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y(has)g(to)f(do)g(with)g(a)g(family)e(of)i(Sc)o(hr\366)q (dinger-t)o(yp)q(e)g(op)q(erators)h(whic)o(h)e(implies)e(di\033eren)o (t)257 2014 y(conceptual)16 b(di\036culties.)331 2092 y(In)f([18)q(])h(w)o(e)f(regarded)i(a)f(Sc)o(hr\366dinger-t)o(yp)q(e)g (op)q(erator)h(with)g(non-selfadjoin)o(t)257 2158 y(b)q(oundary)24 b(conditions.)42 b(The)23 b(corresp)q(onding)g(op)q(en)h(quan)o(tum)d (system)h(is)h(driv)o(en)257 2224 y(b)o(y)d(an)i(adjacen)o(t)e(p)q (oten)o(tial)g(\035o)o(w)h(acting)g(on)g(the)f(b)q(oundary)l(.)36 b(This)20 b(approac)o(h)i(has)257 2291 y(the)c(adv)m(an)o(tage)h(that)f (exactly)f(one)h(Sc)o(hr\366)q(dinger)f(op)q(erator)i(describ)q(es)f (the)f(ph)o(ysical)257 2357 y(system.)26 b(Ho)o(w)o(ev)o(er,)16 b(the)i(price)f(whic)o(h)g(one)h(has)h(to)f(pa)o(y)g(is)g(the)g (non-selfadjoin)o(tness)257 2423 y(of)f(the)f(Hamiltonian.)331 2501 y(F)l(ollo)o(wing)c(this)h(line)e(of)j(in)o(v)o(estigation,)d(w)o (e)i(consider)f(a)i(Sc)o(hr\366)q(dinger-t)o(yp)q(e)e(op)q(era-)1055 2626 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Fr(;)8 b Fn(\001)p Fq(])18 b Fw(a)i(maximal)c (sectorial)j(op)q(erator)i Fr(H)t Fw(.)32 b(If)19 b(either)f Fr(\024)1438 1678 y Fo(a)1479 1671 y Fn(2)i Fv(C)1564 1678 y Fp(+)1617 1671 y Fw(or)g Fr(\024)1708 1678 y Fo(b)1744 1671 y Fn(2)g Fv(C)1830 1678 y Fp(+)1863 1671 y Fw(,)257 1737 y(then)e(the)f(op)q(erator)h Fr(H)k Fw(is)17 b(dissipativ)o(e,)f (i.e.,)f Fn(=)p Fw(m)o Fq(\()p Fr(H)t(g)r(;)8 b(g)r Fq(\))16 b Fn(\024)f Fq(0)j Fw(for)f Fr(g)h Fn(2)e Fw(dom)o Fq(\()p Fr(H)t Fq(\))p Fw(.)25 b(In)257 1803 y([18)q(])15 b(suc)o(h)h(an)h(op)q (erator)h(is)e(called)f(an)o(ti-dissipativ)o(e.)20 b(Indeed,)15 b(from)g(\(2.1\))h(w)o(e)g(get)282 1924 y Fq(\()p Fr(H)t(g)r(;)8 b(g)r Fq(\))14 b(=)g Fn(\000)p Fr(\024)569 1931 y Fo(a)590 1924 y Fn(j)p Fr(g)r Fq(\()p Fr(a)p Fq(\))p Fn(j)707 1903 y Fp(2)734 1924 y Fn(\000)8 b Fr(\024)809 1931 y Fo(b)826 1924 y Fn(j)p Fr(g)r Fq(\()p Fr(b)p Fq(\))p Fn(j)938 1903 y Fp(2)965 1924 y Fq(+)1011 1856 y Fl(Z)1061 1869 y Fo(b)1039 1969 y(a)1086 1924 y Fr(dx)1217 1890 y Fq(1)p 1158 1912 142 2 v 1158 1958 a(2)g Fr(m)p Fq(\()p Fr(x)p Fq(\))1304 1924 y Fn(j)p Fr(g)1343 1903 y Fm(0)1355 1924 y Fq(\()p Fr(x)p Fq(\))p Fn(j)1435 1903 y Fp(2)1462 1924 y Fq(+)g Fr(V)j Fq(\()p Fr(x)p Fq(\))p Fn(j)p Fr(g)r Fq(\()p Fr(x)p Fq(\))p Fn(j)1732 1903 y Fp(2)1776 1924 y Fw(\(2.2\))257 2038 y(whic)o(h)16 b(yields)g Fn(=)p Fw(m)o Fq(\()p Fr(H)t(g)r(;)8 b(g)r Fq(\))15 b Fn(\024)f Fq(0)j Fw(for)g Fr(g)g Fn(2)e Fw(dom)o Fq(\()p Fr(H)t Fq(\))p Fw(.)23 b(In)17 b(our)g(one)g(dimensional)e(situa-)257 2104 y(tion)c(this)g(dissipativ)o(e)f(op)q(erator)j Fr(H)i Fw(admits)10 b(an)i(explicit)d(description)h(whic)o(h)g(coincides)257 2170 y(with)16 b(that)h(of)f(\(1.1\)-\(1.3\).)22 b(A)15 b(dissipativ)o(e)g(op)q(erator)i(is)f(called)f(maximal)e(dissipativ)o (e)257 2236 y(if)19 b(it)h(do)q(es)g(not)g(admit)e(an)o(y)i(prop)q(er)g (dissipativ)o(e)e(extension.)31 b(Since)18 b Fr(H)24 b Fw(is)19 b(maximal)257 2302 y(sectorial)d(the)g(op)q(erator)h(is)f (also)h(maximal)c(dissipativ)o(e.)331 2369 y(The)19 b(sp)q(ectrum)g(of) h(the)f(op)q(erator)i Fr(H)j Fw(is)c(discrete)f(and)h(the)g(only)f (accum)o(ulation)257 2435 y(p)q(oin)o(t)d(is)g(in\034nit)o(y)l(.)j(F)l (urthermore,)14 b(the)i(op)q(erator)g Fr(H)k Fw(p)q(ossesses)e(a)e (Riesz)f(basis.)21 b(F)l(or)16 b(a)257 2501 y(detailed)d(analysis)i(of) f(the)g(sp)q(ectral)g(prop)q(erties)g(of)g Fr(H)19 b Fw(the)14 b(reader)g(is)f(referred)g(to)i([18].)1055 2626 y(4)p eop %%Page: 5 5 5 4 bop 144 274 a Fw(Eac)o(h)24 b(maximal)d(dissipativ)o(e)i(op)q (erator)j Fr(L)e Fw(admits)f(a)i(unique)f(orthogonal)h(de-)71 340 y(comp)q(osition)15 b(in)o(to)h(a)g(self-adjoin)o(t)g(op)q(erator)h Fr(L)962 347 y Fo(s)996 340 y Fw(and)g(a)f(completely)d(non-selfadjoin) o(t)71 406 y(op)q(erator)k Fr(L)300 413 y Fo(c:n:s)375 406 y Fw(,)f(i.e.)709 473 y Fr(L)e Fq(=)g Fr(L)841 480 y Fo(s)870 473 y Fn(\010)d Fr(L)953 480 y Fo(c:n:s:)1038 473 y Fr(:)537 b Fw(\(2.3\))71 565 y(The)15 b(op)q(erator)h Fr(L)g Fw(is)f(called)f(purely)h(maxim)o(al)e(dissipativ)o(e)h(op)q (erator)i(if)f Fr(L)1457 572 y Fo(s)1491 565 y Fw(is)g(absen)o(t.)71 631 y(In)23 b(this)g(sense)h(the)f(op)q(erator)i Fr(H)i Fw(is)d(purely)e(maximal)f(dissipativ)o(e,)i(cf.)43 b([18].)g(In)71 697 y(particular,)15 b(this)h(yields)f(that)i Fr(H)j Fw(has)d(no)g(real)f(eigen)o(v)m(alues.)144 764 y(In)f(order)h(to)h (compute)d(the)i(resolv)o(en)o(t)e(of)i Fr(H)k Fw(let)15 b(us)i(in)o(tro)q(duce)e(elemen)o(tary)e(solu-)71 830 y(tions)j Fr(v)214 837 y Fo(a)234 830 y Fq(\()p Fr(x;)8 b(z)r Fq(\))16 b Fw(and)h Fr(v)482 837 y Fo(b)499 830 y Fq(\()p Fr(x;)8 b(z)r Fq(\))15 b Fw(whic)o(h)h(are)g(de\034ned)g(b)o (y)186 960 y Fr(l)q Fq(\()p Fr(v)245 967 y Fo(a)265 960 y Fq(\()p Fr(x;)8 b(z)r Fq(\)\))i Fn(\000)h Fr(z)r(v)506 967 y Fo(a)526 960 y Fq(\()p Fr(x;)d(z)r Fq(\))13 b(=)h(0)p Fr(;)57 b(v)823 967 y Fo(a)844 960 y Fq(\()p Fr(a;)8 b(z)r Fq(\))13 b(=)g(1)p Fr(;)1173 926 y Fq(1)p 1119 949 131 2 v 1119 994 a(2)p Fr(m)p Fq(\()p Fr(a)p Fq(\))1255 960 y Fr(v)1281 940 y Fm(0)1279 972 y Fo(a)1299 960 y Fq(\()p Fr(a;)8 b(z)r Fq(\))14 b(=)f Fn(\000)p Fr(\024)1542 967 y Fo(a)1589 960 y Fw(\(2.4\))186 1089 y Fr(l)q Fq(\()p Fr(v)245 1096 y Fo(b)261 1089 y Fq(\()p Fr(x;)8 b(z)r Fq(\)\))i Fn(\000)h Fr(z)r(v)502 1096 y Fo(b)519 1089 y Fq(\()p Fr(x;)d(z)r Fq(\))13 b(=)h(0)p Fr(;)57 b(v)816 1096 y Fo(b)833 1089 y Fq(\()p Fr(b;)8 b(z)r Fq(\))13 b(=)h(1)p Fr(;)1154 1055 y Fq(1)p 1104 1078 126 2 v 1104 1123 a(2)p Fr(m)p Fq(\()p Fr(b)p Fq(\))1234 1089 y Fr(v)1260 1069 y Fm(0)1258 1101 y Fo(b)1275 1089 y Fq(\()p Fr(b;)8 b(z)r Fq(\))13 b(=)h Fr(\024)1474 1096 y Fo(b)1491 1089 y Fr(:)84 b Fw(\(2.5\))71 1222 y(The)20 b(existence)g(of)h(these)f (solutions)i(for)f(eac)o(h)f Fr(z)k Fn(2)d Fv(C)33 b Fw(can)21 b(b)q(e)g(pro)o(v)o(ed)f(b)o(y)h(writing)71 1288 y(\(2.4\))16 b(and)h(\(2.5\))f(in)g(in)o(tegral)g(form)102 1428 y Fr(v)126 1435 y Fo(a)147 1428 y Fq(\()p Fr(x;)8 b(z)r Fq(\))13 b(=)h(1)d Fn(\000)g Fq(2)p Fr(\024)462 1435 y Fo(a)492 1360 y Fl(Z)542 1373 y Fo(x)519 1473 y(a)572 1428 y Fr(dt)d(m)p Fq(\()p Fr(t)p Fq(\))i(+)h(2)813 1360 y Fl(Z)864 1373 y Fo(x)842 1473 y(a)894 1428 y Fr(dt)d(m)p Fq(\()p Fr(t)p Fq(\))1052 1360 y Fl(Z)1102 1373 y Fo(t)1080 1473 y(a)1125 1428 y Fr(ds)g Fq(\()p Fr(V)k Fq(\()p Fr(s)p Fq(\))f Fn(\000)f Fr(z)r Fq(\))p Fr(v)1429 1435 y Fo(a)1450 1428 y Fq(\()p Fr(s;)e(z)r Fq(\))31 b Fw(\(2.6\))71 1559 y(and)105 1689 y Fr(v)129 1696 y Fo(b)146 1689 y Fq(\()p Fr(x;)8 b(z)r Fq(\))13 b(=)h(1)d Fn(\000)g Fq(2)p Fr(\024)461 1696 y Fo(b)487 1621 y Fl(Z)537 1634 y Fo(b)514 1734 y(x)562 1689 y Fr(dt)d(m)p Fq(\()p Fr(t)p Fq(\))j(+)g(2)804 1621 y Fl(Z)854 1634 y Fo(b)832 1734 y(x)880 1689 y Fr(dt)d(m)p Fq(\()p Fr(t)p Fq(\))1038 1621 y Fl(Z)1087 1634 y Fo(b)1065 1734 y(t)1113 1689 y Fr(ds)g Fq(\()p Fr(V)j Fq(\()p Fr(s)p Fq(\))g Fn(\000)g Fr(z)r Fq(\))p Fr(v)1417 1696 y Fo(b)1434 1689 y Fq(\()p Fr(s;)d(z)r Fq(\))p Fr(:)33 b Fw(\(2.7\))71 1823 y(Since)21 b(\(2.6\))h(and)h(\(2.7\))f(are)g(V)l(olterra-t)o(yp)q (e)f(equations)h(they)g(are)g(alw)o(a)o(ys)f(soluble.)71 1890 y(Moreo)o(v)o(er,)16 b(one)j(gets)f(that)g Fr(v)625 1897 y Fo(a)664 1890 y Fw(and)h Fr(v)785 1897 y Fo(b)819 1890 y Fw(as)g(w)o(ell)e(as)1067 1870 y Fp(1)p 1048 1878 55 2 v 1048 1907 a(2)6 b Fo(m)1108 1890 y Fr(v)1134 1872 y Fm(0)1132 1902 y Fo(a)1171 1890 y Fw(and)1291 