y(con)m(v)m(ergence)35 b(\(3.26\))d(implies)e(the)j(w)m(eak)h(con)m(v)m(ergence.)1371 b Fb(\003)178 2558 y Fs(3.4)131 b(Carrier)45 b(densities)178 2790 y Fr(Prop)s(osition)36 b(3.5)49 b Fc(L)-5 b(et)30 b Fp(V)49 b Fn(2)28 b Fp(L)1405 2754 y Fk(1)1480 2790 y Fc(,)j Fp(V)1598 2805 y Fj(n)1673 2790 y Fn(2)d Fp(L)1833 2754 y Fk(1)1908 2790 y Fc(,)j Fp(n)d Fq(=)f(1)p Fp(;)17 b Fq(2)p Fp(;)g(:)g(:)g(:)o Fc(.)43 b(If)29 b Fp(\045)p Fq(\()p Fn(\001)p Fq(\))h Fc(satis\034es)g(the)g(c)-5 b(ondition)178 2946 y(\(2.56\))34 b(and)g Fp(V)718 2961 y Fj(n)817 2889 y(L)865 2866 y Fh(1)793 2946 y Fn(\000)-16 b(!)27 b Fp(V)57 b Fc(as)34 b Fp(n)28 b Fn(!)f(1)p Fc(,)35 b(then)f Fp(u)1875 2961 y Fl(^)-41 b Fj(\045)1909 2946 y Fq(\()p Fp(V)2004 2961 y Fj(n)2051 2946 y Fq(\))2155 2889 y Fj(L)2203 2866 y Ff(1)2116 2946 y Fn(\000)-16 b(!)28 b Fp(u)2367 2961 y Fl(^)-41 b Fj(\045)2400 2946 y Fq(\()p Fp(V)22 b Fq(\))35 b Fc(as)f Fp(n)28 b Fn(!)f(1)p Fc(,)35 b(i.e.)1145 3234 y Fq(lim)1121 3294 y Fj(n)p Fk(!1)1322 3099 y Fm(Z)1421 3125 y Fj(b)1377 3324 y(a)1472 3234 y Fp(dx)28 b Fn(j)p Fp(u)1696 3249 y Fl(^)-41 b Fj(\045)1730 3234 y Fq(\()p Fp(V)1825 3249 y Fj(n)1871 3234 y Fq(\)\()p Fp(x)p Fq(\))23 b Fn(\000)f Fp(u)2224 3249 y Fl(^)-41 b Fj(\045)2258 3234 y Fq(\()p Fp(V)22 b Fq(\)\()p Fp(x)p Fq(\))p Fn(j)27 b Fq(=)h(0)p Fp(:)693 b FA(\(3.27\))178 3508 y Fc(In)35 b(p)-5 b(articular,)34 b(the)h(c)-5 b(arrier)34 b(density)h(op)-5 b(er)g(ator)35 b Fn(N)2071 3523 y Fl(^)-41 b Fj(\045)2105 3508 y Fq(\()p Fn(\001)p Fq(\))27 b(:)h Fp(L)2357 3472 y Fk(1)2459 3508 y Fn(\000)-16 b(!)28 b Fp(L)2714 3472 y Fl(1)2788 3508 y Fc(is)35 b(c)-5 b(ontinuous.)178 3781 y Fr(Pro)s(of.)44 b FA(By)33 b(\(2.59\))e(w)m(e)j(ha)m(v)m(e)g(the)f(represen)m(tation) 512 3931 y Fm(Z)612 3957 y Fj(b)567 4157 y(a)663 4067 y Fp(dx)27 b Fq(\()p Fp(u)896 4082 y Fl(^)-41 b Fj(\045)930 4067 y Fq(\()p Fp(V)1025 4082 y Fj(n)1072 4067 y Fq(\)\()p Fp(x)p Fq(\))22 b Fn(\000)h Fp(u)1425 4082 y Fl(^)-41 b Fj(\045)1459 4067 y Fq(\()p Fp(V)21 b Fq(\)\()p Fp(x)p Fq(\)\))p Fp(h)p Fq(\()p Fp(x)p Fq(\))29 b(=)e FA(tr)17 b Fq(\(\()p Fp(\045)p Fq(\()p Fp(V)2415 4082 y Fj(n)2462 4067 y Fq(\))22 b Fn(\000)g Fp(\045)p Fq(\()p Fp(V)g Fq(\)\))p Fp(M)10 b Fq(\()p Fp(h)p Fq(\)\))334 b FA(\(3.28\))178 4331 y(for)32 b(eac)m(h)i Fp(h)28 b Fn(2)g Fp(L)791 4295 y Fk(1)898 4331 y FA(and)33 b Fp(n)28 b Fq(=)f(1)p Fp(;)17 b Fq(2)p Fp(;)g(:)g(:)g(:)o FA(.)43 b(Since)724 4549 y(tr)16 b Fq(\()p Fp(\045)p Fq(\()p Fp(V)22 b Fq(\))p Fp(M)10 b Fq(\()p Fp(h)p Fq(\)\))28 b(=)g FA(tr)1557 4468 y Fm(\000)1603 4549 y Fp(\045)p Fq(\()p Fp(V)22 b Fq(\)\()p Fp(K)7 b Fq(\()p Fp(V)21 b Fq(\))h(+)h Fp(i)p Fq(\)\()p Fp(K)7 b Fq(\()p Fp(V)21 b Fq(\))h(+)g Fp(i)p Fq(\))2755 4507 y Fk(\000)p Fl(1)2850 4549 y Fp(M)10 b Fq(\()p Fp(h)p Fq(\))3086 4468 y Fm(\001)3149 4549 y Fp(:)296 b FA(\(3.29\))178 4766 y(one)33 b(gets)447 4983 y(tr)16 b Fq(\(\()p Fp(\045)p Fq(\()p Fp(V)760 4998 y Fj(n)807 4983 y Fq(\))22 b Fn(\000)h Fp(\045)p Fq(\()p Fp(V)f Fq(\)\))p Fp(M)10 b Fq(\()p Fp(h)p Fq(\)\))1988 b FA(\(3.30\))577 5129 y Fq(=)83 b FA(tr)828 5048 y Fm(\000)874 5129 y Fp(\045)p Fq(\()p Fp(V)1019 5144 y Fj(n)1066 5129 y Fq(\)\()p Fp(K)7 b Fq(\()p Fp(V)1327 5144 y Fj(n)1374 5129 y Fq(\))22 b(+)g Fp(i)p Fq(\))1620 5048 y Fm(\010)1678 5129 y Fq(\()p Fp(K)7 b Fq(\()p Fp(V)1901 5144 y Fj(n)1948 5129 y Fq(\))22 b(+)g Fp(i)p Fq(\))2177 5087 y Fk(\000)p Fl(1)2294 5129 y Fn(\000)g Fq(\()p Fp(K)7 b Fq(\()p Fp(V)22 b Fq(\))g(+)g Fp(i)p Fq(\))2867 5087 y Fk(\000)p Fl(1)2961 5048 y Fm(\011)3036 5129 y Fp(M)10 b Fq(\()p Fp(h)p Fq(\))3272 5048 y Fm(\001)736 5274 y Fq(+)p FA(tr)904 5193 y Fm(\000)950 5274 y Fq(\()p Fp(\045)p Fq(\()p Fp(V)1133 5289 y Fj(n)1180 5274 y Fq(\)\()p Fp(K)d Fq(\()p Fp(V)1441 5289 y Fj(n)1488 5274 y Fq(\))22 b(+)g Fp(i)p Fq(\))g Fn(\000)h Fp(\045)p Fq(\()p Fp(V)f Fq(\)\()p Fp(K)7 b Fq(\()p Fp(V)21 b Fq(\))h(+)g Fp(i)p Fq(\)\)\()p Fp(K)7 b Fq(\()p Fp(V)22 b Fq(\))g(+)g Fp(i)p Fq(\))3029 5233 y Fk(\000)p Fl(1)3124 5274 y Fp(M)10 b Fq(\()p Fp(h)p Fq(\))3360 5193 y Fm(\001)178 5491 y FA(for)32 b Fp(n)c Fq(=)g(1)p Fp(;)17 b Fq(2)p Fp(;)g(:)g(:)g(:)n FA(.)44 b(By)1338 5612 y Fn(k)o Fp(\045)p Fq(\()p Fp(V)1532 5627 y Fj(n)1580 5612 y Fq(\)\()p Fp(K)7 b Fq(\()p Fp(V)1841 5627 y Fj(n)1887 5612 y Fq(\))22 b(+)g Fp(i)p Fq(\))p Fn(k)2166 5641 y Fk(B)r Fl(\()p Fd(K)p Fl(\))2347 5612 y Fn(\024)28 b Fp(C)2528 5627 y Fl(^)-41 b Fj(\045)3472 5612 y FA(\(3.31\))p eop %%Page: 21 23 21 22 bop 178 46 a Ft(Dissip)-8 b(a)g(tive)36 b(Schr)m (\366dinger-Poisson)f(systems)1383 b FA(21)178 295 y(w)m(e)34 b(obtain)344 419 y Fm(\014)344 479 y(\014)377 504 y FA(tr)470 423 y Fm(\000)515 504 y Fp(\045)p Fq(\()p Fp(V)660 519 y Fj(n)707 504 y Fq(\)\()p Fp(K)7 b Fq(\()p Fp(V)968 519 y Fj(n)1015 504 y Fq(\))22 b(+)g Fp(i)p Fq(\))1261 423 y Fm(\010)1319 504 y Fq(\()p Fp(K)7 b Fq(\()p Fp(V)1542 519 y Fj(n)1589 504 y Fq(\))22 b(+)g Fp(i)p Fq(\))1818 463 y Fk(\000)p Fl(1)1935 504 y Fn(\000)g Fq(\()p Fp(K)7 b Fq(\()p Fp(V)22 b Fq(\))g(+)g Fp(i)p Fq(\))2508 463 y Fk(\000)p Fl(1)2603 423 y Fm(\011)2677 504 y Fp(M)10 b Fq(\()p Fp(h)p Fq(\))2913 423 y Fm(\001)2960 419 y(\014)2960 479 y(\014)3472 504 y FA(\(3.32\))982 649 y Fn(\024)29 b Fp(C)1164 664 y Fl(^)-41 b Fj(\045)1214 565 y Fm(\015)1214 624 y(\015)1270 649 y Fq(\()p Fp(K)7 b Fq(\()p Fp(V)1493 664 y Fj(n)1539 649 y Fq(\))23 b(+)f Fp(i)p Fq(\))1769 608 y Fk(\000)p Fl(1)1885 649 y Fn(\000)h Fq(\()p Fp(K)7 b Fq(\()p Fp(V)21 b Fq(\))i(+)f Fp(i)p Fq(\))2459 608 y Fk(\000)p Fl(1)2553 565 y Fm(\015)2553 624 y(\015)2608 688 y Fd(L)2654 697 y Ff(1)2689 688 y Fl(\()p Fd(K)p Fl(\))2811 649 y Fn(k)p Fp(h)p Fn(k)2967 664 y Fj(L)3015 645 y Fh(1)178 871 y FA(for)32 b Fp(n)c Fq(=)g(1)p Fp(;)17 b Fq(2)p Fp(;)g(:)g(:)g(:)n FA(.)44 b(Setting)1333 1080 y Fp(k)s Fq(\()p Fp(\025)p Fq(\))28 b(:=)f Fp(\045)p Fq(\()p Fp(\025)p Fq(\)\()p Fp(\025)c Fq(+)f Fp(i)p Fq(\))p Fp(;)114 b(\025)28 b Fn(2)g Fo(R)5 b Fp(;)911 b FA(\(3.33\))178 1290 y(and)31 b(denoting)f(b)m(y)901 1263 y Fq(^)898 1290 y Fp(k)k FA(the)c(m)m(ultiplication)c(op)s(erator)k(induced)h(b)m (y)g Fp(k)s Fq(\()p Fn(\001)p Fq(\))f FA(on)g Fp(L)3044 1253 y Fl(2)3084 1290 y Fq(\()p Fo(R)5 b Fp(;)17 b Fo(C)3304 1253 y Fl(2)3349 1290 y Fq(\))31 b FA(w)m(e)g(\034nd)178 1410 y(that)1123 1530 y Fp(\045)p Fq(\()p Fp(V)1268 1545 y Fj(n)1315 1530 y Fq(\)\()p Fp(K)7 b Fq(\()p Fp(V)1576 1545 y Fj(n)1623 1530 y Fq(\))22 b(+)g Fp(i)p Fq(\))28 b(=)f(\010)2053 1545 y Fk(\000)2113 1530 y Fq(\()p Fp(V)2208 1545 y Fj(n)2255 1530 y Fq(\))2293 1489 y Fk(\000)p Fl(1)2389 1504 y Fq(^)2387 1530 y Fp(k)s Fq(\010)2511 1545 y Fk(\000)2570 1530 y Fq(\()p Fp(V)2665 1545 y Fj(n)2712 1530 y Fq(\))p Fp(;)695 b FA(\(3.34\))178 1700 y Fp(n)28 b Fq(=)g(1)p Fp(;)17 b Fq(2)p Fp(;)g(:)g(:)g(:)n FA(,)33 b(and)1173 1821 y Fp(\045)p Fq(\()p Fp(V)22 b Fq(\)\()p Fp(K)7 b Fq(\()p Fp(V)22 b Fq(\))g(+)g Fp(i)p Fq(\))28 b(=)f(\010)2053 1836 y Fk(\000)2113 1821 y Fq(\()p Fp(V)21 b Fq(\))2267 1779 y Fk(\000)p Fl(1)2364 1794 y Fq(^)2361 1821 y Fp(k)t Fq(\010)2486 1836 y Fk(\000)2545 1821 y Fq(\()p Fp(V)h Fq(\))p Fp(:)745 b FA(\(3.35\))178 1990 y(Hence)34 b(the)f(represen)m (tation)297 2200 y Fp(\045)p Fq(\()p Fp(V)442 2215 y Fj(n)489 2200 y Fq(\)\()p Fp(K)7 b Fq(\()p Fp(V)750 2215 y Fj(n)797 2200 y Fq(\))22 b(+)g Fp(i)p Fq(\))g Fn(\000)h Fp(\045)p Fq(\()p Fp(V)f Fq(\)\()p Fp(K)7 b Fq(\()p Fp(V)21 b Fq(\))h(+)g Fp(i)p Fq(\))1646 b FA(\(3.36\))616 2345 y Fq(=)83 b(\(\010)883 2360 y Fk(\000)943 2345 y Fq(\()p Fp(V)1038 2360 y Fj(n)1084 2345 y Fq(\))1122 2304 y Fk(\000)p Fl(1)1239 2345 y Fn(\000)22 b Fq(\010)1408 2360 y Fk(\000)1468 2345 y Fq(\()p Fp(V)f Fq(\))1622 2304 y Fk(\000)p Fl(1)1717 2345 y Fq(\))1758 2319 y(^)1755 2345 y Fp(k)s Fq(\010)1879 2360 y Fk(\000)1938 2345 y Fq(\()p Fp(V)2033 2360 y Fj(n)2080 2345 y Fq(\))h(+)g(\010)2308 2360 y Fk(\000)2367 2345 y Fq(\()p Fp(V)g Fq(\))2525 2319 y(^)2522 2345 y Fp(k)s Fq(\(\010)2684 2360 y Fk(\000)2743 2345 y Fq(\()p Fp(V)2838 2360 y Fj(n)2885 2345 y Fq(\))g Fn(\000)h Fq(\010)3115 2360 y Fk(\000)3174 2345 y Fq(\()p Fp(V)f Fq(\)\))178 2554 