1870 y Fp(1)p 1272 1878 V 1272 1907 a(2)g Fo(m)1332 1890 y Fr(v)1358 1872 y Fm(0)1356 1903 y Fo(b)1391 1890 y Fw(are)18 b(absolutely)71 1956 y(con)o(tin)o(uous.)28 b(F)l(urther,)18 b(let)f Fr(W)7 b Fq(\()p Fr(z)r Fq(\))18 b(:=)f Fr(W)7 b Fq(\()p Fr(v)905 1963 y Fo(a)925 1956 y Fq(\()p Fr(x;)h(z)r Fq(\))p Fr(;)g(v)1084 1963 y Fo(b)1100 1956 y Fq(\()p Fr(x;)g(z)r Fq(\)\))18 b Fw(b)q(e)h(the)f(W)l(ronskian)h(of)71 2022 y(these)c(solutions,)i(i.e.)293 2150 y Fr(W)7 b Fq(\()p Fr(z)r Fq(\))13 b(:=)h Fr(v)512 2157 y Fo(a)532 2150 y Fq(\()p Fr(x;)8 b(z)r Fq(\))704 2116 y(1)p 650 2138 133 2 v 650 2184 a(2)p Fr(m)p Fq(\()p Fr(x)p Fq(\))787 2150 y Fr(v)813 2129 y Fm(0)811 2162 y Fo(b)828 2150 y Fq(\()p Fr(x;)g(z)r Fq(\))i Fn(\000)h Fr(v)1025 2157 y Fo(b)1042 2150 y Fq(\()p Fr(x;)d(z)r Fq(\))1213 2116 y(1)p 1160 2138 V 1160 2184 a(2)p Fr(m)p Fq(\()p Fr(x)p Fq(\))1297 2150 y Fr(v)1323 2129 y Fm(0)1321 2162 y Fo(a)1341 2150 y Fq(\()p Fr(x;)g(z)r Fq(\))p Fr(:)121 b Fw(\(2.8\))71 2286 y(W)l(e)14 b(note)i(that)f Fr(W)7 b Fq(\()p Fr(z)r Fq(\))15 b Fw(dep)q(ends)g(only)g(on)h Fr(z)h Fw(and)e(is)g(indep)q (enden)o(t)f(from)g Fr(x)p Fw(.)21 b(In)15 b(partic-)71 2352 y(ular,)g(for)i Fr(x)c Fq(=)h Fr(a)i Fw(and)h Fr(x)c Fq(=)h Fr(b)i Fw(one)h(gets)153 2479 y Fr(W)7 b Fq(\()p Fr(z)r Fq(\))14 b(=)393 2446 y(1)p 339 2468 131 2 v 339 2513 a(2)p Fr(m)p Fq(\()p Fr(a)p Fq(\))475 2479 y Fr(v)501 2459 y Fm(0)499 2492 y Fo(b)516 2479 y Fq(\()p Fr(a;)8 b(z)r Fq(\))i(+)h Fr(\024)714 2486 y Fo(a)735 2479 y Fr(v)759 2486 y Fo(b)776 2479 y Fq(\()p Fr(a;)d(z)r Fq(\))13 b(=)h Fr(\024)980 2486 y Fo(b)997 2479 y Fr(v)1021 2486 y Fo(a)1041 2479 y Fq(\()p Fr(b;)8 b(z)r Fq(\))j Fn(\000)1263 2446 y Fq(1)p 1213 2468 126 2 v 1213 2513 a(2)p Fr(m)p Fq(\()p Fr(b)p Fq(\))1343 2479 y Fr(v)1369 2459 y Fm(0)1367 2492 y Fo(a)1388 2479 y Fq(\()p Fr(b;)d(z)r Fq(\))p Fr(:)81 b Fw(\(2.9\))868 2626 y(5)p eop %%Page: 6 6 6 5 bop 257 274 a Fw(F)l(urthermore,)14 b(the)i(functions)565 398 y Fr(v)589 405 y Fm(\003)p Fo(a)627 398 y Fq(\()p Fr(x;)8 b(z)r Fq(\))14 b(:=)p 819 354 157 2 v 13 w Fr(v)843 405 y Fo(a)863 398 y Fq(\()p Fr(x;)p 932 370 25 2 v 8 w(z)r Fq(\))48 b Fw(and)i Fr(v)1176 405 y Fm(\003)p Fo(b)1210 398 y Fq(\()p Fr(x;)8 b(z)r Fq(\))13 b(:=)p 1402 354 154 2 v 14 w Fr(v)1426 405 y Fo(b)1442 398 y Fq(\()p Fr(x;)p 1511 370 25 2 v 8 w(z)r Fq(\))p Fr(;)183 b Fw(\(2.10\))257 521 y Fr(x)14 b Fn(2)g Fq([)p Fr(a;)8 b(b)p Fq(])15 b Fw(and)h Fr(z)g Fn(2)e Fv(C)28 b Fw(are)17 b(solutions)f(of)313 661 y Fr(l)q Fq(\()p Fr(v)372 668 y Fm(\003)p Fo(a)409 661 y Fq(\()p Fr(x;)8 b(z)r Fq(\)\))j Fn(\000)g Fr(z)r(v)651 668 y Fm(\003)p Fo(a)688 661 y Fq(\()p Fr(x;)d(z)r Fq(\))14 b(=)f(0)50 b Fr(v)964 668 y Fm(\003)p Fo(a)1002 661 y Fq(\()p Fr(a;)8 b(z)r Fq(\))13 b(=)h(1)1309 628 y(1)p 1256 650 131 2 v 1256 695 a(2)p Fr(m)p Fq(\()p Fr(a)p Fq(\))1391 661 y Fr(v)1417 641 y Fm(0)1415 674 y(\003)p Fo(a)1453 661 y Fq(\()p Fr(a;)8 b(z)r Fq(\))13 b(=)h Fn(\000)p 1668 634 49 2 v Fr(\024)1696 668 y Fo(a)1717 661 y Fr(;)21 b Fw(\(2.11\))313 814 y Fr(l)q Fq(\()p Fr(v)372 821 y Fm(\003)p Fo(b)406 814 y Fq(\()p Fr(x;)8 b(z)r Fq(\)\))i Fn(\000)h Fr(z)r(v)647 821 y Fm(\003)p Fo(b)681 814 y Fq(\()p Fr(x;)d(z)r Fq(\))13 b(=)h(0)49 b Fr(v)956 821 y Fm(\003)p Fo(b)991 814 y Fq(\()p Fr(b;)8 b(z)r Fq(\))13 b(=)h(1)1291 780 y(1)p 1240 802 126 2 v 1240 848 a(2)p Fr(m)p Fq(\()p Fr(b)p Fq(\))1371 814 y Fr(v)1397 793 y Fm(0)1395 826 y(\003)p Fo(b)1429 814 y Fq(\()p Fr(b;)8 b(z)r Fq(\))13 b(=)p 1600 786 46 2 v 14 w Fr(\024)1628 821 y Fo(b)1645 814 y Fr(:)93 b Fw(\(2.12\))257 959 y(Ob)o(viously)l(,)15 b(for)i(the)f(W)l(ronskian)g Fr(W)943 966 y Fm(\003)963 959 y Fq(\()p Fr(z)r Fq(\))g Fw(one)g(gets)h(that)285 1099 y Fr(W)331 1106 y Fm(\003)351 1099 y Fq(\()p Fr(z)r Fq(\))c(:=)h Fr(v)517 1106 y Fm(\003)p Fo(a)555 1099 y Fq(\()p Fr(x;)8 b(z)r Fq(\))726 1066 y(1)p 673 1088 133 2 v 673 1134 a(2)p Fr(m)p Fq(\()p Fr(x)p Fq(\))810 1099 y Fr(v)836 1079 y Fm(0)834 1112 y(\003)p Fo(b)868 1099 y Fq(\()p Fr(x;)g(z)r Fq(\))j Fn(\000)f Fr(v)1065 1106 y Fm(\003)p Fo(b)1100 1099 y Fq(\()p Fr(x;)e(z)r Fq(\))1271 1066 y(1)p 1218 1088 V 1218 1134 a(2)p Fr(m)p Fq(\()p Fr(x)p Fq(\))1355 1099 y Fr(v)1381 1079 y Fm(0)1379 1112 y(\003)p Fo(a)1417 1099 y Fq(\()p Fr(x;)g(z)r Fq(\))13 b(=)p 1595 1056 116 2 v 14 w Fr(W)7 b Fq(\()p 1667 1072 25 2 v Fr(z)r Fq(\))p Fr(:)27 b Fw(\(2.13\))257 1245 y(Similarly)14 b(to)i(\(2.9\))h(w)o(e)e (obtain)286 1385 y Fr(W)332 1392 y Fm(\003)352 1385 y Fq(\()p Fr(z)r Fq(\))e(=)538 1351 y(1)p 485 1373 131 2 v 485 1419 a(2)p Fr(m)p Fq(\()p Fr(a)p Fq(\))621 1385 y Fr(v)647 1364 y Fm(0)645 1397 y(\003)p Fo(b)679 1385 y Fq(\()p Fr(a;)8 b(z)r Fq(\))j(+)p 850 1357 49 2 v 11 w Fr(\024)878 1392 y Fo(a)898 1385 y Fr(v)922 1392 y Fm(\003)p Fo(b)957 1385 y Fq(\()p Fr(a;)d(z)r Fq(\))13 b(=)p 1133 1357 46 2 v 14 w Fr(\024)1161 1392 y Fo(b)1178 1385 y Fr(v)1202 1392 y Fm(\003)p Fo(a)1240 1385 y Fq(\()p Fr(b;)8 b(z)r Fq(\))i Fn(\000)1462 1351 y Fq(1)p 1411 1373 126 2 v 1411 1419 a(2)p Fr(m)p Fq(\()p Fr(b)p Fq(\))1542 1385 y Fr(v)1568 1364 y Fm(0)1566 1397 y(\003)p Fo(a)1604 1385 y Fq(\()p Fr(b;)e(z)r Fq(\))p Fr(:)28 b Fw(\(2.14\))257 1527 y(Let)17 b(us)f(de\034ne)g(the)g(follo)o(wing)g(k)o(ernels)311 1677 y Fr(k)r Fq(\()p Fr(x;)8 b(y)r Fq(;)g Fr(z)r Fq(\))k(=)i Fn(\000)652 1644 y Fq(1)p 607 1666 116 2 v 607 1712 a Fr(W)7 b Fq(\()p Fr(z)r Fq(\))736 1607 y Fl(\032)794 1645 y Fr(v)818 1652 y Fo(b)835 1645 y Fq(\()p Fr(x;)h(z)r Fq(\))p Fr(v)972 1652 y Fo(a)991 1645 y Fq(\()p Fr(y)r(;)g(z)r Fq(\))48 b(:)h Fr(y)15 b Fn(\024)f Fr(x)794 1711 y(v)818 1718 y Fo(a)838 1711 y Fq(\()p Fr(x;)8 b(z)r Fq(\))p Fr(v)975 1718 y Fo(b)991 1711 y Fq(\()p Fr(y)r(;)g(z)r Fq(\))48 b(:)h Fr(x)13 b(<)h(y)1366 1677 y(;)22 b Fw(if)48 b Fr(W)7 b Fq(\()p Fr(z)r Fq(\))14 b Fn(6)p Fq(=)g(0)p Fr(;)53 b Fw(\(2.15\))257 1827 y(and)253 1967 y Fr(k)278 1974 y Fm(\003)298 1967 y Fq(\()p Fr(x;)8 b(y)r Fq(;)g Fr(z)r Fq(\))13 b(=)h Fn(\000)620 1934 y Fq(1)p 568 1956 129 2 v 568 2002 a Fr(W)614 2009 y Fm(\003)633 2002 y Fq(\()p Fr(z)r Fq(\))709 1897 y Fl(\032)767 1935 y Fr(v)791 1942 y Fm(\003)p Fo(b)826 1935 y Fq(\()p Fr(x;)8 b(z)r Fq(\))p Fr(v)963 1942 y Fm(\003)p Fo(a)1000 1935 y Fq(\()p Fr(y)r(;)g(z)r Fq(\))48 b(:)h Fr(y)15 b Fn(\024)f Fr(x)767 2001 y(v)791 2008 y Fm(\003)p Fo(a)830 2001 y Fq(\()p Fr(x;)8 b(z)r Fq(\))p Fr(v)967 2008 y Fm(\003)p Fo(b)1000 2001 y Fq(\()p Fr(y)r(;)g(z)r Fq(\))48 b(:)h Fr(x)13 b(<)h(y)1375 1967 y(;)22 b Fw(if)48 b Fr(W)1534 1974 y Fm(\003)1554 1967 y Fq(\()p Fr(z)r Fq(\))14 b Fn(6)p Fq(=)f(0)p Fr(:)32 b Fw(\(2.16\))257 2125 y Fx(Theorem)18 b(2.1)24 b Fj(L)n(et)c Fr(V)30 b Fn(2)20 b Fr(L)822 2107 y Fp(2)842 2125 y Fq(\([)p Fr(a;)8 b(b)p Fq(]\))p Fj(,)19 b Fn(=)p Fw(m)o Fq(\()p Fr(V)11 b Fq(\))19 b(=)g(0)i Fj(and)g Fr(\024)1412 2132 y Fo(a)1432 2125 y Fr(;)8 b(\024)1482 2132 y Fo(b)1519 2125 y Fn(2)19 b Fv(C)1604 2132 y Fp(+)1637 2125 y Fj(.)31 b(Then)21 b(the)257 2191 y(r)n(esolvent)e(of)e(the)h(maximal)g(dissip)n(ative)f(op)n(er)n(ator)f Fr(H)22 b Fj(admits)17 b(the)h(r)n(epr)n(esentation)413 2340 y Fq(\(\()p Fr(H)d Fn(\000)c Fr(z)r Fq(\))600 2319 y Fm(\000)p Fp(1)647 2340 y Fr(f)5 b Fq(\)\()p Fr(x)p Fq(\))14 b(=)827 2272 y Fl(Z)876 2285 y Fo(b)854 2385 y(a)902 2340 y Fr(dy)c(k)r Fq(\()p Fr(x;)e(y)r Fq(;)g Fr(z)r Fq(\))p Fr(f)d Fq(\()p Fr(y)r Fq(\))13 b(=)459 b Fw(\(2.17\))496 2479 y Fn(\000)540 2446 y Fr(v)564 2453 y Fo(b)580 2446 y Fq(\()p Fr(x;)8 b(z)r Fq(\))p 540 2468 