y FA(is)32 b(v)-5 b(alid.)42 b(F)-8 b(rom)31 b(\(3.36\))h(w)m(e)h (deduce)h(the)f(estimate)344 2679 y Fm(\014)344 2739 y(\014)377 2764 y FA(tr)469 2683 y Fm(\000)515 2764 y Fq(\()p Fp(\045)p Fq(\()p Fp(V)698 2779 y Fj(n)745 2764 y Fq(\)\()p Fp(K)7 b Fq(\()p Fp(V)1006 2779 y Fj(n)1053 2764 y Fq(\))22 b(+)g Fp(i)p Fq(\))g Fn(\000)h Fp(\045)p Fq(\()p Fp(V)f Fq(\)\()p Fp(K)7 b Fq(\()p Fp(V)21 b Fq(\))h(+)g Fp(i)p Fq(\)\)\()p Fp(K)7 b Fq(\()p Fp(V)22 b Fq(\))g(+)g Fp(i)p Fq(\))2594 2723 y Fk(\000)p Fl(1)2689 2764 y Fp(M)10 b Fq(\()p Fp(h)p Fq(\))2925 2683 y Fm(\001)2971 2679 y(\014)2971 2739 y(\014)3472 2764 y FA(\(3.37\))781 2919 y Fn(\024)84 b Fp(C)1018 2934 y Fl(^)-41 b Fj(\045)1052 2919 y Fn(k)p Fq(\(\010)1210 2934 y Fk(\000)1269 2919 y Fq(\()p Fp(V)1364 2934 y Fj(n)1411 2919 y Fq(\))22 b Fn(\000)g Fq(\010)1640 2934 y Fk(\000)1700 2919 y Fq(\()p Fp(V)f Fq(\)\))p Fp(P)1969 2878 y Fd(K)1955 2944 y(H)2019 2919 y Fq(\()p Fp(K)7 b Fq(\()p Fp(V)21 b Fq(\))h Fn(\000)h Fp(i)p Fq(\))2494 2878 y Fk(\000)p Fl(1)2589 2919 y Fn(k)2639 2934 y Fd(L)2685 2943 y Ff(1)2719 2934 y Fl(\()p Fd(H)p Fl(\))2828 2919 y Fn(k)p Fp(h)p Fn(k)2984 2934 y Fj(L)3032 2915 y Fh(1)942 3064 y Fq(+)p Fp(C)1094 3079 y Fl(^)-41 b Fj(\045)1127 3064 y Fn(k)p Fq(\(\010)1285 3079 y Fk(\000)1345 3064 y Fq(\()p Fp(V)1440 3079 y Fj(n)1487 3064 y Fq(\))22 b Fn(\000)g Fq(\010)1716 3079 y Fk(\000)1776 3064 y Fq(\()p Fp(V)f Fq(\)\)\()p Fp(K)7 b Fq(\()p Fp(V)22 b Fq(\))g(+)g Fp(i)p Fq(\))2442 3023 y Fk(\000)p Fl(1)2536 3064 y Fp(P)2613 3023 y Fd(K)2599 3089 y(H)2663 3064 y Fn(k)2713 3080 y Fd(L)2759 3089 y Ff(1)2793 3080 y Fl(\()p Fd(H)p Fl(\))2902 3064 y Fn(k)p Fp(h)p Fn(k)3058 3079 y Fj(L)3106 3060 y Fh(1)3175 3064 y Fp(:)178 3274 y FA(T)-8 b(aking)32 b(in)m(to)g(accoun)m(t)i(\(3.30\),)d(\(3.32\))h(and)h(\(3.37\))e(w)m(e) j(\034nally)d(get)i(the)g(estimate)655 3483 y Fn(j)p FA(tr)16 b Fq(\(\()p Fp(\045)p Fq(\()p Fp(V)996 3498 y Fj(n)1043 3483 y Fq(\))22 b Fn(\000)h Fp(\045)p Fq(\()p Fp(V)f Fq(\)\))p Fp(M)10 b Fq(\()p Fp(h)p Fq(\)\))p Fn(j)1724 b FA(\(3.38\))785 3656 y Fn(\024)84 b Fp(C)1022 3671 y Fl(^)-41 b Fj(\045)1073 3545 y Fm(n)1139 3571 y(\015)1139 3631 y(\015)1194 3656 y Fq(\()p Fp(K)7 b Fq(\()p Fp(V)1417 3671 y Fj(n)1464 3656 y Fq(\))22 b(+)g Fp(i)p Fq(\))1693 3614 y Fk(\000)p Fl(1)1810 3656 y Fn(\000)h Fq(\()p Fp(K)7 b Fq(\()p Fp(V)21 b Fq(\))h(+)g Fp(i)p Fq(\))2383 3614 y Fk(\000)p Fl(1)2478 3571 y Fm(\015)2478 3631 y(\015)2533 3695 y Fd(L)2579 3704 y Ff(1)2614 3695 y Fl(\()p 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Fk(\000)p Fl(1)178 5371 y FA(is)29 b(a)f(trace)i(class)f(op)s(erator)f(one)h(gets)g(b)m(y) h(\(3.22\))e(that)h(the)g(second)h(addend)g(of)f(the)g(r.h.s)h(tends) 178 5491 y(to)36 b(zero)h(to)s(o.)54 b(Similarly)32 b(one)37 b(pro)m(v)m(es)h(that)e(the)h(third)e(attend)i(tends)g(to)f(zero.)55 b(This)37 b(pro)m(v)m(es)178 5612 y(\(3.27\).)3189 b Fb(\003)p eop %%Page: 22 24 22 23 bop 178 46 a FA(22)896 b Ft(M.)36 b(Bar)m(o,)f(H.-Ch.)47 b(Kaiser,)36 b(H.)g(Neidhardt,)f(J.)i(Rehber)m(g)178 311 y Fy(4)156 b(Dissipativ)l(e)54 b(Sc)l(hr\366)t(dinger-P)l(oisson)h (systems)178 577 y FA(In)34 b(this)f(section)h(w)m(e)g(in)m(tend)g(to)f (regard)h(the)g(dissipativ)m(e)f(Sc)m(hr\366)s(dinger)g(P)m(oisson)h (system)g(as)g(a)178 697 y(whole.)44 b(Let)32 b(us)h(in)m(tro)s(duce)g (some)f(further)h(notations:)178 936 y Fr(De\034nition)j(4.1)49 b FA(W)-8 b(e)45 b(denote)g(the)f(real)f(part)h(of)g Fp(W)2231 900 y Fl(1)p Fj(;)p Fl(2)2369 936 y FA(b)m(y)h Fp(W)2622 889 y Fl(1)p Fj(;)p Fl(2)2608 963 y Fg(R)2760 936 y FA(and)f(that)g(of)g Fp(C)7 b Fq([)p Fp(a;)17 b(b)p Fq(])45 b FA(b)m(y)178 1056 y Fp(C)248 1071 y Fg(R)300 1056 y Fq([)p Fp(a;)17 b(b)p Fq(])p FA(.)44 b(Let)33 b Fq(\000)28 b Fn(\032)g Fp(@)5 b Fq(\012)34 b FA(b)s(e)e(the)h(\(p)s (ossibly)f(empt)m(y\))h(set)g(of)f(Diric)m(hlet)g(p)s(oin)m(ts.)43 b(W)-8 b(e)33 b(de\034ne)1207 1262 y Fp(W)1313 1215 y Fl(1)p Fj(;)p Fl(2)1299 1289 y(\000)1435 1262 y Fq(:=)28 b Fp(W)1672 1215 y Fl(1)p Fj(;)p Fl(2)1658 1290 y Fg(R)1788 1262 y Fn(\\)22 b(f)p Fp( )59 b Fq(:)d Fp( )t Fq(\(\000\))27 b Fn(\032)h(f)p Fq(0)p Fn(gg)p Fp(:)178 1469 y FA(By)e Fp(W)430 1421 y Fk(\000)p Fl(1)p Fj(;)p Fl(2)416 1496 y(\000)605 1469 y FA(w)m(e)g(denote)g(the)g(dual)f(space)h(of)f Fp(W)1882 1421 y Fl(1)p Fj(;)p Fl(2)1868 1496 y(\000)2002 1469 y FA(and)g(b)m(y)i Fp(<)g Fn(\001)p Fp(;)17 b Fn(\001)27 b Fp(>)2619 1484 y Fl(1)2684 1469 y FA(the)f(dual)e(pairing)g(b)s(et)m (w)m(een)178 1589 y Fp(W)284 1542 y Fl(1)p Fj(;)p Fl(2)270 1616 y(\000)409 1589 y FA(and)30 b 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Fk(\006)2319 2189 y Fq(\()p Fp(V)2414 2204 y Fk(\006)2473 2189 y Fq(\))p FA(,)h(cf.)43 b(\(1.3\),)29 b(for)f(electrons)h(\(sign)178 2309 y(\020-\021\))i(and)i (holes)f(\(sign)g(\020+\021\))g(and)g(P)m(oisson's)i(equation)e (\(1.1\).)43 b(The)33 b(dissipativ)m(e)f(Sc)m(hr\366)s(dinger-)178 2429 y(t)m(yp)s(e)24 b(op)s(erators)e(are)h(determined)f(b)m(y)i(the)f (`e\033ectiv)m(e')h(masses)f Fp(m)2562 2444 y Fk(\006)2622 2429 y FA(,)h(the)g(b)s(oundary)f(co)s(e\036cien)m(ts)178 2550 y Fp(\024)234 2514 y Fk(\006)234 2574 y Fj(a)294 2550 y Fp(;)17 b(\024)394 2509 y Fk(\006)394 2578 y Fj(b)485 2550 y FA(and)33 b(the)g(p)s(oten)m(tials)e Fp(V)1371 2514 y Fk(\006)1463 2550 y FA(whic)m(h)i(are)f(of)h(the)g(form)1651 2756 y Fp(V)1708 2771 y Fk(\006)1795 2756 y Fq(=)27 b Fp(V)1977 2715 y Fk(\006)1955 2781 y Fl(0)2058 2756 y Fn(\006)c Fp(';)1272 b FA(\(4.1\))178 2962 y(where)35 b Fp(V)540 2921 y Fk(\006)518 2986 y Fl(0)633 2962 y FA(are)f(external)g(p)s(oten)m(tials)e(represen)m(ting)j(the)g 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4056 y Fk(\006)1540 4041 y FA(are)f(p)s(ositiv)m(e)g(and)h(ob)s(ey)g Fp(m)2567 4056 y Fk(\006)2626 4041 y Fp(;)2747 4002 y Fl(1)p 2707 4018 115 4 v 2707 4076 a Fj(m)2769 4085 y Fh(\006)2860 4041 y Fn(2)c Fp(L)3020 4005 y Fk(1)3095 4041 y FA(.)43 b(As)32 b(ab)s(o)m(v)m(e,)h(w)m(e)415 4174 y(use)g(the)g(con)m(v)m(en)m(tion)h Fn(k)p Fp(m)1374 4189 y Fk(\006)1433 4174 y Fn(k)27 b Fq(:=)h Fn(k)p Fp(m)1776 4189 y Fk(\006)1835 4174 y Fn(k)1885 4189 y Fj(L)1933 4170 y Fh(1)2002 4174 y FA(.)178 4374 y(\()p Fp(A)289 4333 y Fk(\006)289 4398 y Fl(2)348 4374 y FA(\))h(The)k(b)s(oundary)g (co)s(e\036cien)m(ts)h Fp(\024)1604 4338 y Fk(\006)1604 4398 y Fj(a)1663 4374 y Fp(;)17 b(\024)1763 4333 y Fk(\006)1763 4402 y Fj(b)1854 4374 y FA(are)33 b(from)e(the)i(upp)s(er)g(half)e (plane)h Fo(C)3212 4389 y Fl(+)3277 4374 y FA(.)178 4573 y(\()p Fp(A)289 4532 y Fk(\006)289 4598 y Fl(3)348 4573 y FA(\))d(The)k(external)g(p)s(oten)m(tials)e Fp(V)1520 4532 y Fk(\006)1499 4598 y Fl(0)1612 4573 y FA(b)s(elong)g(to)i Fp(L)2109 4537 y Fk(1)2184 4573 y FA(.)178 4773 y(\()p Fp(A)289 4731 y Fk(\006)289 4797 y Fl(4)348 4773 y FA(\))c(The)k (matrix)e(v)-5 b(alued-functions)32 b Fp(\045)1709 4788 y Fk(\006)1768 4773 y Fq(\()p Fn(\001)p Fq(\))27 b Fn(2)h Fp(L)2059 4737 y Fk(1)2134 4773 y Fq(\()p Fo(R)5 b Fp(;)17 b Fo(C)2354 4737 y Fl(2)2399 4773 y Fq(\))33 b FA(satisfy)h(\(2.56\))o (.)178 4972 y(\()p Fp(A)289 4987 y Fl(5)329 4972 y FA(\))48 b(The)33 b(doping)f(pro\034le)g Fp(C)39 b FA(b)s(elongs)32 b(to)g Fp(W)1917 4925 y Fk(\000)p Fl(1)p Fj(;)p Fl(2)1903 4999 y(\000)2066 4972 y FA(.)178 5172 y(\()p Fp(A)289 5187 y Fl(6)329 5172 y FA(\))48 b(The)41 b(dielectric)e(p)s(ermittivit) m(y)g Fp(\017)i FA(is)f(p)s(ositiv)m(e)g(and)h(ob)s(eys)g Fp(\017;)2771 5132 y Fl(1)p 2771 5149 36 4 v 2774 5206 a Fj(\017)2858 5172 y Fn(2)h Fp(L)3032 5135 y Fk(1)3107 5172 y FA(.)67 b(W)-8 b(e)41 b(set)i Fq(~)-50 b Fp(\017)42 b Fq(:=)415 5292 y(max)o Fn(f)p Fq(1)p Fp(;)17 b Fn(k)799 5253 y Fl(1)p 799 5269 V 802 5326 a Fj(\017)843 5292 y Fn(k)893 5307 y Fj(L)941 5288 y Fh(1)1010 5292 y Fn(g)p FA(.)178 5491 y(\()p Fp(A)289 5506 y Fl(7)329 5491 y FA(\))48 b(The)34 b(set)g Fq(\000)f FA(is)g(not)g(empt)m(y)-8 b(,)34 b(or)f(at)g(least)g(one)h(of)f(the)g(n)m(um)m(b)s(ers)h Fp(k)s Fq(\()p Fp(x)p Fq(\))p FA(,)g Fp(x)c Fn(2)f(f)p Fp(a;)17 b(b)p Fn(g)23 b(n)g Fq(\000)p FA(,)33 b(is)415 5612 y(strictly)e(p)s(ositiv)m(e.)p eop %%Page: 23 25 23 24 bop 178 46 a Ft(Dissip)-8 b(a)g(tive)36 b(Schr)m (\366dinger-Poisson)f(systems)1383 b FA(23)178 302 y(\()p Fp(A)289 317 y Fl(8)329 302 y FA(\))48 b(The)33 b(function)f Fp(')1061 317 y Fl(\000)1142 302 y FA(is)g(from)f(the)i(set)g Fp(W)1896 255 y Fl(1)p Fj(;)p Fl(2)1882 329 y Fg(R)1990 302 y FA(.)178 535 y(Eac)m(h)c(Sc)m(hr\366)s(dinger-t)m(yp)s(e)e(op)s (erator)g Fp(H)1643 499 y Fk(\006)1701 535 y Fq(\()p Fp(V)1796 550 y Fk(\006)1855 535 y Fq(\))g FA(corresp)s(onds)i(a)e (minimal)c(self-adjoin)m(t)j(dilation)178 655 y Fp(K)268 619 y Fk(\006)328 655 y Fq(\()p Fp(V)423 670 y Fk(\006)481 655 y Fq(\))p FA(.)76 b(In)44 b(accordance)g(with)f(section)g(2.5)g (the)h(functions)f Fp(\045)2661 670 y Fk(\006)2720 655 y Fq(\()p Fn(\001)p Fq(\))g FA(de\034ne)h(steady)h(states)178 776 y Fp(\045)228 791 y Fk(\006)288 776 y Fq(\()p Fp(V)383 791 y Fk(\006)442 776 y Fq(\))p FA(,)d(i.e.)68 b(no-negativ)m(e)40 b(self-adjoin)m(t)f(op)s(erators)i(whic)m(h)g(comm)m(ute)f(with)g Fp(K)3255 739 y Fk(\006)3314 776 y Fq(\()p Fp(V)3409 791 y Fk(\006)3468 776 y Fq(\))p FA(.)68 b(By)178 896 y(section)23 b(2.6)g(one)g(can)g(in)m(tro)s(duce)g(carrier)f(densities) h Fp(u)2146 855 y Fk(\006)2152 924 y Fl(^)-41 b Fj(\045)2182 933 y Fh(\006)2238 896 y Fq(\()p Fp(V)2333 911 y Fk(\006)2392 896 y Fq(\))22 b FA(for)h(electrons)g(and)g(holes.)40 b(Notice)178 1029 y(that)g(the)g(electron)g(carrier)f(densit)m(y)i Fp(u)1664 988 y Fk(\000)1670 1057 y Fl(^)-41 b Fj(\045)1700 1066 y Fh(\000)1756 1029 y Fq(\()p Fp(V)1851 1044 y Fk(\000)1910 1029 y Fq(\))40 b FA(is)f(determined)h(b)m(y)h(the)f(electron)g(quan)m (tities)178 1162 y Fp(m)263 1177 y Fk(\000)323 1162 y FA(,)i Fp(\024)448 1126 y Fk(\000)448 1187 y Fj(a)507 1162 y Fp(;)17 b(\024)607 1121 y Fk(\000)607 1190 y Fj(b)706 1162 y FA(and)40 b Fp(V)982 1121 y Fk(\000)960 1187 y Fl(0)1081 1162 y FA(while)f(the)i(hole)f(carrier)f(densit)m(y)i Fp(u)2449 1121 y Fl(+)2455 1190 y(^)-41 b Fj(\045)2485 1199 y Ff(+)2540 1162 y Fq(\()p Fp(V)2635 1177 y Fl(+)2694 1162 y Fq(\))40 b FA(b)m(y)h(the)f(hole)g(quan)m(tities)178 1296 y Fp(m)263 1311 y Fl(+)323 1296 y FA(,)i Fp(\024)448 1260 y Fl(+)448 1320 y Fj(a)507 1296 y Fp(;)17 b(\024)607 1255 y Fl(+)607 1324 y Fj(b)706 1296 y FA(and)40 b Fp(V)981 1255 y Fl(+)960 1320 y(0)1040 1296 y FA(.)66 b(The)41 b(corresp)s(onding)e(carrier)g(densit)m(y)i(op)s(erators)f(are)g (denoted)h(b)m(y)178 1416 y Fn(N)275 1375 y Fk(\006)266 1444 y Fl(^)-41 b Fj(\045)296 1453 y Fh(\006)352 1416 y Fq(\()p Fn(\001)p Fq(\))p FA(.)178 1749 y Fs(4.1)131 b(Rigorous)43 b(de\034nition)178 1981 y FA(A)m(t)29 b(\034rst)f(w)m(e)h (will)d(giv)m(e)i(a)g(rigorous)e(de\034nition)h(of)h(P)m(oisson's)h (equation)f(and)g(afterw)m(ards)g(de\034ne)178 2101 y(what)33 b(w)m(e)h(will)c(call)h(a)h(solution)f(of)h(the)h(dissipativ)m(e)f(Sc)m (hr\366)s(dinger)g(P)m(oisson)h(system.)178 2335 y Fr(De\034nition)j (4.3)49 b FA(W)-8 b(e)33 b(de\034ne)h(the)f(P)m(oisson)g(op)s(erator)f Fn(P)k Fq(:)28 b Fp(W)2522 2287 y Fl(1)p Fj(;)p Fl(2)2508 2362 y Fg(R)2643 2335 y Fn(\000)-16 b(!)27 b Fp(W)2937 2287 y Fk(\000)p Fl(1)p Fj(;)p Fl(2)2923 2362 y(\000)3119 2335 y FA(as)32 b(usual)h(b)m(y)305 2605 y Fp(<)27 b Fn(P)8 b Fp(\035)t(;)17 b(&)35 b(>)732 2620 y Fl(1)772 2605 y Fq(=)875 2469 y Fm(Z)975 2495 y Fj(b)931 2695 y(a)1026 2605 y Fp(dx)28 b(\017)1237 2537 y(d\035)p 1237 2582 107 4 v 1237 2673 a(dx)1369 2537 y(d&)p 1363 2582 V 1363 2673 a(dx)1502 2605 y Fq(+)1690 2510 y Fm(X)1600 2726 y Fj(x)p Fk(2f)p Fj(a;b)p Fk(gn)p Fl(\000)1940 2605 y Fp(k)s Fq(\()p Fp(x)p Fq(\))p Fp(\035)t Fq(\()p Fp(x)p Fq(\))p Fp(&)8 b Fq(\()p Fp(x)p Fq(\))p Fp(;)240 b(\035)31 b Fn(2)d Fp(W)3038 2557 y Fl(1)p Fj(;)p Fl(2)3024 2632 y Fg(R)3132 2605 y Fp(;)44 b(&)36 b Fn(2)28 b Fp(W)3474 2557 y Fl(1)p Fj(;)p Fl(2)3460 2632 y(\000)3568 2605 y Fp(:)3521 2836 y FA(\(4.2\))178 2956 y(The)34 b(restriction)d(of)h Fn(P)41 b FA(to)33 b(the)g(subspace)h Fp(W)1867 2909 y Fl(1)p Fj(;)p Fl(2)1853 2983 y(\000)1993 2956 y FA(will)d(b)s(e)h (denoted)i(b)m(y)f Fn(P)2883 2971 y Fl(0)2923 2956 y FA(.)178 3189 y(W)-8 b(e)33 b(ha)m(v)m(e)670 3481 y Fn(j)27 b Fp(<)h Fn(P)8 b Fp(\035)t(;)17 b(&)35 b(>)1153 3496 y Fl(1)1220 3481 y Fn(j)27 b(\024)1381 3280 y Fm(0)1381 3460 y(@)1468 3481 y Fn(k)p Fp(\017)p Fn(k)1607 3496 y Fj(L)1655 3477 y Fh(1)1746 3481 y Fq(+)1934 3386 y Fm(X)1844 3602 y Fj(x)p Fk(2f)p Fj(a;b)p Fk(gn)p Fl(\000)2185 3481 y Fp(k)s Fq(\()p Fp(x)p Fq(\))17 b Fp(")2433 3439 y Fl(2)2433 3505 y Fj(c)2472 3280 y Fm(1)2472 3460 y(A)2575 3481 y Fn(k)p Fp(\035)t Fn(k)2732 3511 y Fj(W)2809 3478 y Ff(1)p Fe(;)p Ff(2)2799 3534 y Fa(R)2896 3481 y Fn(k)p Fp(&)8 b Fn(k)3039 3511 y Fj(W)3116 3478 y Ff(1)p Fe(;)p Ff(2)3106 3534 y(\000)3203 3481 y Fp(:)178 3791 y FA(Hence)34 b Fn(P)41 b FA(is)32 b(con)m(tin)m(uous.)44 b(F)-8 b(urthermore)32 b(w)m(e)i(get)266 4102 y Fn(k)p Fp(')p Fn(k)430 4061 y Fl(2)430 4141 y Fj(W)507 4109 y Ff(1)p Fe(;)p Ff(2)497 4165 y(\000)622 4102 y Fn(\024)28 b Fq(\(1)22 b(+)g Fp(\015)985 4117 y Fj(k)1027 4102 y Fq(\))1082 3902 y Fm(0)1082 4081 y(@)1169 3966 y(Z)1269 3993 y Fj(b)1224 4192 y(a)1319 4102 y Fn(j)p Fp(')1411 4061 y Fk(0)1434 4102 y Fq(\()p Fp(x)p Fq(\))p Fn(j)1593 4061 y Fl(2)1649 4102 y Fp(dx)h Fq(+)1966 4007 y Fm(X)1876 4223 y Fj(x)p Fk(2f)p Fj(a;b)p Fk(gn)p Fl(\000)2216 4102 y Fp(k)s Fq(\()p Fp(x)p Fq(\))p Fn(j)p Fp(')p Fq(\()p Fp(x)p Fq(\))p Fn(j)2652 4061 y Fl(2)2692 3902 y Fm(1)2692 4081 y(A)2795 4102 y Fp(;)115 b FA(for)32 b(all)e Fp(')e Fn(2)g Fp(W)3513 4055 y Fl(1)p Fj(;)p Fl(2)3499 4129 y(\000)3607 4102 y Fp(;)3521 4339 y FA(\(4.3\))178 4460 y(with)874 4642 y Fp(\015)925 4657 y Fj(k)995 4642 y Fq(:=)127 b(sup)1126 4742 y Fl(0)p Fk(6)p Fl(=)p Fj( )r Fk(2)p Fj(W)1388 4710 y Ff(1)p Fe(;)p Ff(2)1378 4765 y(\000)2070 4488 y Fm(R)2137 4514 y Fj(b)2117 4603 y(a)2188 4568 y Fp( )2255 4532 y Fl(2)2311 4568 y Fp(dx)p 1498 4619 1491 4 v 1498 4652 a Fm(R)1565 4678 y Fj(b)1545 4766 y(a)1599 4651 y Fm(\000)1655 4688 y Fj(d )p 1655 4709 85 4 v 1659 4766 a(dx)1749 4651 y Fm(\001)1795 4673 y Fl(2)1834 4732 y Fp(dx)23 b Fq(+)2061 4657 y Fm(P)2166 4761 y Fj(x)p Fk(2f)p Fj(a;b)p Fk(gn)p Fl(\000)2511 4732 y Fp(k)s Fq(\()p Fp(x)p Fq(\))p Fn(j)p Fp( )t Fq(\()p Fp(x)p Fq(\))p Fn(j)2950 4703 y Fl(2)2999 4642 y Fp(:)495 b FA(\(4.4\))178 4906 y(Because)37 b(the)e(case)h(of)f(purely)g (homogeneous)g(Neumann)g(conditions)f(is)g(excluded)i(b)m(y)g(\()p Fp(A)3617 4921 y Fl(7)3657 4906 y FA(\),)178 5026 y(the)d(constan)m(t)h Fp(\015)791 5041 y Fj(k)865 5026 y FA(is)f(indeed)f(\034nite.)44 b(Th)m(us)34 b(w)m(e)f(get)g(b)m(y)i(\(4.3\))1199 5225 y Fn(k)p Fp(')p Fn(k)1363 5184 y Fl(2)1363 5265 y Fj(W)1440 5232 y Ff(1)p Fe(;)p Ff(2)1430 5288 y(\000)1554 5225 y Fn(\024)29 b Fq(~)-50 b Fp(\017)q Fq(\(1)22 b(+)g Fp(\015)1957 5240 y Fj(k)1999 5225 y Fq(\))p Fn(j)27 b Fp(<)h Fn(P)2265 5240 y Fl(0)2304 5225 y Fp(';)17 b(')28 b(>)2580 5240 y Fl(1)2647 5225 y Fn(j)p Fp(:)296 5491 y FA(Therefore)j(w)m(e)f(get)g (b)m(y)g(the)g(Lax-Milgram)c(lemma)i(that)h(the)h(in)m(v)m(erse)g(of)f Fn(P)3096 5506 y Fl(0)3165 5491 y FA(exists)i(and)e(its)178 5612 y(norm)j(do)s(es)h(not)f(exeed)j Fq(~)-49 b Fp(\017)p Fq(\(1)22 b(+)g Fp(\015)1392 5627 y Fj(k)1434 5612 y Fq(\))p FA(.)p eop %%Page: 24 26 24 25 bop 178 46 a FA(24)896 b Ft(M.)36 b(Bar)m(o,)f(H.