154 2 v 558 2513 a Fr(W)f Fq(\()p Fr(z)r Fq(\))706 2412 y Fl(Z)756 2425 y Fo(x)734 2524 y(a)786 2479 y Fr(dy)j(v)869 2486 y Fo(a)890 2479 y Fq(\()p Fr(y)r(;)e(z)r Fq(\))p Fr(f)d Fq(\()p Fr(y)r Fq(\))10 b Fn(\000)1159 2446 y Fr(v)1183 2453 y Fo(a)1203 2446 y Fq(\()p Fr(x;)e(z)r Fq(\))p 1159 2468 157 2 v 1179 2513 a Fr(W)f Fq(\()p Fr(z)r Fq(\))1329 2412 y Fl(Z)1379 2425 y Fo(b)1356 2524 y(x)1404 2479 y Fr(dy)j(v)1487 2486 y Fo(b)1504 2479 y Fq(\()p Fr(y)r(;)e(z)r Fq(\))p Fr(f)d Fq(\()p Fr(y)r Fq(\))p Fr(;)1055 2626 y Fw(6)p eop %%Page: 7 7 7 6 bop 71 274 a Fr(f)25 b Fn(2)20 b Fr(L)206 256 y Fp(2)226 274 y Fq(\([)p Fr(a;)8 b(b)p Fq(]\))19 b Fj(and)i Fr(z)h Fn(2)e Fr(\045)p Fq(\()p Fr(H)t Fq(\))p Fj(.)33 b(F)l(or)20 b(the)i(r)n(esolvent)g(of)e(the)h(adjoint)g(op)n(er)n(ator)f Fr(H)1670 256 y Fm(\003)71 340 y Fj(one)e(has)f(the)h(r)n(epr)n (esentation)191 476 y Fq(\(\()p Fr(H)273 455 y Fm(\003)304 476 y Fn(\000)11 b Fr(z)r Fq(\))398 455 y Fm(\000)p Fp(1)445 476 y Fr(f)5 b Fq(\)\()p Fr(x)p Fq(\))13 b(=)624 408 y Fl(Z)674 421 y Fo(b)652 520 y(a)700 476 y Fr(dy)d(k)784 483 y Fm(\003)804 476 y Fq(\()p Fr(x;)e(y)r Fq(;)g Fr(z)r Fq(\))p Fr(f)d Fq(\()p Fr(y)r Fq(\))12 b(=)457 b Fw(\(2.18\))274 615 y Fn(\000)318 581 y Fr(v)342 588 y Fm(\003)p Fo(b)376 581 y Fq(\()p Fr(x;)8 b(z)r Fq(\))p 318 603 171 2 v 339 649 a Fr(W)385 656 y Fm(\003)404 649 y Fq(\()p Fr(z)r Fq(\))502 547 y Fl(Z)551 560 y Fo(x)529 660 y(a)582 615 y Fr(dy)i(v)665 622 y Fm(\003)p Fo(a)703 615 y Fq(\()p Fr(y)r(;)e(z)r Fq(\))p Fr(f)d Fq(\()p Fr(y)r Fq(\))10 b Fn(\000)972 581 y Fr(v)996 588 y Fm(\003)p Fo(a)1034 581 y Fq(\()p Fr(x;)e(z)r Fq(\))p 972 603 175 2 v 995 649 a Fr(W)1041 656 y Fm(\003)1061 649 y Fq(\()p Fr(z)r Fq(\))1160 547 y Fl(Z)1209 560 y Fo(b)1187 660 y(x)1235 615 y Fr(dy)i(v)1318 622 y Fm(\003)p Fo(b)1353 615 y Fq(\()p Fr(y)r(;)e(z)r Fq(\))p Fr(f)d Fq(\()p Fr(y)r Fq(\))p Fr(;)71 749 y(f)19 b Fn(2)14 b Fr(L)194 731 y Fp(2)214 749 y Fq(\([)p Fr(a;)8 b(b)p Fq(]\))15 b Fj(and)j Fr(z)e Fn(2)e Fr(\045)p Fq(\()p Fr(H)633 731 y Fm(\003)653 749 y Fq(\))p Fj(.)71 851 y Fw(W)l(e)19 b(omit)g(the)h(pro)q(of.)33 b(Note)20 b(that)g Fr(z)i Fn(2)f Fr(\033)r Fq(\()p Fr(H)t Fq(\))f Fn(,)g Fr(W)7 b Fq(\()p Fr(z)r Fq(\))19 b(=)i(0)f Fw(and)h Fr(z)h Fn(2)e Fr(\033)r Fq(\()p Fr(H)1581 833 y Fm(\003)1600 851 y Fq(\))h Fn(,)71 918 y Fr(W)7 b Fq(\()p Fr(z)r Fq(\))13 b(=)h(0)p Fw(.)71 1092 y Fu(3)77 b(The)25 b(c)n(haracteristic)h(function)71 1208 y Fw(In)15 b(the)g(follo)o(wing) h(w)o(e)f(consider)g(the)g(case)h(that)g(b)q(oth)h(complex)c(n)o(um)o (b)q(ers)h Fr(\024)1500 1215 y Fo(a)1537 1208 y Fw(and)i Fr(\024)1659 1215 y Fo(b)1676 1208 y Fw(,)486 1332 y Fr(\024)514 1339 y Fo(a)549 1332 y Fq(=)e Fr(q)623 1339 y Fo(a)654 1332 y Fq(+)712 1298 y Fr(i)p 708 1320 25 2 v 708 1366 a Fq(2)737 1332 y Fr(\013)768 1311 y Fp(2)768 1344 y Fo(a)838 1332 y Fw(and)49 b Fr(\024)993 1339 y Fo(b)1024 1332 y Fq(=)14 b Fr(q)1098 1339 y Fo(b)1126 1332 y Fq(+)1184 1298 y Fr(i)p 1180 1320 V 1180 1366 a Fq(2)1209 1332 y Fr(\013)1240 1311 y Fp(2)1240 1344 y Fo(b)1260 1332 y Fr(;)315 b Fw(\(3.1\))71 1448 y Fr(q)93 1455 y Fo(a)113 1448 y Fr(;)8 b(q)157 1455 y Fo(b)196 1448 y Fn(2)22 b Fv(R)c Fw(b)q(elong)j(to)g Fv(C)565 1455 y Fp(+)598 1448 y Fw(,)h(i.e.)34 b Fr(\013)763 1455 y Fo(a)784 1448 y Fr(;)8 b(\013)837 1455 y Fo(b)876 1448 y Fr(>)21 b Fq(0)p Fw(.)36 b(The)21 b(other)g(case)g(that)h(only)e Fr(\024)1604 1455 y Fo(a)1646 1448 y Fw(or)71 1514 y Fr(\024)99 1521 y Fo(b)135 1514 y Fw(b)q(elongs)h(to)f Fv(C)409 1521 y Fp(+)462 1514 y Fw(can)f(b)q(e)h(handled)f(m)o(utatis)f (m)o(utandis)h(setting)g(formally)f(either)71 1581 y Fr(\013)102 1588 y Fo(a)136 1581 y Fq(=)c(0)f Fw(or)h Fr(\013)313 1588 y Fo(b)344 1581 y Fq(=)g(0)f Fw(in)g(the)g(form)o (ulas)e(b)q(elo)o(w.)20 b(Let)14 b(us)f(in)o(tro)q(duce)g(the)f(op)q (erator-v)m(alued)71 1647 y(function)k Fr(T)7 b Fq(\()p Fr(z)r Fq(\))13 b(:)g Fn(H)h(\000)-8 b(!)13 b Fv(C)584 1629 y Fp(2)607 1647 y Fw(,)501 1786 y Fr(T)7 b Fq(\()p Fr(z)r Fq(\))p Fr(f)18 b Fq(:=)708 1716 y Fl(\022)789 1753 y Fr(\013)820 1760 y Fo(b)837 1753 y Fq(\(\()p Fr(H)d Fn(\000)c Fr(z)r Fq(\))1024 1735 y Fm(\000)p Fp(1)1071 1753 y Fr(f)5 b Fq(\)\()p Fr(b)p Fq(\))765 1820 y Fn(\000)p Fr(\013)835 1827 y Fo(a)856 1820 y Fq(\(\()p Fr(H)15 b Fn(\000)c Fr(z)r Fq(\))1043 1802 y Fm(\000)p Fp(1)1090 1820 y Fr(f)5 b Fq(\)\()p Fr(a)p Fq(\))1222 1716 y Fl(\023)1589 1786 y Fw(\(3.2\))71 1927 y(for)16 b Fr(z)g Fn(2)e Fr(\045)p Fq(\()p Fr(H)t Fq(\))i Fw(and)h Fr(f)i Fn(2)14 b Fr(L)572 1909 y Fp(2)592 1927 y Fq(\([)p Fr(a;)8 b(b)p Fq(]\))p Fw(.)20 b(Using)c(\(2.17\))h(w)o(e)f(\034nd)423 2074 y Fr(T)7 b Fq(\()p Fr(z)r Fq(\))p Fr(f)19 b Fq(=)667 2041 y(1)p 621 2063 116 2 v 621 2109 a Fr(W)7 b Fq(\()p Fr(z)r Fq(\))750 1989 y Fl( )810 2042 y Fn(\000)p Fr(\013)880 2049 y Fo(b)906 2002 y Fl(R)939 2015 y Fo(b)929 2059 y(a)964 2042 y Fr(dy)16 b(v)1053 2049 y Fo(a)1074 2042 y Fq(\()p Fr(y)r(;)8 b(z)r Fq(\))p Fr(f)d Fq(\()p Fr(y)r Fq(\))830 2112 y Fr(\013)861 2119 y Fo(a)890 2072 y Fl(R)923 2085 y Fo(b)914 2129 y(a)949 2112 y Fr(dy)15 b(v)1037 2119 y Fo(b)1054 2112 y Fq(\()p Fr(y)r(;)8 b(z)r Fq(\))p Fr(f)d Fq(\()p Fr(y)r Fq(\))1297 1989 y Fl(!)1589 2074 y Fw(\(3.3\))71 2224 y(for)16 b Fr(f)j Fn(2)14 b Fr(L)268 2206 y Fp(2)288 2224 y Fq(\([)p Fr(a;)8 b(b)p Fq(]\))p Fw(.)20 b(The)c(adjoin)o(t)g(op)q(erator)i(is)e(giv)o(en)f(b)o(y)282 2350 y Fq(\()p Fr(T)7 b Fq(\()p Fr(z)r Fq(\))400 2330 y Fm(\003)419 2350 y Fr(\030)r Fq(\))i(\()p Fr(x)p Fq(\))41 b(=)713 2317 y(1)p 661 2339 129 2 v 661 2384 a Fr(W)707 2391 y Fm(\003)727 2384 y Fq(\()p 746 2357 25 2 v Fr(z)r Fq(\))803 2350 y(\()p Fn(\000)p Fr(\013)892 2357 y Fo(b)909 2350 y Fr(v)933 2357 y Fm(\003)p Fo(a)971 2350 y Fq(\()p Fr(x;)p 1040 2323 V 8 w(z)q Fq(\))p Fr(;)8 b(\013)1136 2357 y Fo(a)1157 2350 y Fr(v)1181 2357 y Fm(\003)p Fo(b)1216 2350 y Fq(\()p Fr(x;)p 1285 2323 V 8 w(z)q Fq(\)\))g Fr(\030)213 b Fw(\(3.4\))577 2479 y Fq(=)713 2446 y(1)p 661 2468 129 2 v 661 2513 a Fr(W)707 2520 y Fm(\003)727 2513 y Fq(\()p 746 2486 25 2 v Fr(z)r Fq(\))803 2439 y Fl(\000)826 2479 y Fn(\000)p Fr(\013)896 2486 y Fo(b)913 2479 y Fr(v)937 2486 y Fm(\003)p Fo(a)975 2479 y Fq(\()p Fr(x;)p 1044 2452 V 8 w(z)q Fq(\))p Fr(\030)1110 2459 y Fo(b)1139 2479 y Fq(+)11 b Fr(\013)1219 2486 y Fo(a)1240 2479 y Fr(v)1264 2486 y Fm(\003)p Fo(b)1299 2479 y Fq(\()p Fr(x;)p 1368 2452 V 8 w(z)q Fq(\))p Fr(\030)1434 2459 y Fo(a)1455 2439 y Fl(\001)868 2626 y Fw(7)p eop %%Page: 8 8 8 7 bop 257 274 a Fr(x)14 b Fn(2)g Fq([)p Fr(a;)8 b(b)p Fq(])p Fw(,)14 b(where)877 369 y Fr(\030)j Fq(=)967 299 y Fl(\022)1026 337 y Fr(\030)1049 319 y Fo(b)1024 403 y Fr(\030)1047 385 y Fo(a)1089 299 y Fl(\023)1140 369 y Fn(2)d Fv(C)1220 349 y Fp(2)1243 369 y Fr(:)519 b Fw(\(3.5\))257 482 y(Similarly)l(,)13 b(w)o(e)j(set)671 608 y Fr(T)700 615 y Fm(\003)720 608 y Fq(\()p Fr(z)r Fq(\))p Fr(f)j Fq(:=)891 538 y Fl(\022)972 576 y Fr(\013)1003 583 y Fo(b)1020 576 y Fq(\(\()p Fr(H)1102 558 y Fm(\003)1133 576 y Fn(\000)11 b Fr(z)r Fq(\))1227 558 y Fm(\000)p Fp(1)1274 576 y Fr(f)5 b Fq(\)\()p Fr(b)p Fq(\))949 642 y Fn(\000)p Fr(\013)1019 649 y Fo(a)1039 642 y Fq(\(\()p Fr(H)1121 624 y Fm(\003)1152 642 y Fn(\000)11 b Fr(z)r Fq(\))1246 624 y Fm(\000)p Fp(1)1293 642 y Fr(f)5 b Fq(\)\()p Fr(a)p Fq(\))1426 538 y Fl(\023)1776 608 y Fw(\(3.6\))257 739 y(for)17 b Fr(z)f Fn(2)e Fr(\045)p Fq(\()p Fr(H)506 721 y Fm(\003)526 739 y Fq(\))i Fw(and)h Fr(f)i Fn(2)14 b Fr(L)779 721 