-Ch.)47 b(Kaiser,)36 b(H.)g(Neidhardt,)f(J.)i(Rehber)m(g)296 295 y FA(W)-8 b(e)33 b(denote)h(b)m(y)49 b Fq(~)-65 b Fp(')978 310 y Fl(\000)1059 295 y FA(the)33 b(form)1038 564 y Fp(\035)d Fn(7\000)-16 b(!)1310 429 y Fm(Z)1409 455 y Fj(b)1365 654 y(a)1460 564 y Fp(dx)28 b(\017)p Fq(\()p Fp(x)p Fq(\))p Fp(')1828 523 y Fk(0)1828 589 y Fl(\000)1877 564 y Fq(\()p Fp(x)p Fq(\))p Fp(\035)2065 523 y Fk(0)2087 564 y Fq(\()p Fp(x)p Fq(\))p Fp(;)213 b(\035)30 b Fn(2)e Fp(W)2741 517 y Fl(1)p Fj(;)p Fl(2)2727 591 y(\000)2835 564 y Fp(:)178 872 y Fr(De\034nition)36 b(4.4)49 b FA(Assume)44 b Fp(u)1321 835 y Fk(\006)1424 872 y Fn(2)h Fp(L)1601 835 y Fl(1)1641 872 y FA(.)73 b(W)-8 b(e)43 b(sa)m(y)g Fp(')i Fn(2)g Fp(W)2423 824 y Fl(1)p Fj(;)p Fl(2)2409 899 y Fg(R)2559 872 y FA(satis\034es)f(P)m (oisson's)f(equation)178 992 y(\(1.1\))32 b(i\033)g Fp(')22 b Fn(\000)g Fp(')777 1007 y Fl(\000)853 992 y Fn(2)28 b Fp(W)1053 945 y Fl(1)p Fj(;)p Fl(2)1039 1019 y(\000)1180 992 y FA(satis\034es)1289 1199 y Fn(P)1358 1214 y Fl(0)1398 1199 y Fq(\()p Fp(')22 b Fn(\000)g Fp(')1685 1214 y Fl(\000)1733 1199 y Fq(\))28 b(=)f Fp(D)e Fq(+)d Fp(q)f(u)2226 1158 y Fl(+)2306 1199 y Fn(\000)i Fp(q)d(u)2525 1158 y Fk(\000)2584 1199 y Fp(;)910 b FA(\(4.5\))178 1406 y(where)34 b Fp(D)c Fq(:=)e Fp(q)t(C)h Fn(\000)38 b Fq(~)-64 b Fp(')1012 1421 y Fl(\000)1060 1406 y FA(.)178 1645 y(Of)35 b(course,)h(the)g (righ)m(t)e(hand)h(side)g(of)f(\(4.5\))g(is)h(to)f(b)s(e)h(understo)s (o)s(d)h(in)e Fp(W)2940 1598 y Fk(\000)p Fl(1)p Fj(;)p Fl(2)2926 1672 y(\000)3123 1645 y FA(b)m(y)i(em)m(b)s(edding)178 1765 y Fp(u)234 1729 y Fl(+)293 1765 y Fp(;)d(u)409 1729 y Fk(\000)495 1765 y Fn(2)c Fp(L)656 1729 y Fl(1)723 1765 y Fp(,)-17 b Fn(!)28 b Fp(W)967 1718 y Fk(\000)p Fl(1)p Fj(;)p Fl(2)953 1792 y(\000)1116 1765 y FA(.)178 2005 y Fr(De\034nition)36 b(4.5)49 b FA(W)-8 b(e)49 b(sa)m(y)g(a)f (triple)f Fq(\()p Fp(';)17 b(u)1836 1969 y Fl(+)1894 2005 y Fp(;)g(u)1994 1969 y Fk(\000)2052 2005 y Fq(\))54 b Fn(2)h Fp(W)2371 1957 y Fl(1)p Fj(;)p Fl(2)2357 2032 y Fg(R)2498 2005 y Fn(\002)33 b Fp(L)2674 1969 y Fl(1)2747 2005 y Fn(\002)g Fp(L)2923 1969 y Fl(1)3011 2005 y FA(satis\034es)49 b(the)f(dis-)178 2125 y(sipativ)m(e)j(Sc)m(hr\366)s(dinger)h(P)m (oisson)f(system)i(if)d Fp(')h FA(satis\034es)i(P)m(oisson's)f (equation)f(as)h(w)m(ell)e(as)178 2245 y Fp(u)234 2209 y Fl(+)321 2245 y Fq(=)27 b Fp(u)480 2204 y Fl(+)486 2273 y(^)-41 b Fj(\045)516 2282 y Ff(+)571 2245 y Fq(\()p Fp(V)687 2204 y Fl(+)666 2270 y(0)768 2245 y Fq(+)22 b Fp(')p Fq(\))33 b FA(and)f Fp(u)1246 2209 y Fk(\000)1333 2245 y Fq(=)27 b Fp(u)1492 2204 y Fk(\000)1498 2273 y Fl(^)-41 b Fj(\045)1528 2282 y Fh(\000)1584 2245 y Fq(\()p Fp(V)1701 2204 y Fk(\000)1679 2270 y Fl(0)1782 2245 y Fn(\000)22 b Fp(')p Fq(\))p FA(.)178 2580 y Fs(4.2)131 b(Existence)45 b(of)f(solutions)g(and)g(a)f(priori)i(estimates)178 2811 y FA(The)35 b(aim)c(of)i(this)g(section)g(is)g(to)g(pro)m(v)m(e)h (that)g(the)f(dissipativ)m(e)g(Sc)m(hr\366)s(dinger)g(P)m(oisson)h (system)178 2932 y(alw)m(a)m(ys)j(admits)e(a)h(solution)e(and)i(to)g (in)m(v)m(estigate)g(these)h(solutions.)53 b(A)m(t)37 b(\034rst)f(in)g(accordance)178 3052 y(with)e([5])g(w)m(e)g(de\034ne)h (a)f(mapping)e(whose)j(\034xed)g(p)s(oin)m(ts)f(exactly)g(determine)f (the)h(solutions)f(of)178 3173 y(the)g(dissipativ)m(e)f(Sc)m(hr\366)s (dinger)h(P)m(oisson)f(system.)296 3340 y(By)f Fn(J)45 b Fq(:)28 b Fp(L)681 3304 y Fl(1)737 3340 y Fn(\002)16 b Fp(L)896 3304 y Fl(1)964 3340 y Fn(\000)-16 b(!)28 b Fp(W)1259 3293 y Fl(1)p Fj(;)p Fl(2)1245 3368 y Fg(R)1382 3340 y FA(w)m(e)j(denote)f(the)g(map)f(whic)m(h)h(assigns)g(to)g Fq(\()p Fp(u)3024 3304 y Fl(+)3082 3340 y Fp(;)17 b(u)3182 3304 y Fk(\000)3240 3340 y Fq(\))28 b Fn(2)g Fp(L)3466 3304 y Fl(1)3522 3340 y Fn(\002)16 b Fp(L)3681 3304 y Fl(1)178 3461 y FA(the)39 b(solution)e(of)h(P)m(oisson's)h(equation.)61 b(Ob)m(viously)-8 b(,)39 b(the)g(map)f Fn(J)56 b FA(is)38 b(con)m(tin)m(uous.)62 b(F)-8 b(urther,)178 3581 y(w)m(e)34 b(de\034ne)g Fq(\011)27 b(:)h Fp(L)828 3545 y Fk(1)931 3581 y Fn(\000)-16 b(!)27 b Fp(W)1225 3534 y Fl(1)p Fj(;)p Fl(2)1211 3608 y Fg(R)1351 3581 y FA(b)m(y)251 3788 y Fq(\011)h(:)g Fp(V)49 b Fn(\000)-16 b(!)704 3707 y Fm(\000)750 3788 y Fn(N)847 3747 y Fl(+)838 3816 y(^)-41 b Fj(\045)868 3825 y Ff(+)922 3788 y Fq(\()p Fp(V)1039 3747 y Fl(+)1017 3812 y(0)1120 3788 y Fq(+)22 b Fp(V)g Fq(\))p Fp(;)17 b Fn(N)1476 3747 y Fk(\000)1467 3816 y Fl(^)-41 b Fj(\045)1497 3825 y Fh(\000)1552 3788 y Fq(\()p Fp(V)1669 3747 y Fk(\000)1647 3812 y Fl(0)1750 3788 y Fn(\000)23 b Fp(V)e Fq(\))1966 3707 y Fm(\001)2040 3788 y Fn(\000)-17 b(!)28 b(J)2314 3707 y Fm(\000)2359 3788 y Fn(N)2456 3747 y Fl(+)2447 3816 y(^)-41 b Fj(\045)2477 3825 y Ff(+)2532 3788 y Fq(\()p Fp(V)2649 3747 y Fl(+)2627 3812 y(0)2730 3788 y Fq(+)22 b Fp(V)f Fq(\))p Fp(;)c Fn(N)3085 3747 y Fk(\000)3076 3816 y Fl(^)-41 b Fj(\045)3106 3825 y Fh(\000)3162 3788 y Fq(\()p Fp(V)3279 3747 y Fk(\000)3257 3812 y Fl(0)3360 3788 y Fn(\000)22 b Fp(V)g Fq(\))3576 3707 y Fm(\001)3622 3788 y Fp(:)296 4064 y FA(Since)h(the)g(map)f Fn(J)46 b Fq(:)28 b Fp(L)1141 4027 y Fl(1)1183 4064 y Fn(\002)r Fp(L)1328 4027 y Fl(1)1396 4064 y Fn(\000)-16 b(!)27 b Fp(W)1690 4016 y Fl(1)p Fj(;)p Fl(2)1676 4091 y Fg(R)1807 4064 y FA(is)22 b(con)m(tin)m(uous)h(and)g(b)m(y)h(Prop)s(osition)d (3.5)h(the)h(maps)178 4184 y Fn(N)275 4148 y Fk(\006)334 4184 y Fq(\()p Fn(\001)p Fq(\))36 b(:)h Fp(L)604 4148 y Fk(1)717 4184 y Fn(\000)-17 b(!)37 b Fp(L)980 4148 y Fl(1)1058 4184 y FA(are)h(also)f(con)m(tin)m(uous,)j(the)f(map)e Fq(\011)g(:)g Fp(L)2588 4148 y Fk(1)2700 4184 y Fn(\000)-16 b(!)36 b Fp(W)3003 4137 y Fl(1)p Fj(;)p Fl(2)2989 4211 y Fg(R)3135 4184 y FA(is)i(con)m(tin)m(uous,)178 4304 y(to)s(o.)43 b(Let)33 b Fp(E)634 4319 y Fk(1)736 4304 y Fq(:)28 b Fp(W)897 4257 y Fl(1)p Fj(;)p Fl(2)883 4332 y Fg(R)1019 4304 y Fn(\000)-16 b(!)27 b Fp(L)1273 4268 y Fk(1)1381 4304 y FA(denote)33 b(the)g(em)m(b)s(edding)f(op)s(erator)g (of)g Fp(W)2966 4257 y Fl(1)p Fj(;)p Fl(2)2952 4332 y Fg(R)3093 4304 y FA(in)m(to)g Fp(L)3357 4268 y Fk(1)3432 4304 y FA(.)43 b(With)178 4425 y Fq(\011)33 b FA(w)m(e)g(asso)s(ciate)f (the)h(map)f Fq(\011)1301 4440 y Fk(1)1403 4425 y Fq(:)c Fp(L)1524 4388 y Fk(1)1627 4425 y Fn(\000)-16 b(!)27 b Fp(L)1881 4388 y Fk(1)1956 4425 y FA(,)1671 4631 y Fq(\011)1747 4646 y Fk(1)1849 4631 y Fq(:=)h Fp(E)2052 4646 y Fk(1)2126 4631 y Fq(\011)p Fp(;)178 4838 y FA(whic)m(h)34 b(is)f(also)g(con)m(tin)m(uous.)47 b(Moreo)m(v)m(er,)35 b(since)f Fp(E)2051 4853 y Fk(1)2159 4838 y FA(is)f(compact)h(the)f (map)g Fq(\011)3112 4853 y Fk(1)3220 4838 y FA(is)g(also)g(com-)178 4959 y(pact.)178 5198 y Fr(Lemma)k(4.6)49 b Fc(A)n(n)35 b(element)f Fp(')28 b Fn(2)g Fp(L)1540 5162 y Fk(1)1650 5198 y Fc(is)34 b(a)h(\034xe)-5 b(d)34 b(p)-5 b(oint)34 b(of)h Fq(\011)2500 5213 y Fk(1)2609 5198 y Fc(if)g(and)f(only)h(if)g (the)f(triple)997 5405 y Fq(\()p Fp(';)17 b(u)1199 5364 y Fl(+)1257 5405 y Fp(;)g(u)1357 5364 y Fk(\000)1415 5405 y Fq(\))28 b(=)g(\()p Fp(';)17 b(u)1787 5364 y Fl(+)1793 5433 y(^)-41 b Fj(\045)1823 5442 y Ff(+)1877 5405 y Fq(\()p Fp(V)1993 5364 y Fl(+)1972 5430 y(0)2074 5405 y Fq(+)22 b Fp(')p Fq(\))p Fp(;)17 b(u)2374 5364 y Fk(\000)2380 5433 y Fl(^)-41 b Fj(\045)2410 5442 y Fh(\000)2466 5405 y Fq(\()p Fp(V)2582 5364 y Fk(\000)2561 5430 y Fl(0)2663 5405 y Fn(\000)23 b Fp(')p Fq(\)\))178 5612 y Fc(satis\034es)34 b(the)h(dissip)-5 b(ative)34 b(Schr)-5 b(\366)g(dinger)33 b(Poisson)h(system.)p eop %%Page: 25 27 25 26 bop 178 46 a Ft(Dissip)-8 b(a)g(tive)36 b(Schr)m (\366dinger-Poisson)f(systems)1383 b FA(25)178 295 y(The)35 b(pro)s(of)d(is)h(ob)m(vious.)46 b(T)-8 b(o)33 b(pro)m(v)m(e)i(the)e (cen)m(tral)g(results)h(of)f(this)g(c)m(hapter,)h(w)m(e)h(also)d(need)i (the)178 415 y(follo)m(wing)c(tec)m(hnical)178 647 y Fr(Lemma)37 b(4.7)49 b Fc(L)-5 b(et)32 b Fp(\033)987 662 y Fl(1)1027 647 y Fp(;)17 b(\033)1126 662 y Fl(2)1166 647 y Fp(;)g(\033)1265 662 y Fl(3)1336 647 y Fc(b)-5 b(e)31 b(thr)-5 b(e)g(e)31 b(strictly)h(p)-5 b(ositive)31 b(numb)-5 b(ers)31 b(and)g(let)h Fp(x)3117 662 y Fl(0)3188 647 y Fc(b)-5 b(e)31 b(the)h(smal)5 b(l-)178 767 y(est)32 b(p)-5 b(ositive)31 b(r)-5 b(o)g(ot)32 b(of)g(the)g(p)-5 b(olynomial)31 b Fp(p)c Fq(:)h Fp(x)g Fn(7!)f Fp(x)2025 731 y Fl(4)2081 767 y Fn(\000)16 b Fp(\033)2229 782 y Fl(1)2269 767 y Fp(x)2324 731 y Fl(2)2379 767 y Fn(\000)g Fp(\033)2527 782 y Fl(2)2567 767 y Fp(x)g Fn(\000)g Fp(\033)2786 782 y Fl(3)2826 767 y Fc(.)44 b(Then)31 b(for)h(al)5 b(l)31 b Fp(x)d(>)g(x)3682 782 y Fl(0)178 887 y Fc(one)35 b(has)f Fp(p)p Fq(\()p Fp(x)p Fq(\))28 b Fp(>)f Fq(0)p Fc(.)45 b(In)34 b(p)-5 b(articular,)35 b Fp(p)f Fc(do)-5 b(es)35 b(not)f(admit)h(other)g(p)-5 b(ositive)34 b(r)-5 b(o)g(ots.)178 1119 y Fr(Pro)s(of.)78 b FA(It)44 b(is)g(clear)f(that)h (at)g(least)g(one)g(p)s(ositiv)m(e)f(ro)s(ot)h(m)m(ust)g(exist.)78 