y Fp(2)799 739 y Fq(\([)p Fr(a;)8 b(b)p Fq(]\))p Fw(.)19 b(Using)d(\(2.18\))h(w)o(e)f(\034nd)577 877 y Fr(T)606 884 y Fm(\003)626 877 y Fq(\()p Fr(z)r Fq(\))p Fr(f)j Fq(=)841 844 y(1)p 788 866 129 2 v 788 911 a Fr(W)834 918 y Fm(\003)854 911 y Fq(\()p Fr(z)r Fq(\))930 792 y Fl( )990 845 y Fn(\000)p Fr(\013)1060 852 y Fo(b)1086 805 y Fl(R)1119 818 y Fo(b)1109 862 y(a)1144 845 y Fr(dy)d(v)1233 852 y Fm(\003)p Fo(a)1271 845 y Fq(\()p Fr(y)r(;)8 b(z)r Fq(\))p Fr(f)d Fq(\()p Fr(y)r Fq(\))1012 915 y Fr(\013)1043 922 y Fo(b)1068 875 y Fl(R)1101 888 y Fo(b)1092 932 y(a)1127 915 y Fr(dy)16 b(v)1216 922 y Fm(\003)p Fo(b)1250 915 y Fq(\()p Fr(y)r(;)8 b(z)r Fq(\))p Fr(f)d Fq(\()p Fr(y)r Fq(\))1495 792 y Fl(!)1543 877 y Fr(:)219 b Fw(\(3.7\))257 1012 y(The)17 b(adjoin)o(t)f(op)q (erator)h(has)g(the)f(represen)o(tation)486 1126 y Fq(\()p Fr(T)534 1133 y Fm(\003)554 1126 y Fq(\()p Fr(z)r Fq(\))617 1105 y Fm(\003)636 1126 y Fr(\030)r Fq(\))9 b(\()p Fr(x)p Fq(\))41 b(=)924 1092 y(1)p 879 1114 116 2 v 879 1160 a Fr(W)7 b Fq(\()p 951 1132 25 2 v Fr(z)q Fq(\))1008 1126 y(\()p Fn(\000)p Fr(\013)1097 1133 y Fo(b)1113 1126 y Fr(v)1137 1133 y Fo(a)1158 1126 y Fq(\()p Fr(x;)p 1227 1098 V 8 w(z)q Fq(\))p Fr(;)h(\013)1323 1133 y Fo(a)1344 1126 y Fr(v)1368 1133 y Fo(b)1385 1126 y Fq(\()p Fr(x;)p 1454 1098 V 8 w(z)q Fq(\)\))g Fr(\030)231 b Fw(\(3.8\))794 1255 y Fq(=)924 1221 y(1)p 879 1243 116 2 v 879 1289 a Fr(W)7 b Fq(\()p 951 1261 25 2 v Fr(z)q Fq(\))1008 1214 y Fl(\000)1030 1255 y Fn(\000)p Fr(\013)1100 1262 y Fo(b)1117 1255 y Fr(v)1141 1262 y Fo(a)1162 1255 y Fq(\()p Fr(x;)p 1231 1227 V 8 w(z)q Fq(\))p Fr(\030)1297 1234 y Fo(b)1326 1255 y Fq(+)k Fr(\013)1406 1262 y Fo(a)1427 1255 y Fr(v)1451 1262 y Fo(b)1468 1255 y Fq(\()p Fr(x;)p 1537 1227 V 8 w(z)q Fq(\))p Fr(\030)1603 1234 y Fo(a)1625 1214 y Fl(\001)257 1379 y Fr(x)j Fn(2)g Fq([)p Fr(a;)8 b(b)p Fq(])p Fw(,)14 b Fr(\030)j Fn(2)d Fv(C)589 1361 y Fp(2)611 1379 y Fw(.)257 1473 y Fx(Lemma)k(3.1)24 b Fj(L)n(et)16 b Fr(V)26 b Fn(2)14 b Fr(L)769 1455 y Fp(2)789 1473 y Fq(\([)p Fr(a;)8 b(b)p Fq(]\))p Fj(,)15 b Fn(=)p Fw(m)n Fq(\()p Fr(V)d Fq(\))h(=)h(0)k Fj(and)f Fr(\024)1337 1480 y Fo(a)1358 1473 y Fr(;)8 b(\024)1408 1480 y Fo(b)1439 1473 y Fn(2)14 b Fv(C)1519 1480 y Fp(+)1551 1473 y Fj(.)22 b(Then)c(one)g(has)461 1573 y Fq(\()p Fr(H)524 1553 y Fm(\003)555 1573 y Fn(\000)11 b Fr(z)r Fq(\))649 1553 y Fm(\000)p Fp(1)707 1573 y Fn(\000)g Fq(\()p Fr(H)k Fn(\000)c Fr(z)r Fq(\))925 1553 y Fm(\000)p Fp(1)986 1573 y Fq(=)j Fn(\000)p Fr(iT)1123 1580 y Fm(\003)1142 1573 y Fq(\()p 1161 1546 V Fr(z)q Fq(\))1204 1553 y Fm(\003)1224 1573 y Fr(T)1253 1580 y Fm(\003)1272 1573 y Fq(\()p Fr(z)r Fq(\))g(=)g Fn(\000)p Fr(iT)7 b Fq(\()p 1512 1546 V Fr(z)p Fq(\))1554 1553 y Fm(\003)1574 1573 y Fr(T)g Fq(\()p Fr(z)r Fq(\))103 b Fw(\(3.9\))257 1673 y Fj(for)17 b Fr(z)f Fn(2)e Fr(\045)p Fq(\()p Fr(H)t Fq(\))d Fn(\\)h Fr(\045)p Fq(\()p Fr(H)672 1655 y Fm(\003)692 1673 y Fq(\))p Fj(.)257 1767 y(Pr)n(o)n(of:)20 b Fw(T)l(aking)d(in)o(to)f (accoun)o(t)g(the)g(b)q(oundary)h(conditions)g(\(1.1\))f(one)g(gets)356 1868 y Fq(\()p Fr(H)419 1847 y Fm(\003)439 1868 y Fr(f)s(;)8 b(g)r Fq(\))j Fn(\000)g Fq(\()p Fr(f)s(;)d(H)705 1847 y Fm(\003)724 1868 y Fr(g)r Fq(\))14 b(=)g(2)p Fr(i)p Fn(=)p Fw(m)o Fq(\()p Fr(\024)999 1875 y Fo(b)1016 1868 y Fq(\))p Fr(f)5 b Fq(\()p Fr(b)p Fq(\))p 1123 1824 84 2 v Fr(g)r Fq(\()p Fr(b)p Fq(\))11 b(+)g(2)p Fr(i)p Fn(=)p Fw(m)n Fq(\()p Fr(\024)1431 1875 y Fo(a)1452 1868 y Fq(\))p Fr(f)5 b Fq(\()p Fr(a)p Fq(\))p 1564 1824 89 2 v Fr(g)r Fq(\()p Fr(a)p Fq(\))99 b Fw(\(3.10\))257 1968 y(for)17 b Fr(f)s(;)8 b(g)15 b Fn(2)f Fw(dom)p Fq(\()p Fr(H)622 1950 y Fm(\003)641 1968 y Fq(\))p Fw(.)22 b(By)15 b(\(3.1\))i(w)o(e)e (\034nd)554 2068 y Fq(\()p Fr(H)617 2047 y Fm(\003)637 2068 y Fr(f)s(;)8 b(g)r 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1236 y Fm(\003)919 1257 y Fr(T)g Fq(\()p Fr(\025)p Fq(\))14 b(=)g Fr(T)1116 1264 y Fm(\003)1135 1257 y Fq(\()p Fr(\025)p Fq(\))1201 1236 y Fm(\003)1221 1257 y Fr(T)1250 1264 y Fm(\003)1269 1257 y Fq(\()p Fr(\025)p Fq(\))417 b Fw(\(3.24\))257 1354 y(whic)o(h)16 b(yields)f(that)i Fq(\002)678 1361 y Fo(H)711 1354 y Fq(\()p Fr(\025)p Fq(\))g Fw(is)f(unitary)g(for)g (eac)o(h)g Fr(\025)e Fn(2)g Fv(R)p Fw(.)331 1421 y(Similarly)-5 b(,)13 b(one)k(can)f(in)o(tro)q(duce)g(the)g(c)o(haracteristic)e (function)i(of)h Fr(H)1617 1403 y Fm(\003)1653 1421 y Fw(de\034ned)f(b)o(y)360 1540 y Fq(\002)398 1547 y Fo(H)430 1538 y Fi(\003)449 1540 y Fq(\()p Fr(z)r Fq(\))p Fr(T)541 1547 y Fm(\003)560 1540 y Fq(\()p Fr(z)r Fq(\))p Fr(f)j Fq(=)14 b Fr(T)7 b Fq(\()p Fr(z)r Fq(\))p Fr(f)s(;)55 b(z)16 b Fn(2)e Fr(\045)p Fq(\()p Fr(H)t Fq(\))e Fn(\\)f Fr(\045)p Fq(\()p Fr(H)1250 1520 y Fm(\003)1270 1540 y Fq(\))p Fr(;)57 b(f)19 b Fn(2)14 b Fr(L)1483 1520 y Fp(2)1503 1540 y Fq(\([)p Fr(a;)8 b(b)p Fq(]\))p Fr(:)100 b Fw(\(3.25\))257 1659 y(As)17 b(ab)q(o)o(v)o(e)f(one)h(can)g(sho)o(w)g (that)g Fq(\002)913 1666 y Fo(H)945 1657 y Fi(\003)964 1659 y Fq(\()p Fn(\001)p Fq(\))g Fw(is)f(a)h(con)o(tractiv)o(e)e (analytic)h(function)g(in)h(the)257 1726 y(upp)q(er)i(half)f(plane.)26 b(On)18 b(the)f(real)h(axis)g(b)q(oth)h(c)o(haracteristic)d(functions)i (are)g(related)257 1792 y(b)o(y)772 1861 y Fq(\002)810 1868 y Fo(H)842 1859 y Fi(\003)862 1861 y Fq(\()p Fr(\025)p Fq(\))c(=)g(\002)1032 1868 y Fo(H)1066 1861 y Fq(\()p Fr(\025)p Fq(\))1132 1841 y Fm(\003)1152 1861 y Fr(;)56 b(\025)15 b Fn(2)f Fv(R)p Fr(;)387 b Fw(\(3.26\))257 1958 y(whic)o(h)20 b(sho)o(ws)i(that)f(the)g(c)o(haracteristic)e (function)i Fq(\002)1283 1965 y Fo(H)1315 1956 y Fi(\003)1334 1958 y Fq(\()p Fn(\001)p Fq(\))g Fw(is)f(unitary)h(on)g(the)g(real)257 2024 y(axis,)16 b(to)q(o.)22 b(Moreo)o(v)o(er,)15 b(a)i(straigh)o (tforw)o(ard)g(computation)e(sho)o(ws)i(that)651 2144 y Fq(\002)689 2151 y Fo(H)721 2142 y Fi(\003)741 2144 y Fq(\()p Fr(z)r Fq(\))c(=)h(\002)907 2151 y Fo(H)940 2144 y Fq(\()p 959 2116 25 2 v Fr(z)r Fq(\))1003 2123 y Fm(\003)1023 2144 y Fr(;)57 b(z)15 b Fn(2)f Fr(\045)p Fq(\()p Fr(H)t Fq(\))e Fn(\\)f Fr(\045)p Fq(\()p Fr(H)1430 2123 y Fm(\003)1450 2144 y Fq(\))p Fr(:)269 b Fw(\(3.27\))257 2263 y(holds.)22 b(F)l(urthermore,)14 b(from)h(\(3.20\))i(and)g (\(3.25\))g(w)o(e)e(\034nd)637 2382 y Fq(\002)675 2389 y Fo(H)707 2380 y Fi(\003)727 2382 y Fq(\()p Fr(z)r Fq(\))e(=)h(\002) 893 2389 y Fo(H)927 2382 y Fq(\()p Fr(z)r Fq(\))990 2361 y Fm(\000)p Fp(1)1037 2382 y Fr(;)56 b(z)16 b Fn(2)e Fr(\045)p Fq(\()p Fr(H)t Fq(\))e Fn(\\)f Fr(\045)p Fq(\()p Fr(H)1444 2361 y Fm(\003)1464 2382 y Fq(\))p Fr(:)255 b Fw(\(3.28\))257 2501 y(Let)17 b(us)f(no)o(w)h(compute)e(the)h(c)o (haracteristic)e(function)i Fq(\002)1319 2508 y Fo(H)1353 2501 y Fq(\()p Fr(z)r Fq(\))g Fw(of)g Fr(H)t Fw(:)1042 2626 y(10)p eop %%Page: 11 11 11 10 bop 71 274 a Fx(Lemma)17 b(3.3)24 b Fj(L)n(et)e Fr(V)34 b Fn(2)24 b Fr(L)606 256 y Fp(2)625 274 y Fq(\([)p Fr(a;)8 b(b)p Fq(]\))p Fj(,)22 b Fn(=)p Fw(m)o Fq(\()p Fr(V)11 b Fq(\))23 b(=)g(0)g Fj(and)f Fr(\024)1209 281 y Fo(a)1230 274 y Fr(;)8 b(\024)1280 281 y Fo(b)1320 274 y Fn(2)23 b Fv(C)1409 281 y Fp(+)1442 274 y Fj(.)37 b(Then)23 b(the)71 340 y(char)n(acteristic)17 b(function)i Fq(\002)603 347 y Fo(H)636 340 y Fq(\()p Fr(z)r Fq(\))f Fj(is)f(given)i(by)160 482 y Fq(\002)198 489 y Fo(H)232 482 y Fq(\()p Fr(z)r Fq(\))13 b(=)417 449 y(1)p 365 471 129 2 v 365 516 a Fr(W)411 523 y Fm(\003)431 516 y Fq(\()p Fr(z)r Fq(\))507 412 y Fl(\022)564 450 y Fr(W)7 b Fq(\()p Fr(z)r Fq(\))k Fn(\000)g Fr(i\013)789 432 y Fp(2)789 462 y Fo(a)809 450 y Fr(v)833 457 y Fo(b)850 450 y Fq(\()p Fr(a;)d(z)r Fq(\))160 b Fn(\000)p Fr(i\013)1208 457 y Fo(a)1228 450 y Fr(\013)1259 457 y Fo(b)685 516 y Fn(\000)p Fr(i\013)772 523 y Fo(a)792 516 y Fr(\013)823 523 y Fo(b)1002 516 y Fr(W)7 b Fq(\()p Fr(z)r Fq(\))k Fn(\000)f Fr(i\013)1226 498 y Fp(2)1226 529 y Fo(b)1246 516 y Fr(v)1270 523 y Fo(a)1291 516 y Fq(\()p Fr(b;)e(z)r Fq(\))1417 412 y Fl(\023)1462 482 y Fr(:)89 b Fw(\(3.29\))71 624 y Fj(for)16 b Fr(z)g Fn(2)e Fr(\045)p Fq(\()p Fr(H)t Fq(\))e Fn(\\)f Fr(\045)p Fq(\()p Fr(H)485 606 y Fm(\003)505 624 y Fq(\))p Fj(.)71 730 y(Pr)n(o)n(of:)20 b Fw(Using)c(\(3.20\))g(w)o(e)g(ha)o(v)o (e)g(to)g(\034nd)h(a)f(t)o(w)o(o-b)o(y-t)o(w)o(o)g(matrix-v)m(alued)f (function)563 872 y Fq(\002)601 879 y Fo(H)635 872 y Fq(\()p Fr(z)r Fq(\))f(=)763 801 y Fl(\022)823 839 y Fq(\002)861 846 y Fo(bb)893 839 y Fq(\()p Fr(z)r Fq(\))45 b(\002)1039 846 y Fo(ba)1075 839 y Fq(\()p Fr(z)r Fq(\))821 906 y(\002)859 913 y Fo(ab)895 906 y Fq(\()p Fr(z)r Fq(\))c(\002)1037 913 y Fo(aa)1076 906 y Fq(\()p Fr(z)r Fq(\))1160 801 y Fl(\023)1565 872 y Fw(\(3.30\))71 1011 y(suc)o(h)16 b(that)94 1150 y Fq(\002)132 1157 y Fo(H)165 1150 y Fq(\()p Fr(z)r Fq(\))236 1065 y Fl( )297 1112 y Fn(\000)363 1090 y Fo(\013)386 1096 y Fh(b)p 340 1100 84 2 v 340 1129 a Fo(W)5 b Fp(\()p Fo(z)q Fp(\))437 1072 y Fl(R)470 1085 y Fo(b)461 1129 y(a)496 1112 y Fr(dy)10 b(v)579 1119 y Fo(a)599 1112 y Fq(\()p Fr(y)r(;)e(z)r Fq(\))p Fr(f)d Fq(\()p Fr(y)r Fq(\))344 1168 y Fo(\013)367 1172 y Fh(a)p 323 1176 V 323 1204 a Fo(W)g Fp(\()p Fo(z)q Fp(\))420 1147 y Fl(R)453 1160 y Fo(b)443 1205 y(a)478 1187 y Fr(dy)11 b(v)562 1194 y Fo(b)578 1187 y Fq(\()p Fr(y)r(;)d(z)r Fq(\))p Fr(f)d Fq(\()p Fr(y)r Fq(\))823 1065 y Fl(!)877 1150 y Fq(=)928 1065 y Fl( )989 1112 y Fn(\000)1061 1090 y Fo(\013)1084 1096 y Fh(b)p 1032 1100 97 2 v 1032 1129 a Fo(W)1065 1133 y Fi(\003)1084 1129 y Fp(\()p Fo(z)q Fp(\))1142 1072 y Fl(R)1175 1085 y Fo(b)1166 1129 y(a)1201 1112 y Fr(dy)10 b(v)1284 1119 y Fm(\003)p Fo(a)1322 1112 y Fq(\()p Fr(y)r(;)e(z)r Fq(\))p Fr(f)d Fq(\()p Fr(y)r Fq(\))1042 1168 y Fo(\013)1065 1172 y Fh(a)p 1015 1176 V 1015 1204 a Fo(W)1048 1208 y Fi(\003)1066 1204 y Fp(\()p Fo(z)q Fp(\))1125 1147 y Fl(R)1158 1160 y Fo(b)1148 1205 y(a)1183 1187 y Fr(dy)10 b(v)1266 1194 y Fm(\003)p Fo(b)1301 1187 y Fq(\()p Fr(y)r(;)e(z)r Fq(\))p Fr(f)d Fq(\()p Fr(y)r Fq(\))1546 1065 y Fl(!)1594 1150 y Fr(:)1565 1254 y Fw(\(3.31\))71 1321 y(Since)11 b Fr(v)218 1328 y Fo(a)239 1321 y Fq(\()p Fr(x;)d(z)r Fq(\))k Fw(and)h Fr(v)479 1328 y Fo(b)496 1321 y Fq(\()p Fr(x;)8 b(z)r Fq(\))k Fw(as)i(w)o(ell)d(as)j Fr(v)854 1328 y Fm(\003)p Fo(a)892 1321 y Fq(\()p Fr(x;)8 b(z)r Fq(\))k Fw(and)h Fr(v)1132 1328 y Fm(\003)p Fo(b)1166 1321 y Fq(\()p Fr(x;)8 b(z)r Fq(\))13 b Fw(are)f(solutions)i(of)f(the)71 1387 y(same)18 b(second)h(order)g(di\033eren)o(tial)e(equation)i(the)g(solutions)h Fr(v)1252 1394 y Fm(\003)p Fo(a)1290 1387 y Fq(\()p Fr(x;)8 b(z)r Fq(\))18 b Fw(and)i Fr(v)1543 1394 y Fm(\003)p Fo(b)1577 1387 y Fq(\()p Fr(x;)8 b(z)r Fq(\))71 1453 y Fw(are)17 b(linear)g(com)o(binations)f(of)i(the)f(solutions)h Fr(v)959 1460 y Fo(a)979 1453 y Fq(\()p Fr(x;)8 b(z)r Fq(\))17 b Fw(and)h Fr(v)1229 1460 y Fo(b)1246 1453 y Fq(\()p Fr(x;)8 b(z)r Fq(\))p Fw(.)24 b(A)17 b(straigh)o(tfor-)71 1519 y(w)o(ard)f(computation)g(pro)o(v)o(es)f(that)165 1646 y Fr(v)189 1653 y Fm(\003)p Fo(a)228 1646 y Fq(\()p Fr(x;)8 b(z)r Fq(\))13 b(=)456 1612 y(1)p 411 1634 116 2 v 411 1680 a Fr(W)7 b Fq(\()p Fr(z)r Fq(\))539 1605 y Fl(\010)568 1646 y Fq(\()p Fr(W)g Fq(\()p Fr(z)r Fq(\))k Fn(\000)g Fr(i\013)812 1625 y Fp(2)812 1658 y Fo(a)832 1646 y Fr(v)856 1653 y Fo(b)873 1646 y Fq(\()p Fr(a;)d(z)r Fq(\)\))p Fr(v)1027 1653 y Fo(a)1047 1646 y Fq(\()p Fr(x;)g(z)r Fq(\))i(+)h Fr(i\013)1267 1625 y Fp(2)1267 1658 y Fo(a)1288 1646 y Fr(v)1312 1653 y Fo(b)1329 1646 y Fq(\()p Fr(x;)d(z)r Fq(\))1442 1605 y Fl(\011)1565 1646 y Fw(\(3.32\))71 1778 y(and)158 1891 y Fr(v)182 1898 y Fm(\003)p Fo(b)216 1891 y Fq(\()p Fr(x;)g(z)r Fq(\))13 b(=)445 1857 y(1)p 399 1879 V 399 1925 a Fr(W)7 b Fq(\()p Fr(z)r Fq(\))528 1850 y Fl(\010)557 1891 y Fr(i\013)605 1870 y Fp(2)605 1903 y Fo(b)625 1891 y Fr(v)649 1898 y Fo(a)669 1891 y Fq(\()p Fr(x;)h(z)r Fq(\))j(+)g(\()p Fr(W)c Fq(\()p Fr(z)r Fq(\))j Fn(\000)h Fr(i\013)1085 1870 y Fp(2)1085 1903 y Fo(b)1105 1891 y Fr(v)1129 1898 y Fo(a)1149 1891 y Fq(\()p Fr(b;)d(z)r Fq(\)\))p Fr(v)1298 1898 y Fo(b)1314 1891 y Fq(\()p Fr(x;)g(z)r Fq(\))1427 1850 y Fl(\011)1464 1891 y Fr(:)87 b Fw(\(3.33\))71 2026 y(Inserting)15 b(\(3.32\))i(and)g (\(3.33\))g(in)o(to)f(\(3.31\))h(w)o(e)e(obtain)224 2177 y Fr(T)253 2184 y Fm(\003)272 2177 y Fq(\()p Fr(z)r Fq(\))p Fr(f)k Fq(=)430 2092 y Fl( )490 2139 y Fn(\000)563 2118 y Fo(\013)586 2124 y Fh(b)p 534 2128 97 2 v 534 2156 a Fo(W)567 2160 y Fi(\003)585 2156 y Fp(\()p Fo(z)q Fp(\))644 2099 y Fl(R)677 2112 y Fo(b)667 2156 y(a)702 2139 y Fr(dy)11 b(v)786 2146 y Fm(\003)p Fo(a)824 2139 y Fq(\()p Fr(y)r(;)d(z)r Fq(\))p Fr(f)d Fq(\()p Fr(y)r Fq(\))544 2195 y Fo(\013)567 2199 y Fh(a)p 516 2203 V 516 2232 a Fo(W)549 2236 y Fi(\003)568 2232 y Fp(\()p Fo(z)q Fp(\))626 2174 y Fl(R)659 2188 y Fo(b)650 2232 y(a)685 2215 y Fr(dy)10 b(v)768 2222 y Fm(\003)p Fo(b)803 2215 y Fq(\()p Fr(y)r(;)e(z)r Fq(\))p Fr(f)d Fq(\()p Fr(y)r Fq(\))1048 2092 y Fl(!)1101 2177 y Fq(=)426 b Fw(\(3.34\))364 2325 y Fq(1)p 312 2347 129 2 v 312 2393 a Fr(W)358 2400 y Fm(\003)378 2393 y Fq(\()p Fr(z)r Fq(\))454 2289 y Fl(\022)511 2327 y Fr(W)7 b Fq(\()p Fr(z)r Fq(\))k Fn(\000)g Fr(i\013)736 2309 y Fp(2)736 2339 y Fo(a)756 2327 y Fr(v)780 2334 y Fo(b)797 2327 y Fq(\()p Fr(a;)d(z)r Fq(\))160 b Fn(\000)p Fr(i\013)1155 2334 y Fo(b)1172 2327 y Fr(\013)1203 2334 y Fo(a)632 2393 y Fn(\000)p Fr(i\013)719 2400 y Fo(a)739 2393 y Fr(\013)770 2400 y Fo(b)949 2393 y Fr(W)7 b Fq(\()p Fr(z)r Fq(\))k Fn(\000)f Fr(i\013)1173 2375 y Fp(2)1173 2406 y Fo(b)1193 2393 y Fr(v)1217 2400 y Fo(a)1237 2393 y Fq(\()p Fr(b;)e(z)r Fq(\))1364 2289 y Fl(\023)1409 2359 y Fr(T)f Fq(\()p Fr(z)r Fq(\))p Fr(f)71 2501 y Fw(whic)o(h)15 b(v)o(eri\034es)g(\(3.29\).)1134 b Fn(4)856 2626 y Fw(11)p eop %%Page: 12 12 12 11 bop 331 274 a Fw(Using)16 b(\(2.9\))g(and)h(\(3.32\))g(w)o(e)f (\034nd)g(that)617 377 y Fr(W)663 384 y Fm(\003)683 377 y Fq(\()p Fr(z)r Fq(\))11 b(+)g Fr(i\013)854 357 y Fp(2)854 390 y Fo(b)874 377 y Fr(v)898 384 y Fm(\003)p Fo(a)936 377 y Fq(\()p Fr(b;)d(z)r Fq(\))13 b(=)h Fr(W)7 b Fq(\()p Fr(z)r Fq(\))j Fn(\000)h Fr(i\013)1331 357 y Fp(2)1331 390 y Fo(a)1352 377 y Fr(v)1376 384 y Fo(b)1393 377 y Fq(\()p Fr(a;)d(z)r Fq(\))p Fr(:)234 b Fw(\(3.35\))257 481 y(Similarly)l(,)13 b(from)i(\(2.9\))i(and)g(\(3.33\))f(w)o(e)g(get) 617 584 y Fr(W)663 591 y Fm(\003)683 584 y Fq(\()p Fr(z)r Fq(\))11 b(+)g Fr(i\013)854 564 y Fp(2)854 597 y Fo(a)874 584 y Fr(v)898 591 y Fm(\003)p Fo(b)933 584 y Fq(\()p Fr(a;)d(z)r Fq(\))13 b(=)h Fr(W)7 b Fq(\()p Fr(z)r Fq(\))j Fn(\000)h Fr(i\013)1333 564 y Fp(2)1333 597 y Fo(b)1353 584 y Fr(v)1377 591 y Fo(a)1397 584 y Fq(\()p Fr(b;)d(z)r Fq(\))p Fr(:)235 b Fw(\(3.36\))257 688 y(Inserting)16 b(\(3.35\))h(and)g(\(3.36\))g(in)o(to)f(\(3.29\))g(w)o(e)g(obtain)514 821 y Fq(\002)552 828 y Fo(H)586 821 y Fq(\()p Fr(z)r Fq(\))d(=)h Fr(I)736 829 y Fg(C)756 819 y Ff(2)790 821 y Fq(+)d Fr(i)913 787 y Fq(1)p 861 809 129 2 v 861 855 a Fr(W)907 862 y Fm(\003)926 855 y Fq(\()p Fr(z)r Fq(\))1002 