b(Then)45 b(one)g(has)f(for)178 1239 y Fp(x)28 b Fq(=)g Fp(tx)455 1254 y Fl(0)528 1239 y FA(with)k Fp(t)c(>)f Fq(1)p FA(:)595 1437 y Fp(p)p Fq(\()p Fp(x)p Fq(\))h(=)g Fp(t)942 1396 y Fl(4)982 1437 y Fp(x)1037 1396 y Fl(4)1037 1462 y(0)1099 1437 y Fn(\000)22 b Fp(\033)1253 1452 y Fl(1)1293 1437 y Fp(t)1328 1396 y Fl(2)1368 1437 y Fp(x)1423 1396 y Fl(2)1423 1462 y(0)1485 1437 y Fn(\000)g Fp(\033)1639 1452 y Fl(2)1679 1437 y Fp(tx)1769 1452 y Fl(0)1831 1437 y Fn(\000)h Fp(\033)1986 1452 y Fl(3)915 1582 y Fq(=)83 b Fp(t)1109 1541 y Fl(4)1148 1582 y Fq(\()p Fp(\033)1241 1597 y Fl(1)1281 1582 y Fp(x)1336 1541 y Fl(2)1336 1607 y(0)1398 1582 y Fq(+)22 b Fp(\033)1551 1597 y Fl(2)1591 1582 y Fp(x)1646 1597 y Fl(0)1708 1582 y Fq(+)g Fp(\033)1861 1597 y Fl(3)1901 1582 y Fq(\))g Fn(\000)g Fp(\033)2115 1597 y Fl(1)2155 1582 y Fp(t)2190 1541 y Fl(2)2230 1582 y Fp(x)2285 1541 y Fl(2)2285 1607 y(0)2347 1582 y Fn(\000)h Fp(\033)2502 1597 y Fl(2)2542 1582 y Fp(tx)2632 1597 y Fl(0)2694 1582 y Fn(\000)f Fp(\033)2848 1597 y Fl(3)915 1728 y Fq(=)83 b Fp(\033)1129 1743 y Fl(1)1168 1728 y Fp(t)1203 1686 y Fl(2)1243 1728 y Fq(\()p Fp(t)1316 1686 y Fl(2)1378 1728 y Fn(\000)22 b Fq(1\))p Fp(x)1619 1686 y Fl(2)1619 1752 y(0)1681 1728 y Fq(+)g Fp(\033)1834 1743 y Fl(2)1874 1728 y Fp(t)p Fq(\()p Fp(t)1982 1686 y Fl(3)2044 1728 y Fn(\000)h Fq(1\))p Fp(x)2286 1743 y Fl(0)2347 1728 y Fq(+)f Fp(\033)2500 1743 y Fl(3)2540 1728 y Fq(\()p Fp(t)2613 1686 y Fl(4)2675 1728 y Fn(\000)h Fq(1\))k Fp(>)h Fq(0)p Fp(:)178 1925 y FA(This)33 b(sho)m(ws)h(that)e (a)h(p)s(ositiv)m(e)f(ro)s(ot)f(larger)h(than)g Fp(x)2102 1940 y Fl(0)2174 1925 y FA(do)s(es)h(not)g(exists.)813 b Fb(\003)178 2093 y FA(Let)33 b Fp(c)395 2108 y Fk(\006)487 2093 y FA(b)s(e)f(the)h(constan)m(ts)h(de\034ned)g(b)m(y)g(\(2.21\).)42 b(W)-8 b(e)33 b(sp)s(ecify)g Fp(\033)2556 2108 y Fl(1)2596 2093 y Fp(;)17 b(\033)2695 2108 y Fl(2)2734 2093 y Fp(;)g(\033)2833 2108 y Fl(3)2906 2093 y FA(to)535 2318 y Fp(\033)590 2333 y Fl(1)713 2318 y Fq(:=)83 b(8)948 2231 y Fn(p)p 1031 2231 49 4 v 87 x Fq(2)p Fp(q)t Fq(\()p Fp(b)22 b Fn(\000)h Fp(a)p Fq(\))p Fp(")1463 2333 y Fl(1)1502 2318 y Fp(")1548 2333 y Fk(1)1623 2318 y Fq(~)-50 b Fp(\017)q Fq(\(1)22 b(+)g Fp(\015)1920 2333 y Fj(k)1962 2318 y Fq(\))g Fn(\001)2072 2207 y Fm(\020)2131 2318 y Fp(C)2207 2333 y Fl(^)-41 b Fj(\045)2237 2342 y Ff(+)2292 2228 y Fm(p)p 2392 2228 244 4 v 90 x Fn(k)p Fp(m)2527 2333 y Fl(+)2586 2318 y Fn(k)22 b Fq(+)g Fp(C)2832 2333 y Fl(^)-41 b Fj(\045)2862 2342 y Fh(\000)2918 2228 y Fm(p)p 3018 2228 V 90 x Fn(k)p Fp(m)3153 2333 y Fk(\000)3212 2318 y Fn(k)3262 2207 y Fm(\021)3321 2318 y Fp(;)173 b FA(\(4.6\))535 2501 y Fp(\033)590 2516 y Fl(2)713 2501 y Fq(:=)83 b(16)27 b Fp(q)t Fi(g)1121 2516 y Fl(1)1161 2501 y Fp(")1207 2516 y Fl(1)1246 2501 y Fp(")1292 2516 y Fk(1)1367 2501 y Fq(~)-50 b Fp(\017)q Fq(\(1)21 b(+)h Fp(\015)1663 2516 y Fj(k)1706 2501 y Fq(\))g Fn(\001)1816 2420 y Fm(\000)1861 2501 y Fp(C)1937 2516 y Fl(^)-41 b Fj(\045)1967 2525 y Ff(+)2022 2501 y Fp(p)2071 2516 y Fl(+)2152 2501 y Fq(+)22 b Fp(C)2326 2516 y Fl(^)-41 b Fj(\045)2356 2525 y Fh(\000)2412 2501 y Fp(p)2461 2516 y Fk(\000)2520 2420 y Fm(\001)2583 2501 y Fp(;)911 b FA(\(4.7\))535 2646 y Fp(\033)590 2661 y Fl(3)713 2646 y Fq(:=)83 b Fp(")945 2661 y Fk(1)1042 2646 y Fn(\001)21 b(k)p Fp(')1205 2661 y Fl(\000)1253 2646 y Fn(k)1303 2676 y Fj(W)1380 2644 y Ff(1)p Fe(;)p Ff(2)1370 2700 y Fa(R)1490 2646 y Fq(+)h Fp(")1634 2661 y Fk(1)1709 2646 y Fq(~)-50 b Fp(\017)q Fq(\(1)21 b(+)h Fp(\015)2005 2661 y Fj(k)2048 2646 y Fq(\))g Fn(\001)g(k)p Fp(D)s Fn(k)2342 2676 y Fj(W)2419 2644 y Fh(\000)p Ff(1)p Fe(;)p Ff(2)2409 2699 y(\000)2575 2646 y Fq(+)870 b FA(\(4.8\))899 2808 y Fq(+)p Fp(q)t(")1068 2823 y Fl(1)1107 2808 y Fp(")1153 2823 y Fk(1)1228 2808 y Fq(~)-50 b Fp(\017)q Fq(\(1)22 b(+)g Fp(\015)1525 2823 y Fj(k)1567 2808 y Fq(\))g Fn(\001)g Fq(\()p Fp(C)1791 2823 y Fl(^)-41 b Fj(\045)1821 2832 y Ff(+)1875 2808 y Fp(r)1919 2823 y Fl(+)2001 2808 y Fq(+)22 b Fp(C)2175 2823 y Fl(^)-41 b Fj(\045)2205 2832 y Fh(\000)2261 2808 y Fp(r)2305 2823 y Fk(\000)2364 2808 y Fq(\))178 3005 y FA(where)1001 3126 y Fp(p)1050 3141 y Fk(\006)1137 3126 y Fq(:=)27 b(\(\()p Fp(\013)1406 3084 y Fk(\006)1405 3150 y Fj(a)1465 3126 y Fq(\))1503 3084 y Fl(2)1564 3126 y Fq(+)22 b(\()p Fp(\013)1763 3084 y Fk(\006)1762 3153 y Fj(b)1822 3126 y Fq(\))1860 3084 y Fl(2)1899 3126 y Fq(\))1937 3084 y Fl(1)p Fj(=)p Fl(2)2047 3126 y Fn(k)p Fq(\()p Fp(H)2224 3084 y Fk(\006)2216 3150 y Fl(0)2283 3126 y Fq(\))2321 3084 y Fk(\000)p Fl(1)p Fj(=)p Fl(2)2486 3126 y Fn(k)2536 3075 y Fl(1)p Fj(=)p Fl(2)2536 3158 y Fk(B)r Fl(\()p Fd(H)p Fj(;W)2758 3139 y Ff(1)p Fe(;)p Ff(2)2840 3158 y Fl(\))2872 3126 y Fp(;)622 b FA(\(4.9\))261 3436 y Fp(r)305 3451 y Fk(\006)447 3436 y Fq(:=)83 b(8)22 b(+)g(4)851 3349 y Fn(p)p 934 3349 49 4 v 87 x Fq(2)27 b(\()p Fp(b)c Fn(\000)g Fp(a)p Fq(\))1301 3346 y Fm(p)p 1400 3346 244 4 v 1400 3436 a Fn(k)p Fp(m)1535 3451 y Fk(\006)1594 3436 y Fn(k)f(\001)1716 3313 y Fm(q)p 1816 3313 843 4 v 123 x Fq(8)p Fp(c)1907 3451 y Fk(\006)1988 3436 y Fq(+)g(5)g(+)g(4)p Fn(k)p Fp(V)2432 3395 y Fk(\006)2411 3460 y Fl(0)2491 3436 y Fn(k)2541 3451 y Fj(L)2589 3432 y Fh(1)2697 3436 y Fq(+)699 b FA(\(4.10\))633 3622 y Fq(+8)758 3535 y Fn(p)p 841 3535 49 4 v 87 x Fq(2)p Fi(g)940 3637 y Fl(1)979 3622 y Fq(\(\()p Fp(\013)1118 3581 y Fk(\006)1117 3647 y Fj(a)1177 3622 y Fq(\))1215 3581 y Fl(2)1276 3622 y Fq(+)22 b(\()p Fp(\013)1475 3581 y Fk(\006)1474 3650 y Fj(b)1534 3622 y Fq(\))1572 3581 y Fl(2)1611 3622 y Fq(\))1649 3581 y Fl(1)p Fj(=)p Fl(2)1759 3622 y Fn(k)p Fq(\()p Fp(H)1936 3581 y Fk(\006)1928 3647 y Fl(0)1995 3622 y Fq(\))2033 3581 y Fk(\000)p Fl(1)p Fj(=)p Fl(2)2197 3622 y Fn(k)2247 3571 y Fl(1)p Fj(=)p Fl(2)2247 3655 y Fk(B)r Fl(\()p Fd(H)p Fj(;W)2469 3636 y Ff(1)p Fe(;)p Ff(2)2552 3655 y Fl(\))2584 3622 y Fq(\(8)p Fp(c)2713 3637 y Fk(\006)2794 3622 y Fq(+)g(5)g(+)g(4)p Fn(k)p Fp(V)3238 3581 y Fk(\006)3217 3647 y Fl(0)3297 3622 y Fn(k)3347 3637 y Fj(L)3395 3618 y Fh(1)3464 3622 y Fq(\))3502 3581 y Fl(1)p Fj(=)p Fl(4)3612 3622 y Fp(:)178 3854 y Fr(Theorem)37 b(4.8)49 b Fc(The)34 b(fol)5 b(lowing)34 b(statements)g(ar)-5 b(e)35 b(true:)264 4085 y(\(i\))48 b(The)34 b(mapping)g Fq(\011)1091 4100 y Fk(1)1193 4085 y Fq(:)28 b Fp(L)1314 4049 y Fk(1)1417 4085 y Fn(\000)-16 b(!)27 b Fp(L)1671 4049 y Fk(1)1781 4085 y Fc(always)34 b(admits)g(a)h(\034xe)-5 b(d)34 b(p)-5 b(oint.)234 4282 y(\(ii\))48 b(If)30 b Fp(x)575 4297 y Fl(0)646 4282 y Fc(is)g(the)h(\(unique\))f(p)-5 b(ositive)30 b(r)-5 b(o)g(ot)31 b(of)f(the)h(p)-5 b(olynomial)29 b Fp(p)e Fq(:)h Fp(x)g Fn(\000)-16 b(!)28 b Fp(x)3046 4246 y Fl(4)3098 4282 y Fn(\000)13 b Fp(\033)3243 4297 y Fl(1)3283 4282 y Fp(x)3338 4246 y Fl(2)3391 4282 y Fn(\000)g Fp(\033)3536 4297 y Fl(2)3576 4282 y Fp(x)g Fn(\000)422 4402 y Fp(\033)477 4417 y Fl(3)517 4402 y Fc(,)35 b(then)f(for)h(any)g(\034xe)-5 b(d)34 b(p)-5 b(oint)34 b Fp(V)57 b Fc(of)34 b Fq(\011)1915 4417 y Fk(1)2025 4402 y Fc(the)h(ine)-5 b(quality)1810 4600 y Fn(k)p Fp(V)22 b Fn(k)1989 4615 y Fj(L)2037 4596 y Fh(1)2133 4600 y Fn(\024)29 b Fp(x)2294 4559 y Fl(4)2294 4625 y(0)3472 4600 y FA(\(4.11\))422 4798 y Fc(holds.)178 5029 y Fr(Pro)s(of.)44 b FA(One)32 b(has)503 5227 y Fn(kJ)18 b Fq(\()p Fp(u)733 5186 y Fl(+)791 5227 y Fp(;)f(u)891 5186 y Fk(\000)950 5227 y Fq(\))p Fn(k)1038 5257 y Fj(W)1115 5225 y Ff(1)p Fe(;)p Ff(2)1105 5281 y Fa(R)1229 5227 y Fn(\024)29 b(k)p Fp(')1449 5242 y Fl(\000)1497 5227 y Fn(k)1547 5257 y Fj(W)1624 5225 y Ff(1)p Fe(;)p Ff(2)1614 5281 y Fa(R)1733 5227 y Fq(+)22 b Fn(kP)1958 5186 y Fk(\000)p Fl(1)1950 5252 y(0)2053 5227 y Fq(\()p Fp(D)i Fq(+)e Fp(q)t(u)2397 5186 y Fl(+)2478 5227 y Fn(\000)h Fp(q)t(u)2681 5186 y Fk(\000)2739 5227 y Fq(\))p Fn(k)2827 5257 y Fj(W)2904 5225 y Ff(1)p Fe(;)p Ff(2)2894 5280 y(\000)633 5395 y Fn(\024)84 b(k)p Fp(')908 5410 y Fl(\000)956 5395 y Fn(k)1006 5426 y Fj(W)1083 5393 y Ff(1)p Fe(;)p Ff(2)1073 5449 y Fa(R)1192 5395 y Fq(+)23 b(~)-50 b Fp(\017)q Fq(\(1)21 b(+)h Fp(\015)1587 5410 y Fj(k)1630 5395 y Fq(\))g Fn(\001)g(k)p Fp(D)i Fq(+)e Fp(q)t(u)2096 5354 y Fl(+)2177 5395 y Fn(\000)g Fp(q)t(u)2379 5354 y Fk(\000)2438 5395 y Fn(k)2488 5426 y Fj(W)2565 5393 y Fh(\000)p Ff(1)p Fe(;)p Ff(2)2555 5449 y(\000)633 5597 y Fn(\024)84 b(k)p Fp(')908 5612 y Fl(\000)956 5597 y Fn(k)1006 5627 y Fj(W)1083 5595 y Ff(1)p Fe(;)p Ff(2)1073 5651 y Fa(R)1220 5597 y Fq(+)23 b(~)-50 b Fp(\017)p Fq(\(1)22 b(+)g Fp(\015)1615 5612 y Fj(k)1657 5597 y Fq(\))h Fn(\001)1767 5486 y Fm(h)1814 5597 y Fn(k)p Fp(D)s Fn(k)1998 5627 y Fj(W)2075 5595 y Fh(\000)p Ff(1)p Fe(;)p Ff(2)2065 5650 y(\000)2232 5597 y Fq(+)f Fp(q)t(")2423 5612 y Fl(1)2462 5516 y Fm(\000)2508 5597 y Fn(k)p Fp(u)2614 5556 y Fl(+)2672 5597 y Fn(k)2722 5613 y Fj(L)2770 5595 y Ff(1)2831 5597 y Fq(+)g Fn(k)p Fp(u)3035 5556 y Fk(\000)3093 5597 y Fn(k)3143 5613 y Fj(L)3191 5595 y Ff(1)3230 5516 y Fm(\001)3276 5486 y(i)3323 5597 y Fp(;)p eop %%Page: 26 28 26 27 bop 178 46 a FA(26)896 b Ft(M.)36 b(Bar)m(o,)f(H.-Ch.)47 b(Kaiser,)36 b(H.)g(Neidhardt,)f(J.)i(Rehber)m(g)178 295 y FA(whic)m(h)c(implies)257 500 y Fn(k)p Fp(E)379 515 y Fk(1)454 500 y Fn(J)18 b Fq(\()p Fp(u)634 459 y Fl(+)693 500 y Fp(;)f(u)793 459 y Fk(\000)851 500 y Fq(\))p Fn(k)939 515 y Fj(L)987 496 y Fh(1)3472 500 y FA(\(4.12\))416 673 y Fn(\024)167 b Fp(")706 688 y Fk(1)780 673 y Fn(k)p Fp(')894 688 y Fl(\000)942 673 y Fn(k)992 703 y Fj(W)1069 670 y Ff(1)p Fe(;)p Ff(2)1059 726 y Fa(R)1206 673 y Fq(+)50 b Fp(")1378 688 y Fk(1)1453 673 y Fq(~)-50 b Fp(\017)q Fq(\(1)21 b(+)h Fp(\015)1749 688 y Fj(k)1792 673 y Fq(\))g Fn(\001)1902 562 y Fm(h)1949 673 y Fn(k)p Fp(D)s Fn(k)2133 703 y Fj(W)2210 670 y Fh(\000)p Ff(1)p Fe(;)p Ff(2)2200 726 y(\000)2366 673 y Fq(+)g Fp(q)t(")2557 688 y Fl(1)2596 592 y Fm(\000)2642 673 y Fn(k)p Fp(u)2748 631 y Fl(+)2806 673 y Fn(k)2856 689 y Fj(L)2904 670 y Ff(1)2965 673 y Fq(+)g Fn(k)p Fp(u)3169 631 y Fk(\000)3228 673 y Fn(k)3278 689 y Fj(L)3326 670 y Ff(1)3364 592 y Fm(\001)3410 562 y(i)3457 673 y Fp(:)178 912 y FA(Since)33 b Fp(u)489 876 y Fk(\006)575 912 y Fq(=)28 b Fn(N)776 871 y Fk(\006)767 940 y Fl(^)-41 b Fj(\045)797 949 y Fh(\006)853 912 y Fq(\()p Fp(V)948 927 y Fk(\006)1007 912 y Fq(\))32 b FA(one)h(gets)g(from)e(\(2.61\))h(the)h(estimate)115 1163 y Fn(k)p Fp(u)221 1122 y Fk(\006)280 1163 y Fn(k)330 1180 y Fj(L)378 1161 y Ff(1)444 1163 y Fn(\024)28 b Fp(C)625 1178 y Fl(^)-41 b Fj(\045)655 1187 y Fh(\006)733 1163 y Fn(\001)783 1052 y Fm(h)830 1163 y Fq(8)22 b(+)g(4)1048 1076 y Fn(p)p 1131 1076 49 4 v 87 x Fq(2)27 b(\()p Fp(b)c Fn(\000)g Fp(a)p Fq(\))1498 1073 y Fm(p)p 1597 1073 244 4 v 1597 1163 a Fn(k)p Fp(m)1732 1178 y Fk(\006)1791 1163 y Fn(k)f(\001)1913 1073 y Fm(p)p 2013 1073 821 4 v 90 x Fq(8)p Fp(c)2104 1178 y Fk(\006)2185 1163 y Fq(+)g(5)g(+)g(4)p Fn(k)p Fp(V)2608 1178 y Fk(\006)2666 1163 y Fn(k)2716 1178 y Fj(L)2764 1159 y Fh(1)434 1376 y Fq(+)83 b(8)642 1288 y Fn(p)p 725 1288 49 4 v 88 x Fq(2)p Fi(g)824 1391 y Fl(1)864 1376 y Fq(\(\()p Fp(\013)1003 1334 y Fk(\006)1002 1400 y Fj(a)1061 1376 y Fq(\))1099 1334 y Fl(2)1161 1376 y Fq(+)22 b(\()p Fp(\013)1360 1334 y Fk(\006)1359 1403 y Fj(b)1418 1376 y Fq(\))1456 1334 y Fl(2)1496 1376 y Fq(\))1534 1334 y Fl(1)p Fj(=)p Fl(2)1644 1376 y Fn(k)p Fq(\()p Fp(H)1821 1334 y Fk(\006)1813 1400 y Fl(0)1879 1376 y Fq(\))1917 1334 y Fk(\000)p Fl(1)p Fj(=)p Fl(2)2082 1376 y Fn(k)2132 1325 y Fl(1)p Fj(=)p Fl(2)2132 1408 y Fk(B)r Fl(\()p Fd(H)p Fj(;W)2354 1389 y Ff(1)p Fe(;)p Ff(2)2437 1408 y Fl(\))2468 1376 y Fq(\(8)p Fp(c)2597 1391 y Fk(\006)2678 1376 y Fq(+)g(5)g(+)g(4)p Fn(k)p Fp(V)3101 1391 y Fk(\006)3159 1376 y Fn(k)3209 1391 y Fj(L)3257 1372 y Fh(1)3326 1376 y Fq(\))3364 1334 y Fl(1)p Fj(=)p Fl(4)3474 1265 y Fm(i)3521 1376 y Fp(:)178 1615 y FA(By)33 b Fp(V)388 1630 y Fk(\006)475 1615 y Fq(=)28 b Fp(V)657 1574 y Fk(\006)636 1639 y Fl(0)738 1615 y Fn(\006)23 b Fp(V)54 b FA(w)m(e)34 b(obtain)168 1820 y Fn(k)p Fp(u)274 1779 y Fk(\006)332 1820 y Fn(k)382 1837 y Fj(L)430 1818 y Ff(1)496 1820 y Fn(\024)2899 b FA(\(4.13\))358 2009 y Fp(C)434 2024 y Fl(^)-41 b Fj(\045)464 2033 y Fh(\006)542 2009 y Fn(\001)592 1899 y Fm(h)639 2009 y Fq(8)22 b(+)g(4)857 1922 y Fn(p)p 939 1922 V 939 2009 a Fq(2)28 b(\()p Fp(b)22 b Fn(\000)h Fp(a)p Fq(\))1306 1919 y Fm(p)p 1406 1919 244 4 v 90 x Fn(k)p Fp(m)1541 2024 y Fk(\006)1600 2009 y Fn(k)f(\001)1722 1886 y Fm(q)p 1821 1886 1307 4 v 1821 2009 a Fq(8)p Fp(c)1912 2024 y Fk(\006)1993 2009 y Fq(+)g(5)g(+)g(4)p Fn(k)p Fp(V)2437 1968 y Fk(\006)2416 2034 y Fl(0)2496 2009 y Fn(k)2546 2024 y Fj(L)2594 2005 y Fh(1)2686 2009 y Fq(+)g(4)p Fn(k)p Fp(V)f Fn(k)3011 2024 y Fj(L)3059 2005 y Fh(1)192 2222 y Fq(+8)317 2134 y Fn(p)p 399 2134 49 4 v 399 2222 a Fq(2)p Fi(g)498 2237 y Fl(1)538 2222 y Fq(\(\()p Fp(\013)677 2181 y Fk(\006)676 2246 y Fj(a)735 2222 y Fq(\))773 2181 y Fl(2)835 2222 y Fq(+)h(\()p Fp(\013)1034 2181 y Fk(\006)1033 2250 y Fj(b)1092 2222 y Fq(\))1130 2181 y Fl(2)1170 2222 y Fq(\))1208 2181 y Fl(1)p Fj(=)p Fl(2)1318 2222 y Fn(k)p Fq(\()p Fp(H)1495 2181 y Fk(\006)1487 2246 y Fl(0)1553 2222 y Fq(\))1591 2181 y Fk(\000)p Fl(1)p Fj(=)p Fl(2)1756 2222 y Fn(k)1806 2171 y Fl(1)p Fj(=)p Fl(2)1806 2254 y Fk(B)r Fl(\()p Fd(H)p Fj(;W)2028 2235 y Ff(1)p Fe(;)p Ff(2)2111 2254 y Fl(\))2142 2222 y Fq(\(8)p Fp(c)2271 2237 y Fk(\006)2352 2222 y Fq(+)g(5)g(+)g(4)p Fn(k)p Fp(V)2796 2181 y Fk(\006)2775 2246 y Fl(0)2855 2222 y Fn(k)2905 2237 y Fj(L)2953 2218 y Fh(1)3044 2222 y Fq(+)g(4)p Fn(k)p Fp(V)f Fn(k)3369 2237 y Fj(L)3417 2218 y Fh(1)3486 2222 y Fq(\))3524 2181 y Fl(1)p Fj(=)p Fl(4)3634 2111 y Fm(i)3681 2222 y Fp(:)178 2448 y FA(Using)32 b(the)h(estimates)428 2569 y Fm(q)p 527 2569 1307 4 v 527 2692 a Fq(8)p Fp(c)618 2707 y Fk(\006)699 2692 y Fq(+)22 b(5)g(+)g(4)p Fn(k)p Fp(V)1143 2651 y Fk(\006)1122 2716 y Fl(0)1203 2692 y Fn(k)1253 2707 y Fj(L)1301 2688 y Fh(1)1392 2692 y Fq(+)g(4)p Fn(k)p Fp(V)f Fn(k)1717 2707 y Fj(L)1765 2688 y Fh(1)1861 2692 y Fn(\024)1967 2569 y Fm(q)p 2066 2569 843 4 v 2066 2692 a Fq(8)p Fp(c)2157 2707 y Fk(\006)2238 2692 y Fq(+)h(5)g(+)g(4)p Fn(k)p Fp(V)2682 2651 y Fk(\006)2661 2716 y Fl(0)2741 2692 y Fn(k)2791 2707 y Fj(L)2839 2688 y Fh(1)2930 2692 y Fq(+)g(2)3077 2602 y Fm(p)p 3177 2602 296 4 v 90 x Fn(k)p Fp(V)f Fn(k)3355 2707 y Fj(L)3403 2688 y Fh(1)178 2906 y FA(and)350 3111 y Fq(\(8)p Fp(c)479 3126 y Fk(\006)560 3111 y Fq(+)h(5)g(+)g(4)p Fn(k)p Fp(V)1004 3070 y Fk(\006)983 3135 y Fl(0)1063 3111 y Fn(k)1113 3126 y Fj(L)1161 3107 y Fh(1)1252 3111 y Fq(+)g(4)p Fn(k)p Fp(V)f Fn(k)1577 3126 y Fj(L)1625 3107 y Fh(1)1694 3111 y Fq(\))1732 3070 y Fl(1)p Fj(=)p Fl(4)1870 3111 y Fn(\024)28 b Fq(\(8)p Fp(c)2104 3126 y Fk(\006)2185 3111 y Fq(+)22 b(5)g(+)g(4)p Fn(k)p Fp(V)2629 3070 y Fk(\006)2608 3135 y Fl(0)2688 3111 y Fn(k)2738 3126 y Fj(L)2786 3107 y Fh(1)2855 3111 y Fq(\))2893 3070 y Fl(1)p Fj(=)p Fl(4)3025 3111 y Fq(+)3123 3024 y Fn(p)p 3206 3024 49 4 v 87 x Fq(2)p Fn(k)p Fp(V)f Fn(k)3433 3060 y Fl(1)p Fj(=)p Fl(4)3433 3138 y Fj(L)3481 3119 y Fh(1)178 3316 y FA(w)m(e)34 b(get)1127 3436 y Fn(k)p Fp(u)1233 3395 y Fk(\006)1291 3436 y Fn(k)1341 3453 y Fj(L)1389 3434 y Ff(1)1456 3436 y Fn(\024)28 b Fp(\033)1620 3395 y Fk(\006)1616 3461 y Fl(1)1679 3436 y Fn(k)p Fp(V)21 b Fn(k)1857 3385 y Fl(1)p Fj(=)p Fl(2)1857 3463 y Fj(L)1905 3444 y Fh(1)1996 3436 y Fq(+)h Fp(\033)2153 3395 y Fk(\006)2149 3461 y Fl(2)2212 3436 y Fn(k)p Fp(V)g Fn(k)2391 3385 y Fl(1)p Fj(=)p Fl(4)2391 3463 y Fj(L)2439 3444 y Fh(1)2530 3436 y Fq(+)g Fp(\033)2687 3395 y Fk(\006)2683 3461 y Fl(3)2746 3436 y Fp(;)699 b FA(\(4.14\))178 3604 y(where)1196 3809 y Fp(\033)1255 3768 y Fk(\006)1251 3834 y Fl(1)1398 3809 y Fq(:=)83 b(8)1633 3722 y Fn(p)p 1715 3722 V 1715 3809 a Fq(2)28 b Fp(C)1868 3824 y Fl(^)-41 b Fj(\045)1898 3833 y Fh(\006)1976 3809 y Fn(\001)22 b Fq(\()p Fp(b)g Fn(\000)h Fp(a)p Fq(\))2316 3719 y Fm(p)p 2416 3719 244 4 v 90 x Fn(k)p Fp(m)2551 3824 y Fk(\006)2610 3809 y Fn(k)p Fp(;)1196 3969 y(\033)1255 3928 y Fk(\006)1251 3993 y Fl(2)1398 3969 y Fq(:=)83 b(16)1709 3881 y Fn(p)p 1792 3881 49 4 v 88 x Fq(2)27 b Fi(g)1918 3984 y Fl(1)1986 3969 y Fp(C)2062 3984 y Fl(^)-41 b Fj(\045)2092 3993 y