750 y Fl(\022)1060 788 y Fr(\013)1091 770 y Fp(2)1091 801 y Fo(b)1111 788 y Fr(v)1135 795 y Fm(\003)p Fo(a)1173 788 y Fq(\()p Fr(b;)d(z)r Fq(\))82 b Fn(\000)p Fr(\013)1431 795 y Fo(b)1448 788 y Fr(\013)1479 795 y Fo(a)1100 854 y Fn(\000)p Fr(\013)1170 861 y Fo(b)1187 854 y Fr(\013)1218 861 y Fo(a)1320 854 y Fr(\013)1351 836 y Fp(2)1351 867 y Fo(a)1372 854 y Fr(v)1396 861 y Fm(\003)p Fo(b)1430 854 y Fq(\()p Fr(a;)8 b(z)r Fq(\))1561 750 y Fl(\023)1606 821 y Fr(:)132 b Fw(\(3.37\))257 950 y(On)19 b(the)f(formal)f(lev)o(el) f(the)i(c)o(haracteristic)f(function)h(can)h(b)q(e)f(expressed)g(as)h (follo)o(ws.)257 1016 y(W)l(e)d(in)o(tro)q(duce)g(the)g(unclosed)g(op)q (erator)h Fr(\013)e Fq(:)e Fn(H)h(\000)-8 b(!)14 b Fv(C)1292 998 y Fp(2)1315 1016 y Fw(.)539 1146 y Fr(\013f)20 b Fq(=)666 1076 y Fl(\022)747 1114 y Fr(\013)778 1121 y Fo(b)795 1114 y Fr(f)5 b Fq(\()p Fr(b)p Fq(\))723 1180 y Fn(\000)p Fr(\013)793 1187 y Fo(a)814 1180 y Fr(f)g Fq(\()p Fr(a)p Fq(\))927 1076 y Fl(\023)972 1146 y Fr(;)106 b(f)19 b Fn(2)14 b Fw(dom)o Fq(\()p Fr(\013)p Fq(\))g(=)g Fr(C)t Fq(\([)p Fr(a;)8 b(b)p Fq(]\))p Fr(;)155 b Fw(\(3.38\))257 1276 y(one)17 b(gets)f(the)g(represen)o(tation)814 1379 y Fq(\002)852 1386 y Fo(H)886 1379 y Fq(\()p Fr(z)r Fq(\))d(=)h Fr(I)1036 1387 y Fg(C)1057 1378 y Ff(2)1090 1379 y Fn(\000)d Fr(i\013T)c Fq(\()p 1243 1352 25 2 v Fr(z)q Fq(\))1286 1358 y Fm(\003)1306 1379 y Fr(:)432 b Fw(\(3.39\))257 1482 y(If)14 b(w)o(e)g(assume)f(for)i(a)f(momen)o(t)e(that)i(the)g(op)q (erator)i Fr(\013)1249 1464 y Fm(\003)1283 1482 y Fw(mak)o(es)c(sense)j (then)f(w)o(e)f(get)i(the)257 1549 y(form)o(ula)589 1615 y Fq(\002)627 1622 y Fo(H)660 1615 y Fq(\()p Fr(z)r Fq(\))f(=)g Fr(I)811 1623 y Fg(C)831 1614 y Ff(2)864 1615 y Fn(\000)d Fr(i\013)p Fq(\()p Fr(H)1025 1594 y Fm(\003)1056 1615 y Fn(\000)g Fr(z)r Fq(\))1150 1594 y Fm(\000)p Fp(1)1197 1615 y Fr(\013)1228 1594 y Fm(\003)1248 1615 y Fr(;)57 b(z)15 b Fn(2)g Fr(\045)p Fq(\()p Fr(H)t Fq(\))1512 1594 y Fm(\003)1532 1615 y Fr(;)206 b Fw(\(3.40\))257 1703 y(whic)o(h)17 b(is)g(exp)q(ected.)23 b(Indeed,)17 b(if)g(the)g (imaginary)f(part)h(of)h(the)f(dissipativ)o(e)f(op)q(erator)257 1769 y Fr(H)21 b Fw(is)16 b(a)h(b)q(ounded)g(op)q(erator,)g(then)f (form)o(ula)e(\(3.40\))j(is)f(w)o(ell-kno)o(wn,)f(e.g.)21 b([1].)257 1943 y Fu(4)77 b(Dilations)257 2058 y Fw(Since)22 b Fr(H)k Fw(is)c(a)h(maximal)c(dissipativ)o(e)i(op)q(erator)j(there)d (is)h(a)h(larger)f(Hilb)q(ert)f(space)257 2124 y Fk(K)15 b Fn(\023)e Fk(H)j Fw(and)h(a)g(self-adjoin)o(t)f(op)q(erator)h Fr(K)j Fw(on)d Fk(K)g Fw(suc)o(h)f(that)g(one)h(has)644 2228 y Fr(P)682 2207 y Fe(K)675 2240 y(H)708 2228 y Fq(\()p Fr(K)e Fn(\000)c Fr(z)r Fq(\))877 2207 y Fm(\000)p Fp(1)924 2228 y Fn(j)p Fk(H)i Fq(=)h(\()p Fr(H)h Fn(\000)c Fr(z)r Fq(\))1207 2207 y Fm(\000)p Fp(1)1254 2228 y Fr(;)57 b(z)15 b Fn(2)f Fv(C)1443 2235 y Fp(+)1476 2228 y Fr(:)286 b Fw(\(4.1\))257 2331 y(The)20 b(op)q(erator)i Fr(K)i Fw(is)c(called)f(a)h(self-adjoin)o(t)g(dilation)f(of)i(the)f(maximal)d (dissipativ)o(e)257 2398 y(op)q(erator)h Fr(H)t Fw(.)j(Ob)o(viously)l (,)15 b(from)g(the)h(condition)g(\(4.1\))h(one)f(gets)634 2501 y Fr(P)672 2480 y Fe(K)665 2513 y(H)698 2501 y Fq(\()p Fr(K)f Fn(\000)c Fr(z)r Fq(\))867 2480 y Fm(\000)p Fp(1)914 2501 y Fn(j)p Fk(H)i Fq(=)h(\()p Fr(H)1092 2480 y Fm(\003)1123 2501 y Fn(\000)d Fr(z)r Fq(\))1217 2480 y Fm(\000)p Fp(1)1264 2501 y Fr(;)56 b(z)16 b Fn(2)e Fv(C)1453 2508 y Fm(\000)1486 2501 y Fr(:)276 b Fw(\(4.2\))1042 2626 y(12)p eop %%Page: 13 13 13 12 bop 71 274 a Fw(If)15 b(the)h(condition)677 297 y Fl(_)651 405 y Fo(z)q Fm(2)p Fg(C)t Fm(n)p Fg(R)758 345 y Fq(\()p Fr(K)f Fn(\000)c Fr(z)r Fq(\))927 324 y Fm(\000)p Fp(1)974 345 y Fk(H)j Fq(=)g Fk(K)480 b Fw(\(4.3\))71 489 y(is)18 b(satis\034ed,)h(then)g Fr(K)k Fw(is)c(called)e(a)j(minim)o (al)c(self-adjoin)o(t)i(dilation)h(of)g Fr(H)t Fw(.)29 b(Minimal)71 555 y(self-adjoin)o(t)12 b(dilations)i(of)f(maximal)d (dissipativ)o(e)i(op)q(erators)j(are)e(determined)e(up)i(to)h(a)71 622 y(certain)g(isomorphism,)f(in)i(particular,)f(all)h(minim)o(al)d (self-adjoin)o(t)j(dilations)g(are)h(uni-)71 688 y(tarily)e(equiv)m (alen)o(t.)19 b(The)c(minimal)d(self-adjoin)o(t)i(dilation)h(of)g(a)h (self-adjoin)o(t)e(op)q(erator)71 754 y(is)21 b(the)g(self-adjoin)o(t)f (op)q(erator)j(itself,)e(th)o(us)g(of)g(real)g(in)o(terest)f(is)h(only) g(the)g(minimal)71 820 y(self-adjoin)o(t)15 b(dilation)h(of)g(the)g (completely)d(non-selfadjoin)o(t)j(part)h(of)f(a)g(maximal)d(dis-)71 886 y(sipativ)o(e)j(op)q(erator.)27 b(Since)17 b Fr(H)22 b Fw(is)17 b(purely)g(maxim)e(dissipativ)o(e)h(w)o(e)i(are)g(in)o (terested)e(in)71 953 y(the)g(minim)o(al)d(self-adjoin)o(t)j(dilation)g (of)g Fr(H)21 b Fw(only)l(.)144 1020 y(Our)11 b(next)f(aim)g(is)h(to)g (obtain)h(an)g(explicit)d(description)h(of)h(the)g(self-adjoin)o(t)g (dilation)71 1087 y(of)16 b Fr(H)t Fw(.)22 b(T)l(o)16 b(this)g(end)h(w)o(e)e(in)o(tro)q(duce)h(the)g(Hilb)q(ert)f(space)h Fk(K)h Fw(giv)o(en)e(b)o(y)684 1207 y Fk(K)f Fq(=)g Fn(D)821 1214 y Fm(\000)862 1207 y Fn(\010)d Fk(H)f Fn(\010)h(D)1046 1214 y Fp(+)1589 1207 y Fw(\(4.4\))71 1328 y(where)k Fn(D)249 1335 y Fm(\006)293 1328 y Fq(:=)f Fr(L)392 1310 y Fp(2)411 1328 y Fq(\()p Fv(R)469 1335 y Fm(\006)496 1328 y Fr(;)8 b Fv(C)551 1310 y Fp(2)573 1328 y Fq(\))p Fw(.)22 b(In)o(tro)q(ducing)16 b(the)g(domain)1155 1315 y Fq(^)1149 1328 y(\012)p 444 1479 333 4 v 610 1457 a Fv(R)649 1464 y Fm(\000)p 776 1479 V 942 1457 a Fv(R)981 1464 y Fp(+)p 444 1645 V 610 1623 a Fv(R)649 1630 y Fm(\000)p 776 1645 V 942 1623 a Fv(R)981 1630 y Fp(+)p 775 1644 4 167 v 797 1569 a Fq([)p Fr(a;)8 b(b)p Fq(])71 1800 y Fw(one)16 b(can)g(write)g(the)g(Hilb)q(ert)f(space)h Fk(K)h Fw(as)g Fr(L)899 1782 y Fp(2)919 1800 y Fq(\()943 1788 y(^)938 1800 y(\012)p Fr(;)8 b(dx)p Fq(\))p Fw(.)21 b(F)l(urther,)15 b(w)o(e)h(de\034ne)699 1921 y Fr(~)-21 b(g)16 b Fq(:=)d Fr(g)829 1928 y Fm(\000)870 1921 y Fn(\010)e Fr(g)i Fn(\010)e Fr(g)1029 1928 y Fp(+)1589 1921 y Fw(\(4.5\))71 2041 y(where)332 2129 y Fr(g)355 2136 y Fm(\000)385 2129 y Fq(\()p Fr(x)p Fq(\))j(:=)530 2058 y Fl(\022)588 2096 y Fr(g)613 2078 y Fo(b)611 2109 y Fm(\000)640 2096 y Fq(\()p Fr(x)p Fq(\))588 2163 y Fr(g)613 2144 y Fo(a)611 2175 y Fm(\000)640 2163 y Fq(\()p Fr(x)p Fq(\))727 2058 y Fl(\023)869 2129 y Fw(and)50 b Fr(g)1020 2136 y Fp(+)1049 2129 y Fq(\()p Fr(x)p Fq(\))14 b(:=)1194 2058 y Fl(\022)1252 2096 y Fr(g)1277 2078 y Fo(b)1275 2109 y Fp(+)1305 2096 y Fq(\()p Fr(x)p Fq(\))1252 2163 y Fr(g)1277 2144 y Fo(a)1275 2175 y Fp(+)1305 2163 y Fq(\()p Fr(x)p Fq(\))1391 2058 y Fl(\023)1589 2129 y Fw(\(4.6\))71 2253 y(for)i Fr(x)e Fn(2)g Fv(R)273 2260 y Fm(\000)316 2253 y Fw(and)j Fr(x)d Fn(2)g Fv(R)539 2260 y Fp(+)566 2253 y Fw(,)i(resp)q(ectiv)o(ely)l(.)j (Let)d(us)h(in)o(tro)q(duce)f(the)g(matrices)f Fr(K)1569 2235 y Fo(a)1565 2265 y Fm(\006)1611 2253 y Fw(and)71 2319 y Fr(K)116 2301 y Fo(b)112 2332 y Fm(\006)158 2319 y Fw(whic)o(h)g(are)i(de\034ned)f(b)o(y)289 2472 y Fr(K)334 2452 y Fo(a)330 2485 y Fm(\000)374 2472 y Fq(:=)458 2438 y(1)p 444 2461 52 2 v 444 2506 a Fr(\013)475 2513 y Fo(a)509 2402 y Fl(\022)567 2440 y Fq(0)54 b(0)567 2506 y(1)42 b Fr(\024)661 2513 y Fo(a)702 2402 y Fl(\023)845 2472 y Fw(and)98 b Fr(K)1066 2452 y Fo(a)1062 2485 y Fp(+)1106 2472 y Fq(:=)1190 2438 y(1)p 1176 2461 V 1176 2506 a Fr(\013)1207 2513 y Fo(a)1241 2402 y Fl(\022)1299 2440 y Fq(0)54 b(0)1299 2506 y(1)p 1365 2479 49 2 v 42 w Fr(\024)1393 2513 y Fo(a)1434 2402 y Fl(\023)1589 2472 y Fw(\(4.7\))856 2626 y(13)p eop %%Page: 14 14 14 13 bop 257 274 a Fw(as)17 b(w)o(ell)e(as)383 394 y Fr(K)428 374 y Fo(b)424 407 y Fm(\000)468 394 y Fq(:=)551 361 y(1)p 539 383 49 2 v 539 429 a Fr(\013)570 436 y Fo(b)600 324 y Fl(\022)658 362 y Fq(1)41 b Fn(\000)p Fr(\024)790 369 y Fo(b)658 428 y Fq(0)71 b(0)828 324 y Fl(\023)971 394 y Fw(and)98 b Fr(K)1192 374 y Fo(b)1188 407 y Fp(+)1232 394 y Fq(:=)1314 361 y(1)p 1302 383 V 1302 429 a Fr(\013)1333 436 y Fo(b)1363 324 y Fl(\022)1421 362 y Fq(1)42 b Fn(\000)p 1526 335 46 2 v Fr(\024)1554 369 y Fo(b)1421 428 y Fq(0)72 b(0)1591 324 y Fl(\023)1636 394 y Fr(:)126 b Fw(\(4.8\))257 524 y(W)l(e)16 b(note)h(that)472 628 y Fr(K)517 608 y Fo(a)p Fm(\003)513 640 y(\000)555 628 y Fr(K)600 608 y Fo(a)596 640 y Fm(\000)637 628 y Fn(\000)11 b Fr(K)732 608 y Fo(a)p Fm(\003)728 640 y Fp(+)771 628 y Fr(K)816 608 y Fo(a)812 640 y Fp(+)856 628 y Fq(=)i Fr(iE)52 b Fw(and)d Fr(K)1184 608 y Fo(b)p Fm(\003)1180 640 y(\000)1219 628 y Fr(K)1264 608 y Fo(b)1260 640 y Fm(\000)1301 628 y Fn(\000)11 b Fr(K)1396 608 y Fo(b)p Fm(\003)1392 640 y Fp(+)1431 628 y Fr(K)1476 608 y Fo(b)1472 640 y Fp(+)1516 628 y Fq(=)i Fn(\000)p Fr(iE)117 b Fw(\(4.9\))257 732 y(as)17 b(w)o(ell)e(as)511 836 y Fr(K)556 815 y Fo(b)p Fm(\003)552 848 y(\000)591 836 y Fr(K)636 815 y Fo(a)632 848 y Fm(\000)676 836 y Fq(=)e Fr(K)772 815 y Fo(a)p Fm(\003)768 848 y(\000)811 836 y Fr(K)856 815 y Fo(b)852 848 y Fm(\000)896 836 y Fq(=)h(0)49 b Fw(and)g Fr(K)1193 815 y Fo(b)p Fm(\003)1189 848 y Fp(+)1228 836 y Fr(K)1273 815 y Fo(a)1269 848 y Fp(+)1313 836 y Fq(=)14 b Fr(K)1410 815 y Fo(a)p Fm(\003)1406 848 y Fp(+)1448 836 y Fr(K)1493 815 y Fo(b)1489 848 y Fp(+)1533 836 y Fq(=)g(0)p Fr(;)129 b Fw(\(4.10\))257 940 y(where)875 1022 y Fr(E)17 b Fq(:=)993 952 y Fl(\022)1070 990 y Fq(0)61 b(1)1050 1056 y Fn(\000)p Fq(1)42 b(0)1200 952 y Fl(\023)1245 1022 y Fr(:)493 b Fw(\(4.11\))257 1136 y(F)l(urther)16 b(w)o(e)g(need)g(the)g(relation)609 1240 y Fq(\()p Fr(l)q Fq(\()p Fr(g)r Fq(\))p Fr(;)8 b(f)d Fq(\))11 b Fn(\000)g Fq(\()p Fr(g)r(;)d(l)q Fq(\()p Fr(f)d Fq(\)\))14 b(=)g Fn(h)p Fr(E)s(g)1153 1247 y Fo(b)1170 1240 y Fr(;)8 b(f)1216 1247 y Fo(b)1234 1240 y Fn(i)j(\000)g(h)p Fr(E)s(g)1395 1247 y Fo(a)1416 1240 y Fr(;)d(f)1462 1247 y Fo(a)1483 1240 y Fn(i)h Fr(;)227 b Fw(\(4.12\))257 1344 y(whic)o(h)16 b(holds)g(for)h Fr(f)s(;)659 1324 y Fp(1)p 652 1333 32 2 v 652 1361 a Fo(m)689 1344 y Fr(f)s(;)8 b(g)r(;)796 1324 y Fp(1)p 789 1333 V 789 1361 a Fo(m)825 1344 y Fr(g)16 b Fn(2)e Fr(W)964 1326 y Fp(1)p Fo(;)p Fp(2)1011 1344 y Fq(\([)p Fr(a;)8 b(b)p Fq(]\))15 b Fw(where)534 1490 y Fr(g)557 1497 y Fo(a)592 1490 y Fq(=)644 1405 y Fl( )748 1435 y Fp(1)p 709 1444 96 2 v 709 1472 a(2)p Fo(m)p Fp(\()p Fo(a)p Fp(\))809 1455 y Fr(g)834 1437 y Fm(0)846 1455 y Fq(\()p Fr(a)p Fq(\))763 1527 y Fr(g)r Fq(\()p Fr(a)p Fq(\))930 1405 y Fl(!)1027 1490 y Fw(and)49 b Fr(g)1177 1497 y Fo(b)1209 1490 y Fq(=)1260 1405 y Fl( )1362 1435 y Fp(1)p 1325 1444 92 2 v 1325 1472 a(2)p Fo(m)p Fp(\()p Fo(b)p Fp(\))1422 1455 y Fr(g)1447 1437 y Fm(0)1459 1455 y Fq(\()p Fr(b)p Fq(\))1377 1527 y Fr(g)r Fq(\()p Fr(b)p Fq(\))1538 1405 y Fl(!)1586 1490 y Fr(:)152 b Fw(\(4.13\))257 1641 y(Similarly)l(,)13 b(one)k(de\034nes)f Fr(f)748 1648 y Fo(a)785 1641 y Fw(and)h Fr(f)904 1648 y Fo(b)922 1641 y Fw(.)k(On)c(the)f(Hilb)q(ert)f(space)i Fr(L)1455 1623 y Fp(2)1474 1641 y Fq(\()1498 1629 y(^)1493 1641 y(\012)q Fr(;)8 b(dx)p Fq(\))16 b Fw(w)o(e)g(de\034ne)g(a)257 1708 y(self-adjoin)o(t)d(op)q(erator)i Fr(K)j Fw(whic)o(h)12 b(should)i(pla)o(y)f(the)g(role)g(of)h(a)g(self-adjoin)o(t)f(dilation)g (of)257 1774 y(the)j(maximal)d(dissipativ)o(e)i(op)q(erator)j Fr(H)t Fw(.)j(W)l(e)16 b(c)o(ho)q(ose)h(the)f(op)q(erator)h Fr(K)k Fw(in)16 b(the)g(form)p 340 2007 416 4 v 382 1964 a Fn(\000)p Fr(i)452 1944 y Fo(d)p 443 1953 39 2 v 443 1981 a(dx)485 1964 y Fr(g)510 1946 y Fo(b)508 1976 y Fm(\000)756 2014 y Fq(\))672 1931 y Fr(g)697 1913 y Fo(b)695 1943 y Fm(\000)725 1931 y Fq(\(0\))p 1171 2007 416 4 v 1378 1964 a Fn(\000)p Fr(i)1448 1944 y Fo(d)p 1439 1953 39 2 v 1439 1981 a(dx)1482 1964 y Fr(g)1507 1946 y Fo(b)1505 1976 y Fp(+)1162 2014 y Fw(\()1171 1931 y Fr(g)1196 1913 y Fo(b)1194 1943 y Fp(+)1223 1931 y Fq(\(0\))p 340 2173 416 4 v 382 2130 a Fn(\000)p Fr(i)452 2110 y Fo(d)p 443 2119 39 2 v 443 2147 a(dx)485 2130 y Fr(g)510 2112 y Fo(a)508 2142 y Fm(\000)756 2180 y Fq(\))672 2263 y Fr(g)697 2245 y Fo(a)695 2275 y Fm(\000)725 2263 y Fq(\(0\))p 1171 2173 416 4 v 1378 2130 a Fn(\000)p Fr(i)1448 2110 y Fo(d)p 1439 2119 39 2 v 1439 2147 a(dx)1482 2130 y Fr(g)1507 2112 y Fo(a)1505 2142 y Fp(+)1162 2180 y Fq(\()1171 2263 y Fr(g)1196 2245 y Fo(a)1194 2275 y Fp(+)1223 2263 y Fq(\(0\))p 961 2172 4 167 v 1005 2088 a Fr(l)q Fq(\()p Fr(g)r Fq(\))830 1853 y Fd( )929 1880 y Fp(1)p 892 1887 92 2 v 892 1914 a(2)p Fo(m)p Fp(\()p Fo(b)p Fp(\))988 1898 y Fc(g)1012 1881 y Fm(0)1023 1898 y Fb(\()p Fc(b)p Fb(\))943 1966 y Fc(g)r Fb(\()p Fc(b)p Fb(\))1099 1853 y Fd(!)830 2218 y( )931 2245 y Fp(1)p 892 2252 96 2 v 892 2279 a(2)p Fo(m)p Fp(\()p Fo(a)p Fp(\))992 2263 y Fc(g)1016 2247 y Fm(0)1027 2263 y Fb(\()p Fc(a)p Fb(\))945 2331 y Fc(g)r Fb(\()p Fc(a)p Fb(\))1107 2218 y Fd(!)257 2435 y Fw(where)24 b(the)f(problem)f(is)h(to)h(\034nd)g(suitable)f(b)q (oundary)i(conditions)e(suc)o(h)h(that)g(the)257 2501 y(arising)16 b(op)q(erator)h(is)f(self-adjoin)o(t.)21 b(This)16 b(is)f(the)h(con)o(ten)o(t)f(of)h(the)g(follo)o(wing)f (theorem.)1042 2626 y(14)p eop %%Page: 15 15 15 14 bop 71 274 a Fx(Theorem)17 b(4.1)24 b Fj(L)n(et)c Fr(V)31 b Fn(2)19 b Fr(L)635 256 y Fp(2)655 274 y Fq(\([)p Fr(a;)8 b(b)p Fq(]\))p Fj(,)19 b Fn(=)p Fw(m)o Fq(\()p Fr(V)11 b Fq(\))19 b(=)g(0)i Fj(and)g Fr(\024)1225 281 y Fo(a)1246 274 y Fr(;)8 b(\024)1296 281 y Fo(b)1332 274 y Fn(2)19 b Fv(C)1417 281 y Fp(+)1450 274 y Fj(.)31 b(Then)21 b(the)71 340 y(op)n(er)n(ator)15 b Fr(K)22 b Fj(de\034ne)n(d)c(by)125 484 y Fw(dom)o Fq(\()p Fr(K)t Fq(\))c(:=)379 414 y Fl(\032)435 452 y Fr(~)-22 b(g)16 b Fn(2)e(K)h Fq(:)e Fr(g)626 459 y Fm(\006)670 452 y Fn(2)h Fr(W)770 434 y Fp(1)p Fo(;)p Fp(2)817 452 y Fq(\()p Fv(R)875 459 y Fm(\006)901 452 y Fr(;)8 b Fv(C)956 434 y Fp(2)979 452 y Fq(\))p Fr(;)22 b(g)r(;)1092 432 y Fp(1)p 1085 440 32 2 v 1085 469 a Fo(m)1122 452 y Fr(g)1147 434 y Fm(0)1172 452 y Fn(2)14 b Fr(W)1272 434 y Fp(1)p Fo(;)p Fp(2)1319 452 y Fq(\([)p Fr(a;)8 b(b)p Fq(]\))459 518 y Fr(K)504 500 y Fo(a)500 530 y Fm(\000)530 518 y Fr(g)553 525 y Fo(a)586 518 y Fq(+)j Fr(K)680 500 y Fo(b)676 530 y Fm(\000)706 518 y Fr(g)729 525 y Fo(b)760 518 y Fq(=)j Fr(g)835 525 y Fm(\000)865 518 y Fq(\(0\))p Fr(;)22 b(K)1008 500 y Fo(a)1004 530 y Fp(+)1034 518 y Fr(g)1057 525 y Fo(a)1089 518 y Fq(+)11 b Fr(K)1183 500 y Fo(b)1179 530 y Fp(+)1209 518 y Fr(g)1232 525 y Fo(b)1263 518 y Fq(=)j Fr(g)1338 525 y Fp(+)1368 518 y Fq(\(0\))1473 414 y Fl(\033)1565 484 y Fw(\(4.14\))71 628 y Fj(and)336 702 y Fr(K)q(~)-21 b(g)16 b Fq(:=)d Fn(\000)p Fr(i)559 668 y(d)p 546 691 54 2 v 546 736 a(dx)604 702 y(g)627 709 y Fm(\000)667 702 y Fn(\010)e Fr(l)q Fq(\()p Fr(g)r Fq(\))g Fn(\010)g(\000)p Fr(i)931 668 y(d)p 918 691 V 918 736 a(dx)975 702 y(g)998 709 y Fp(+)1028 702 y Fr(;)106 b(~)-22 b(g)16 b Fn(2)e Fw(dom)o Fq(\()p Fr(K)t Fq(\))p Fr(;)140 b Fw(\(4.15\))71 804 y Fj(is)17 b(self-adjoint.)71 914 y(Pr)n(o)n(of:)j Fw(W)l(e)15 b(\034nd)229 1052 y Fq(\()p Fr(K)q(~)-21 b(g)r(;)347 1038 