Fh(\006)2175 3969 y Fp(p)2224 3984 y Fk(\006)2283 3969 y Fp(;)1196 4114 y(\033)1255 4073 y Fk(\006)1251 4139 y Fl(3)1398 4114 y Fq(:=)83 b Fp(C)1660 4129 y Fl(^)-41 b Fj(\045)1690 4138 y Fh(\006)1768 4114 y Fn(\001)49 b Fp(r)1889 4129 y Fk(\006)1948 4114 y Fp(;)178 4319 y FA(where)35 b Fp(p)510 4334 y Fk(\006)602 4319 y FA(and)f Fp(r)837 4334 y Fk(\006)929 4319 y FA(are)f(de\034ned)i(b)m(y)f (\(4.9\))f(and)h(\(4.10\),)f(resp)s(ectiv)m(ely)-8 b(.)46 b(Inserting)34 b(\(4.14\))e(in)m(to)178 4440 y(\(4.12\))g(w)m(e)i (obtain)927 4658 y Fn(k)p Fp(E)1049 4673 y Fk(1)1123 4658 y Fn(J)18 b Fq(\()p Fp(u)1303 4617 y Fl(+)1362 4658 y Fp(;)f(u)1462 4617 y Fk(\000)1520 4658 y Fq(\))p Fn(k)1608 4673 y Fj(L)1656 4654 y Fh(1)1753 4658 y Fn(\024)28 b Fp(\033)1913 4673 y Fl(1)1953 4658 y Fn(k)p Fp(V)21 b Fn(k)2131 4607 y Fl(1)p Fj(=)p Fl(2)2131 4685 y Fj(L)2179 4666 y Fh(1)2270 4658 y Fq(+)h Fp(\033)2423 4673 y Fl(2)2463 4658 y Fn(k)p Fp(V)f Fn(k)2641 4607 y Fl(1)p Fj(=)p Fl(4)2641 4685 y Fj(L)2689 4666 y Fh(1)2781 4658 y Fq(+)h Fp(\033)2934 4673 y Fl(3)178 4863 y FA(where)46 b Fp(\033)527 4878 y Fl(1)567 4863 y Fp(;)17 b(\033)666 4878 y Fl(2)706 4863 y Fp(;)g(\033)805 4878 y Fl(3)889 4863 y FA(are)45 b(the)g(constan)m(ts)h(de\034ned)g(under)f(\(4.6\))f(-)g(\(4.8\).)80 b(Hence)46 b(w)m(e)g(get)e(the)178 4984 y(estimate)1051 5104 y Fn(k)p Fq(\011)1177 5119 y Fk(1)1252 5104 y Fq(\()p Fp(V)21 b Fq(\))p Fn(k)1456 5119 y Fj(L)1504 5100 y Fh(1)1601 5104 y Fn(\024)28 b Fp(\033)1761 5119 y Fl(1)1801 5104 y Fn(k)p Fp(V)21 b Fn(k)1979 5053 y Fl(1)p Fj(=)p Fl(2)1979 5131 y Fj(L)2027 5112 y Fh(1)2118 5104 y Fq(+)h Fp(\033)2271 5119 y Fl(2)2311 5104 y Fn(k)p Fp(V)g Fn(k)2490 5053 y Fl(1)p Fj(=)p Fl(4)2490 5131 y Fj(L)2538 5112 y Fh(1)2629 5104 y Fq(+)g Fp(\033)2782 5119 y Fl(3)2822 5104 y Fp(:)623 b FA(\(4.15\))178 5272 y(If)31 b Fp(x)329 5287 y Fl(0)399 5272 y FA(is)f(the)h(\(unique\))g(p)s(ositiv)m(e)e(ro)s(ot)h(of)g(the)h (p)s(olynomial)26 b Fp(x)j Fn(\000)-17 b(!)28 b Fp(x)2720 5236 y Fl(4)2777 5272 y Fn(\000)19 b Fp(\033)2928 5287 y Fl(1)2968 5272 y Fp(x)3023 5236 y Fl(2)3080 5272 y Fn(\000)g Fp(\033)3231 5287 y Fl(2)3271 5272 y Fp(x)f Fn(\000)g Fp(\033)3494 5287 y Fl(3)3565 5272 y FA(and)178 5393 y Fn(k)p Fp(V)k Fn(k)357 5408 y Fj(L)405 5389 y Fh(1)501 5393 y Fn(\024)29 b Fp(x)662 5356 y Fl(4)662 5417 y(0)701 5393 y FA(,)k(then)g(b)m(y)h(\(4.15\))d(one)i(obtains)449 5612 y Fn(k)p Fq(\011)575 5627 y Fk(1)650 5612 y Fq(\()p Fp(V)21 b Fq(\))p Fn(k)854 5627 y Fj(L)902 5608 y Fh(1)999 5612 y Fn(\024)28 b Fp(\033)1159 5627 y Fl(1)1199 5531 y Fm(\000)1244 5612 y Fn(k)p Fp(V)22 b Fn(k)1423 5561 y Fl(1)p Fj(=)p Fl(4)1423 5639 y Fj(L)1471 5620 y Fh(1)1540 5531 y Fm(\001)1585 5553 y Fl(2)1647 5612 y Fq(+)g Fp(\033)1800 5627 y Fl(2)1840 5612 y Fn(k)p Fp(V)f Fn(k)2018 5561 y Fl(1)p Fj(=)p Fl(4)2018 5639 y Fj(L)2066 5620 y Fh(1)2157 5612 y Fq(+)h Fp(\033)2310 5627 y Fl(3)2378 5612 y Fn(\024)28 b Fp(\033)2538 5627 y Fl(1)2578 5612 y Fp(x)2633 5571 y Fl(2)2633 5636 y(0)2695 5612 y Fq(+)22 b Fp(\033)2848 5627 y Fl(1)2888 5612 y Fp(x)2943 5627 y Fl(0)3005 5612 y Fq(+)g Fp(\033)3158 5627 y Fl(3)3225 5612 y Fq(=)28 b Fp(x)3384 5571 y Fl(4)3384 5636 y(0)3424 5612 y Fp(:)p eop %%Page: 27 29 27 28 bop 178 46 a Ft(Dissip)-8 b(a)g(tive)36 b(Schr)m (\366dinger-Poisson)f(systems)1383 b FA(27)178 295 y(This)24 b(means,)i(that)d Fq(\011)989 310 y Fk(1)1088 295 y FA(maps)g(the)h (ball)e Fn(f)p Fp(V)49 b Fq(:)28 b Fn(k)p Fp(V)21 b Fn(k)2063 310 y Fj(L)2111 291 y Fh(1)2208 295 y Fn(\024)28 b Fp(x)2368 259 y Fl(4)2368 319 y(0)2408 295 y Fn(g)23 b FA(con)m(tin)m(uously)h (in)m(to)f(itself.)40 b(Since)178 415 y Fq(\011)254 430 y Fk(1)370 415 y FA(is)g(compact)h(the)g(image)f(of)g(this)h(ball)e (under)j Fq(\011)2219 430 y Fk(1)2334 415 y FA(is)f(precompact)g(in)f Fp(L)3163 379 y Fk(1)3238 415 y FA(.)69 b(Th)m(us,)45 b(b)m(y)178 535 y(Sc)m(hauder's)39 b(\034xed)g(p)s(oin)m(t)d(theorem)h Fq(\011)1628 550 y Fk(1)1739 535 y FA(m)m(ust)h(ha)m(v)m(e)g(a)f (\034xed)h(p)s(oin)m(t.)57 b(This)37 b(pro)m(v)m(es)i(the)e(\034rst)178 656 y(assertion.)296 823 y(Assume)29 b(that)f(the)g(second)h(assertion) f(is)f(false)g(and)h(a)g(\034xed)h(p)s(oin)m(t)e Fp(V)49 b FA(satisfying)27 b Fn(k)p Fp(V)21 b Fn(k)3501 838 y Fj(L)3549 820 y Fh(1)3646 823 y Fp(>)178 944 y(x)233 908 y Fl(4)233 969 y(0)306 944 y FA(exists.)44 b(Then)34 b(\(4.15\))d(w)m(ould)i(giv)m(e)494 1092 y Fm(\000)540 1173 y Fn(k)p Fp(V)21 b Fn(k)718 1122 y Fl(1)p Fj(=)p Fl(4)718 1200 y Fj(L)766 1181 y Fh(1)835 1092 y Fm(\001)881 1114 y Fl(4)948 1173 y Fq(=)27 b Fn(k)p Fp(V)22 b Fn(k)1230 1188 y Fj(L)1278 1169 y Fh(1)1374 1173 y Fq(=)28 b Fn(k)p Fq(\011)1604 1188 y Fk(1)1678 1173 y Fq(\()p Fp(V)22 b Fq(\))p Fn(k)1883 1188 y Fj(L)1931 1169 y Fh(1)2027 1173 y Fn(\024)29 b Fp(\033)2188 1188 y Fl(1)2227 1092 y Fm(\000)2273 1173 y Fn(k)p Fp(V)21 b Fn(k)2451 1122 y Fl(1)p Fj(=)p Fl(4)2451 1200 y Fj(L)2499 1181 y Fh(1)2568 1092 y Fm(\001)2614 1114 y Fl(2)2676 1173 y Fq(+)h Fp(\033)2829 1188 y Fl(2)2869 1173 y Fn(k)p Fp(V)f Fn(k)3047 1122 y Fl(1)p Fj(=)p Fl(4)3047 1200 y Fj(L)3095 1181 y Fh(1)3186 1173 y Fq(+)h Fp(\033)3339 1188 y Fl(3)3379 1173 y Fp(:)178 1411 y FA(Because)34 b Fn(k)p Fp(V)22 b Fn(k)730 1360 y Fl(1)p Fj(=)p Fl(4)730 1438 y Fj(L)778 1419 y Fh(1)874 1411 y Fp(>)28 b(x)1033 1426 y Fl(0)1105 1411 y FA(this)k(con)m (tradicts)h(Lemma)e(4.7.)1347 b Fb(\003)178 1579 y FA(No)m(w)33 b(w)m(e)h(can)f(state)g(the)g(main)e(result)h(of)g(the)h(pap)s(er:)178 1825 y Fr(Theorem)k(4.9)49 b Fc(Under)35 b(the)g(assumption)f(\()p Fp(A)1931 1784 y Fk(\006)1931 1850 y Fl(1)1990 1825 y Fc(\)-\()p Fp(A)2178 1840 y Fl(8)2217 1825 y Fc(\))h(the)f(fol)5 b(lowing)34 b(statements)g(ar)-5 b(e)35 b(true:)264 2071 y(\(i\))48 b(the)35 b(dissip)-5 b(ative)34 b(Schr)-5 b(\366)g(dinger)33 b(Poisson)h(system)h(always)f(admits)g(a)h(solution) f(and)234 2273 y(\(ii\))48 b(any)31 b(solution)f Fq(\()p Fp(';)17 b(u)1171 2237 y Fl(+)1229 2273 y Fp(;)g(u)1329 2237 y Fk(\000)1387 2273 y Fq(\))31 b Fc(of)f(the)g(dissip)-5 b(ative)30 b(Schr)-5 b(\366)g(dinger)29 b(Poisson)g(system)h (satis\034es)422 2394 y(the)35 b(a)g(priori)f(estimates)1278 2609 y Fn(k)p Fp(')p Fn(k)1442 2624 y Fj(L)1490 2605 y Fh(1)1586 2609 y Fn(\024)28 b Fp(x)1746 2568 y Fl(4)1746 2633 y(0)1886 2609 y Fc(and)99 b Fn(k)p Fp(u)2246 2568 y Fk(\006)2304 2609 y Fn(k)2354 2625 y Fj(L)2402 2606 y Ff(1)2469 2609 y Fn(\024)28 b Fp(C)2650 2624 y Fl(^)-41 b Fj(\045)2680 2633 y Fh(\006)2740 2609 y Fq(^)-53 b Fp(r)2780 2624 y Fk(\006)2839 2609 y Fp(;)606 b FA(\(4.16\))178 2864 y Fc(wher)-5 b(e)39 b Fp(x)513 2879 y Fl(0)592 2864 y Fc(is)f(the)h(unique)g(p)-5 b(ositive)39 b(r)-5 b(o)g(ot)39 b(of)f(the)h(p)-5 b(olynomial)38 b Fp(x)e Fn(\000)-16 b(!)35 b Fp(x)2863 2828 y Fl(4)2928 2864 y Fn(\000)26 b Fp(\033)3086 2879 y Fl(1)3126 2864 y Fp(x)3181 2828 y Fl(2)3246 2864 y Fn(\000)f Fp(\033)3403 2879 y Fl(2)3443 2864 y Fp(x)h Fn(\000)g Fp(\033)3682 2879 y Fl(3)178 2985 y Fc(with)35 b(c)-5 b(o)g(e\036cients)34 b(given)g(by)h (\(4.6\)-\(4.10\))e(and)102 3244 y Fq(^)-54 b Fp(r)141 3259 y Fk(\006)228 3244 y Fq(:=)359 3133 y Fm(h)406 3244 y Fq(8)22 b(+)g(4)624 3156 y Fn(p)p 706 3156 49 4 v 706 3244 a Fq(2)28 b(\()p Fp(b)22 b Fn(\000)h Fp(a)p Fq(\))1073 3154 y Fm(p)p 1173 3154 244 4 v 90 x Fn(k)p Fp(m)1308 3259 y Fk(\006)1367 3244 y Fn(k)f(\001)1489 3120 y Fm(q)p 1588 3120 1058 4 v 1588 3244 a Fq(8)p Fp(c)1679 3259 y Fk(\006)1760 3244 y Fq(+)g(5)g(+)g(4)p Fn(k)p Fp(V)2204 3203 y Fk(\006)2183 3268 y Fl(0)2263 3244 y Fn(k)2313 3259 y Fj(L)2361 3240 y Fh(1)2453 3244 y Fq(+)g Fp(x)2606 3209 y Fl(4)2606 3268 y(0)298 3504 y Fq(+)83 b(8)506 3416 y Fn(p)p 589 3416 49 4 v 88 x Fq(2)28 b Fi(g)716 3519 y Fl(1)755 3504 y Fq(\(\()p Fp(\013)894 3462 y Fk(\006)893 3528 y Fj(a)953 3504 y Fq(\))991 3462 y Fl(2)1052 3504 y Fq(+)22 b(\()p Fp(\013)1251 3462 y Fk(\006)1250 3531 y Fj(b)1310 3504 y Fq(\))1348 3462 y Fl(2)1387 3423 y Fm(\001)1433 3445 y Fl(1)p Fj(=)p Fl(2)1543 3504 y Fn(k)p Fq(\()p Fp(H)1720 3462 y Fk(\006)1712 3528 y Fl(0)1778 3504 y Fq(\))1816 3462 y Fk(\000)p Fl(1)p Fj(=)p