y(~)340 1052 y(f)5 b Fq(\))14 b(=)454 981 y Fl(\034)491 1052 y Fn(\000)p Fr(i)565 1018 y(d)p 552 1040 V 552 1086 a(dx)610 1052 y(g)633 1059 y Fm(\000)663 1052 y Fr(;)8 b(f)709 1059 y Fm(\000)738 981 y Fl(\035)786 1052 y Fq(+)j(\()p Fr(l)q Fq(\()p Fr(g)r Fq(\))p Fr(;)d(f)d Fq(\))11 b(+)1063 981 y Fl(\034)1101 1052 y Fn(\000)p Fr(i)1175 1018 y(d)p 1162 1040 V 1162 1086 a(dx)1219 1052 y(g)1242 1059 y Fp(+)1272 1052 y Fr(;)d(f)1318 1059 y Fp(+)1347 981 y Fl(\035)1393 1052 y Fr(;)158 b Fw(\(4.16\))68 1203 y Fr(~)-21 b(g)r(;)124 1190 y(~)118 1203 y(f)19 b Fn(2)14 b Fw(dom)o Fq(\()p Fr(K)t Fq(\))p Fw(.)21 b(One)16 b(has)322 1273 y Fl(\034)360 1344 y Fn(\000)p Fr(i)434 1310 y(d)p 421 1332 V 421 1378 a(dx)478 1344 y(g)501 1351 y Fm(\000)531 1344 y Fr(;)8 b(f)577 1351 y Fm(\000)606 1273 y Fl(\035)657 1344 y Fq(=)14 b Fn(h)q(\000)p Fr(ig)808 1351 y Fm(\000)837 1344 y Fq(\(0\))p Fr(;)8 b(f)945 1351 y Fm(\000)975 1344 y Fq(\(0\))p Fn(i)j Fq(+)1116 1273 y Fl(\034)1154 1344 y Fr(g)1177 1351 y Fm(\000)1207 1344 y Fr(;)d Fn(\000)p Fr(i)1303 1310 y(d)p 1290 1332 V 1290 1378 a(dx)1347 1344 y(f)1371 1351 y Fm(\000)1400 1273 y Fl(\035)1565 1344 y Fw(\(4.17\))71 1481 y(and)331 1489 y Fl(\034)368 1559 y Fn(\000)p Fr(i)442 1525 y(d)p 429 1547 V 429 1593 a(dx)486 1559 y(g)509 1566 y Fp(+)539 1559 y Fr(;)g(f)585 1566 y Fp(+)615 1489 y Fl(\035)666 1559 y Fq(=)14 b Fn(h)p Fr(ig)777 1566 y Fp(+)807 1559 y Fq(\(0\))p Fr(;)8 b(f)915 1566 y Fp(+)944 1559 y Fq(\(0\))p Fn(i)k Fq(+)1086 1489 y Fl(\034)1123 1559 y Fr(g)1146 1566 y Fp(+)1176 1559 y Fr(;)c Fn(\000)p Fr(i)1272 1525 y(d)p 1259 1547 V 1259 1593 a(dx)1317 1559 y(f)1341 1566 y Fp(+)1370 1489 y Fl(\035)1416 1559 y Fr(:)135 b Fw(\(4.18\))71 1677 y(Inserting)15 b(\(4.12\),)i(\(4.17\))g(and)f(\(4.18\))h(in)o(to)f(\(4.16\))h(w)o(e)f (obtain)216 1794 y Fq(\()p Fr(K)q(~)-21 b(g)r(;)334 1781 y(~)327 1794 y(f)5 b Fq(\))12 b Fn(\000)e Fq(\()n Fr(~)-22 b(g)r(;)8 b(K)554 1781 y(~)547 1794 y(f)e Fq(\))14 b(=)917 b Fw(\(4.19\))299 1873 y Fn(\000)p Fr(i)8 b Fn(h)p Fr(g)405 1880 y Fm(\000)435 1873 y Fq(\(0\))p Fr(;)g(f)543 1880 y Fm(\000)573 1873 y Fq(\(0\))p Fn(i)k Fq(+)f Fn(h)p Fr(E)s(g)796 1880 y Fo(b)813 1873 y Fr(;)d(f)859 1880 y Fo(b)877 1873 y Fn(i)j(\000)g(h)p Fr(E)s(g)1038 1880 y Fo(a)1059 1873 y Fr(;)d(f)1105 1880 y Fo(a)1126 1873 y Fn(i)k Fq(+)f Fr(i)d Fn(h)p Fr(g)1273 1880 y Fp(+)1303 1873 y Fq(\(0\))p Fr(;)g(f)1411 1880 y Fp(+)1440 1873 y Fq(\(0\))p Fn(i)h Fr(:)71 1991 y Fw(Using)16 b(\(4.14\))h(w)o(e)e (\034nd)198 2108 y Fq(\()p Fr(K)q(~)-21 b(g)r(;)316 2095 y(~)309 2108 y(f)5 b Fq(\))11 b Fn(\000)g Fq(\()n Fr(~)-22 b(g)r(;)8 b(K)536 2095 y(~)529 2108 y(f)e Fq(\))13 b(=)h Fn(\000)p Fr(i)707 2068 y Fl(\012)730 2108 y Fn(f)p Fr(K)800 2088 y Fo(a)796 2121 y Fm(\000)826 2108 y Fr(g)849 2115 y Fo(a)881 2108 y Fq(+)d Fr(K)975 2088 y Fo(b)971 2121 y Fm(\000)1002 2108 y Fr(g)1025 2115 y Fo(b)1042 2108 y Fn(g)p Fr(;)d Fn(f)p Fr(K)1159 2088 y Fo(a)1155 2121 y Fm(\000)1185 2108 y Fr(f)1209 2115 y Fo(a)1241 2108 y Fq(+)j Fr(K)1335 2088 y Fo(b)1331 2121 y Fm(\000)1361 2108 y Fr(f)1385 2115 y Fo(b)1402 2108 y Fn(g)1427 2068 y Fl(\013)1459 2108 y Fq(+)68 b Fw(\(4.20\))281 2187 y Fn(h)p Fr(E)s(g)362 2194 y Fo(b)380 2187 y Fr(;)8 b(f)426 2194 y Fo(b)443 2187 y Fn(i)j(\000)g(h)q Fr(E)s(g)605 2194 y Fo(a)626 2187 y Fr(;)d(f)672 2194 y Fo(a)692 2187 y Fn(i)k Fq(+)f Fr(i)797 2147 y Fl(\012)820 2187 y Fn(f)p Fr(K)890 2166 y Fo(a)886 2199 y Fp(+)916 2187 y Fr(g)939 2194 y Fo(a)972 2187 y Fq(+)g Fr(K)1066 2166 y Fo(b)1062 2199 y Fp(+)1092 2187 y Fr(g)1115 2194 y Fo(b)1132 2187 y Fn(g)p Fr(;)d Fn(f)p Fr(K)1249 2166 y Fo(a)1245 2199 y Fp(+)1275 2187 y Fr(f)1299 2194 y Fo(a)1331 2187 y Fq(+)j Fr(K)1425 2166 y Fo(b)1421 2199 y Fp(+)1451 2187 y Fr(f)1475 2194 y Fo(b)1492 2187 y Fn(g)1517 2147 y Fl(\013)1549 2187 y Fr(:)71 2305 y Fw(By)k(\(4.10\))i(and)g(\(2.3\))f (one)h(has)288 2422 y Fq(\()p Fr(K)q(~)-21 b(g)r(;)406 2409 y(~)399 2422 y(f)5 b Fq(\))12 b Fn(\000)e Fq(\()n Fr(~)-22 b(g)r(;)8 b(K)626 2409 y(~)619 2422 y(f)e Fq(\))14 b(=)f Fn(\000)p Fr(i)797 2382 y Fl(\012)821 2422 y Fn(f)p Fr(K)891 2402 y Fo(a)p Fm(\003)887 2435 y(\000)929 2422 y Fr(K)974 2402 y Fo(a)970 2435 y Fm(\000)1011 2422 y Fn(\000)e Fr(K)1106 2402 y Fo(a)p Fm(\003)1102 2435 y Fp(+)1144 2422 y Fr(K)1189 2402 y Fo(a)1185 2435 y Fp(+)1215 2422 y Fn(g)p Fr(g)1263 2429 y Fo(a)1284 2422 y Fr(;)d(f)1330 2429 y Fo(a)1351 2382 y Fl(\013)1383 2422 y Fn(\000)143 b Fw(\(4.21\))371 2501 y Fr(i)396 2461 y Fl(\012)420 2501 y Fn(f)p Fr(K)490 2480 y Fo(b)p Fm(\003)486 2513 y(\000)524 2501 y Fr(K)569 2480 y Fo(b)565 2513 y Fm(\000)606 2501 y Fn(\000)11 b Fr(K)701 2480 y Fo(b)p Fm(\003)697 2513 y Fp(+)736 2501 y Fr(K)781 2480 y Fo(b)777 2513 y Fp(+)807 2501 y Fn(g)p Fr(g)855 2508 y Fo(b)873 2501 y Fr(;)d(f)919 2508 y Fo(b)936 2461 y Fl(\013)970 2501 y Fq(+)j Fn(h)q Fr(E)s(g)1101 2508 y Fo(b)1118 2501 y Fr(;)d(f)1164 2508 y Fo(b)1181 2501 y Fn(i)k(\000)e(h)q Fr(E)s(g)1343 2508 y Fo(a)1364 2501 y Fr(;)e(f)1410 2508 y Fo(a)1431 2501 y Fn(i)g Fr(:)856 2626 y Fw(15)p eop %%Page: 16 16 16 15 bop 257 274 a Fw(Using)14 b(\(4.9\))h(w)o(e)e(obtain)i Fq(\()p Fr(K)q(~)-21 b(g)r(;)843 261 y(~)837 274 y(f)5 b Fq(\))14 b(=)g(\()n Fr(~)-22 b(g)r(;)8 b(K)1068 261 y(~)1062 274 y(f)d Fq(\))14 b Fw(for)e Fr(~)-22 b(g)s(;)1250 261 y(~)1244 274 y(f)19 b Fn(2)14 b Fw(dom)o Fq(\()p Fr(K)t Fq(\))g Fw(whic)o(h)f(pro)o(v)o(es)h(the)257 340 y(symmetry)f(of)k Fr(K)t Fw(.)331 406 y(Next)i(w)o(e)i(are)g(going)g (to)h(v)o(erify)d(that)i Fr(K)k Fw(is)c(self-adjoin)o(t.)35 b(Let)1569 393 y Fr(~)1562 406 y(f)27 b Fn(2)22 b Fw(dom)o Fq(\()p Fr(K)1824 388 y Fm(\003)1844 406 y Fq(\))p Fw(.)257 473 y(It)c(is)g(not)h(hard)g(to)f(see)g(that)h(in)f(this)g(case)g(one)h (has)g Fr(f)1284 480 y Fm(\006)1330 473 y Fn(2)f Fr(W)1434 455 y Fp(1)p Fo(;)p Fp(2)1481 473 y Fq(\()p Fv(R)1539 480 y Fm(\006)1565 473 y Fq(\))h Fw(and)f Fr(f)s(;)1760 453 y Fp(1)p 1753 461 32 2 v 1753 490 a Fo(m)1789 473 y Fr(g)1814 455 y Fm(0)1843 473 y Fn(2)257 539 y Fr(W)310 521 y Fp(1)p Fo(;)p Fp(2)357 539 y Fq(\([)p Fr(a;)8 b(b)p Fq(]\))p Fw(.)27 b(It)18 b(remains)g(to)h(sho)o(w)g(that)g(the)f(b)q (oundary)i(conditions)f(of)g(\(4.14\))g(are)257 605 y(satis\034ed.)32 b(T)l(o)21 b(this)e(end)h(w)o(e)f(note)h(that)g Fq(\()p Fr(K)t(g)r(;)8 b(f)d Fq(\))20 b(=)g(\()p Fr(g)r(;)8 b(K)1388 587 y Fm(\003)1408 605 y Fr(f)d Fq(\))p Fw(,)20 b Fr(g)i Fn(2)e Fw(dom)p Fq(\()p Fr(K)t Fq(\))p Fw(,)g(and)257 671 y(\(4.19\))d(imply)337 781 y Fq(0)d(=)g Fn(\000)p Fr(i)8 b Fn(h)p Fr(g)533 788 y Fm(\000)563 781 y Fq(\(0\))p Fr(;)g(f)671 788 y Fm(\000)701 781 y Fq(\(0\))p Fn(i)k Fq(+)f Fn(h)p Fr(E)s(g)924 788 y Fo(b)941 781 y Fr(;)d(f)987 788 y Fo(b)1005 781 y Fn(i)j(\000)g(h)p Fr(E)s(g)1166 788 y Fo(a)1187 781 y Fr(;)d(f)1233 788 y Fo(a)1254 781 y Fn(i)j Fq(+)g Fr(i)d Fn(h)q Fr(g)1401 788 y Fp(+)1431 781 y Fq(\(0\))p Fr(;)g(f)1539 788 y Fp(+)1568 781 y Fq(\(0\))p Fn(i)h Fr(:)80 b Fw(\(4.22\))257 890 y(Using)16 b(the)g(b)q(oundary)i(conditions)e(\(4.14\))h(w)o(e)f(\034nd)508 1000 y Fq(0)e(=)598 959 y Fl(\012)621 1000 y Fr(K)666 979 y Fo(a)662 1012 y Fm(\000)692 1000 y Fr(g)715 1007 y Fo(a)748 1000 y Fq(+)d Fr(K)842 979 y Fo(b)838 1012 y Fm(\000)868 1000 y Fr(g)891 1007 y Fo(b)908 1000 y Fr(;)d(f)954 1007 y Fm(\000)984 1000 y Fq(\(0\))1046 959 y Fl(\013)1078 1000 y Fq(+)636 b Fw(\(4.23\))591 1078 y Fr(i)8 b Fn(h)p Fr(E)s(g)697 1085 y Fo(b)715 1078 y Fr(;)g(f)761 1085 y Fo(b)778 1078 y Fn(i)j(\000)