Fl(2)1981 3504 y Fn(k)2031 3453 y Fl(1)p Fj(=)p Fl(2)2031 3536 y Fk(B)r Fl(\()p Fd(H)p Fj(;W)2253 3517 y Ff(1)p Fe(;)p Ff(2)2336 3536 y Fl(\))2368 3504 y Fq(\(8)p Fp(c)2497 3519 y Fk(\006)2577 3504 y Fq(+)g(5)g(+)g(4)p Fn(k)p Fp(V)3021 3462 y Fk(\006)3000 3528 y Fl(0)3080 3504 y Fn(k)3130 3519 y Fj(L)3178 3500 y Fh(1)3270 3504 y Fq(+)g Fp(x)3423 3462 y Fl(4)3423 3528 y(0)3463 3504 y Fq(\))3501 3462 y Fl(1)p Fj(=)p Fl(4)3610 3393 y Fm(i)3657 3504 y Fp(:)178 3787 y Fr(Pro)s(of.)75 b FA(The)44 b(\034rst)f(assertion)g (follo)m(ws)f(from)f(Lemma)h(4.6)h(and)g(Theorem)g(4.8.)74 b(The)44 b(\034rst)178 3907 y(inequalit)m(y)f(of)h(\(4.16\))f(is)h (implied)d(b)m(y)k(\(4.11\))e(while)h(the)g(second)h(estimates)f(are)g (obtained)178 4028 y(from)32 b(\(4.13\))f(and)i(the)g(\034rst)g (inequalit)m(y)-8 b(.)1955 b Fb(\003)178 4407 y Fy(5)156 b(Remarks)178 4673 y FA(Let)33 b(us)g(commen)m(t)f(the)h(results.)271 4920 y(\(i\))47 b(Theorem)f(4.9)f(sho)m(ws)h(that)g(the)f(dissipativ)m (e)g(Sc)m(hr\366)s(dinger-P)m(oisson)g(system)h(alw)m(a)m(ys)p 3442 4955 280 4 v 422 5049 a(admits)31 b(a)i(solution,)e(if)g(the)i (assumptions)f(\()p Fp(A)2142 5007 y Fk(\006)2142 5073 y Fl(1)2202 5049 y FA(\)-\()p Fp(A)2384 5064 y Fl(8)2422 5049 y FA(\))h(are)f(satis\034ed.)243 5251 y(\(ii\))47 b(Solutions)26 b Fq(\()p Fp(';)17 b(u)1042 5214 y Fl(+)1100 5251 y Fp(;)g(u)1200 5214 y Fk(\000)1258 5251 y Fq(\))27 b FA(of)f(dissipativ)m(e)g(Sc)m(hr\366)s(dinger-P)m(oisson)h(systems)h (admit)d(b)s(ounds)422 5371 y(whic)m(h)32 b(only)p 700 5406 182 4 v 30 w(dep)s(end)g(on)f(the)g(inputs)g Fp(m)1931 5386 y Fk(\006)1991 5371 y FA(,)g Fp(\024)2105 5335 y Fk(\006)2105 5396 y Fj(a)2164 5371 y Fp(;)17 b(\024)2264 5330 y Fk(\006)2264 5399 y Fj(b)2323 5371 y FA(,)31 b Fp(V)2460 5330 y Fk(\006)2438 5395 y Fl(0)2519 5371 y FA(,)g Fp(C)7 b FA(,)32 b Fp(\017)p FA(,)f Fp(')2874 5386 y Fl(\000)2922 5371 y FA(,)h Fp(k)s FA(,)f(and)g(the)g(steady)422 5491 y(state)j Fp(\045)712 5506 y Fk(\006)771 5491 y Fq(\()p Fn(\001)p Fq(\))p FA(.)45 b(In)34 b(particular,)d(the)j(o)s (ccuring)e(n)m(um)m(b)s(er)i Fp(x)2520 5506 y Fl(0)2593 5491 y FA(ma)m(y)f(b)s(e)g(directly)g(calculated)422 5612 y(from)e(the)i(data)g(b)m(y)g(Cardano's)g(form)m(ula.)p eop %%Page: 28 30 28 29 bop 178 46 a FA(28)896 b Ft(M.)36 b(Bar)m(o,)f(H.-Ch.)47 b(Kaiser,)36 b(H.)g(Neidhardt,)f(J.)i(Rehber)m(g)216 295 y FA(\(iii\))46 b(In)35 b(con)m(trast)g(to)f(self-adjoin)m(t)e(Sc)m (hr\366)s(dinger-P)m(oisson)j(systems)g(dissipativ)m(e)f(ones)h(allo)m (w)422 415 y(in)f(general)g(non-trivial)e(curren)m(ts)k Fp(j)1783 374 y Fk(\006)1783 443 y Fl(^)-41 b Fj(\045)1813 452 y Fh(\006)1905 415 y FA(whic)m(h)35 b(are)g(indep)s(enden)m(t)h (from)d Fp(x)f Fn(2)g Fq([)p Fp(a;)17 b(b)p Fq(])35 b FA(pro-)422 535 y(vided)e(the)g(steady)h(states)f Fp(\045)1487 550 y Fk(\006)1546 535 y Fq(\()p Fn(\001)p Fq(\))f FA(ob)s(ey)1616 676 y Fm(Z)1672 901 y Fg(R)1740 811 y Fp(d\025)c FA(tr)p Fq(\()p Fp(\045)2040 826 y Fk(\006)2099 811 y Fq(\()p Fp(\025)p Fq(\)\))f Fp(<)h Fn(1)p Fp(;)993 b FA(\(5.1\))422 1091 y(cf.)44 b([9].)219 1307 y(\(iv\))k(The)c(last)e(fact)h(giv)m(es)h (the)f(p)s(ossibilit)m(y)e(to)i(couple)g(dissipativ)m(e)f(Sc)m(hr\366)s (dinger-P)m(oisson)422 1428 y(systems)h(to)e(drift)g(di\033usion)f(mo)s (dels)h(whic)m(h)h(acts)g(outside)g(the)g(in)m(terv)-5 b(al)40 b Fq([)p Fp(a;)17 b(b)p Fq(])42 b FA(via)f(a)422 1548 y(curren)m(t)46 b(con)m(tin)m(uit)m(y)f(condition.)79 b(In)45 b(a)g(forthcoming)e(pap)s(er)i(w)m(e)h(sho)m(w)g(that)f(this)f (is)422 1668 y(really)31 b(p)s(ossible)h(and,)h(moreo)m(v)m(er,)g(the)g (coupled)g(system)g(admits)e(a)h(solution.)246 1884 y(\(v\))49 b(The)41 b(problem)d(remains)h(op)s(en)h(under)g(whic)m(h)g(conditions) f(the)h(solution,)g(guaran)m(teed)422 2005 y(b)m(y)34 b(Theorem)e(4.9,)h(is)f(unique.)219 2221 y(\(vi\))48 b(The)c(presen)m(t)g(pap)s(er)f(solv)m(es)g(the)g(dissipativ)m(e)f(Sc)m (hr\366)s(dinger-P)m(oisson)h(system)g(in)f(one)422 2341 y(dimension.)g(The)34 b(2D-)e(and)h(3D-problems)e(remain)g(op)s(en.)178 2734 y Fy(A)l(c)l(kno)l(wledgemen)l(t)178 3003 y FA(The)37 b(authors)f(thank)h(Prof.)54 b(P)-8 b(.)36 b(Degond)h(for)e(fruitful)f (discussions.)55 b(The)37 b(\034nancial)d(supp)s(ort)178 3123 y(of)e(DF)m(G)h(\(Gran)m(t)f(RE)h(1480/1-2\))d(is)i(gratefully)f (ac)m(kno)m(wledged.)178 3516 y Fy(References)227 3785 y FA([1])49 b(F)-8 b(oias,)59 b(C.;)67 b(Sz.-Nagy)-8 b(,)61 b(B:)55 b Fc(Harmonic)g(analysis)f(of)i(op)-5 b(er)g(ators)55 b(on)g(Hilb)-5 b(ert)56 b(sp)-5 b(ac)g(es)p FA(.)379 3906 y(Budap)s(est:)58 b(Ak)-5 b(ad\351miai)37 b(Kiad\363;)k(Amsterdam-London:)56 b(North-Holland)37 b(Publishing)379 4026 y(Compan)m(y)-8 b(.)33 b(XI)s(I)s(I)f(\(1970\).) 227 4242 y([2])49 b(Caussignac,)33 b(Ph.;)g(Zimmermann,)d(B.;)i(F)-8 b(erro,)33 b(F.:)43 b(Finite)31 b(elemen)m(t)h(appro)m(ximation)e(of) 379 4362 y(electrostatic)i(p)s(oten)m(tial)f(in)h(one)i(dimensional)c (m)m(ultila)m(y)m(er)h(structures)j(with)f(quan)m(tized)379 4483 y(electronic)f(c)m(harge.)h Fc(Computing)h(1990)227 4698 y FA([3])49 b(Ga)5 b(jewski,)58 b(H.:)85 b Fc(A)n(nalysis)53 b(und)h(Numerik)f(von)g(L)-5 b(adungstr)g(ansp)g(ort)54 b(in)f(Halbleitern)p FA(,)379 4819 y(GAMM)33 b(Mitteilungen)e Fr(16)h FA(\(1993\),)g(35.)227 5035 y([4])49 b(H\344fner,)30 b(D.:)43 b(R\351gularit\351)27 b(Gevrey)k(p)s(our)e(un)h(syst\350me)h (Sc)m(hr\366)s(dinger-P)m(oisson)e(dissipatif.)379 5155 y Fc(C.)34 b(R.)h(A)-5 b(c)g(ad.)35 b(Sci.,)f(Paris)p FA(,)e(S\351r.)h(I,)g(Math.)g Fr(326)p FA(,)f(No.7,)h(829-832)d (\(1998\).)227 5371 y([5])49 b(Kaiser,)33 b(H.-Ch.;)h(Reh)m(b)s(erg,)g (J.:)44 b(Ab)s(out)34 b(a)e(one-dimensional)f(stationary)h(Sc)m(hr\366) s(dinger-)379 5491 y(P)m(oisson)39 b(system)g(with)f(K)m(ohn-Sham)f(p)s (oten)m(tial.)g Fc(Z.)j(A)n(ngew.)f(Math.)i(Phys.)d Fr(50)p FA(,)i(No.3,)379 5612 y(423-458)30 b(\(1999\).)p eop %%Page: 29 31 29 30 bop 178 46 a Ft(Dissip)-8 b(a)g(tive)36 b(Schr)m (\366dinger-Poisson)f(systems)1383 b FA(29)227 295 y([6])49 b(Kaiser,)42 b(H.-Ch.;)j(Reh)m(b)s(erg,)e(J.:)59 b(Ab)s(out)41 b(a)f(stationary)f(Sc)m(hr\366)s(dinger-P)m(oisson)i(system)379 415 y(with)k(K)m(ohn-Sham)g(p)s(oten)m(tial)f(in)h(a)g(b)s(ounded)i(t)m (w)m(o-)f(or)f(three-dimensional)e(domain.)379 535 y Fc(Nonline)-5 b(ar)34 b(A)n(nal.,)g(The)-5 b(ory)34 b(Metho)-5 b(ds)35 b(Appl.)d Fr(41A)p FA(,)h(No.)f(1-2,)g(33-72)f(\(2000\).)227 739 y([7])49 b(Kaiser,)24 b(H-Ch.;)j(Neidhardt,)e(H.;)i(Reh)m(b)s(erg,) e(J.:)39 b Fc(Macrsopic)26 b(curr)-5 b(ent)26 b(induc)-5 b(e)g(d)26 b(b)-5 b(oundary)379 859 y(c)g(onditions)29 b(for)i(Schr)-5 b(\366)g(dinger-typ)g(e)29 b(op)-5 b(er)g(ators)p FA(.)28 b(WIAS-preprin)m(t)g(N)2846 934 y(\011)2906 859 y(o.)g(650,)h(Berlin)e(2001.)227 1063 y([8])49 b(Kaiser,)62 b(H-Ch.;)69 b(Neidhardt,)63 b(H.;)69 b(Reh)m(b)s(erg,)63 b(J.:)93 b Fc(On)56 b(1-dimensional)f(dissip)-5 b(ative)379 1183 y(Schr)g(\366)g(dinger-typ)g(e)33 b(op)-5 b(er)g(ators,)34 b(their)h(dilations)e(and)h(eigenfuction)g(exp)-5 b(ansions)p FA(.)30 b(T)-8 b(o)32 b(ap-)379 1303 y(p)s(ear.)227 1507 y([9])49 b(Kaiser,)28 b(H-Ch.;)i(Neidhardt,)f(H.;)h(Reh)m(b)s(erg,)g (J.:)41 b Fc(Density)30 b(and)h(curr)-5 b(ent)31 b(of)f(a)h(dissip)-5 b(ative)379 1627 y(Schr)g(\366)g(dinger)33 b(op)-5 b(er)g(ator)p FA(.)32 b(T)-8 b(o)33 b(app)s(ear.)178 1831 y([10])49 b(Kato,)25 b(T.:)39 b Fc(Perturb)-5 b(ation)27 b(the)-5 b(ory)27 b(for)f(line)-5 b(ar)26 b(op)-5 b(er)g(ators)p FA(.)24 b(Berlin'Heidelb)s(erg-New)d(Y)-8 b(ork:)379 1951 y(Springer-V)g(erlag.)30 b(XX)j(\(1966\).)178 2154 y([11])49 b(Lange,)25 b(H.;)i(Zw)m(eifel,)d(P)-8 b(.F.:)39 b(Dissipation)23 b(in)f(Wigner-P)m(oisson)h(systems.)i Fc(J.)i(Math.)g(Phys.)379 2275 y Fr(35)p FA(,)32 b(No.4,)h(1513-1521)d (\(1994\).)178 2478 y([12])49 b(Nier,)36 b(F.:)50 b Fc(A)39 b(stationary)f(Schr)-5 b(\366)g(dinger)37 b(Poisson)g(system)h(arising) f(fr)-5 b(om)37 b(the)h(mo)-5 b(del)5 b(ling)379 2599 y(of)34 b(ele)-5 b(ctr)g(onic)34 b(devic)-5 b(es)p FA(,)32 b(F)-8 b(orum)31 b(Mathematicum)g Fr(2)h FA(\(1990\),)g(489-510)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF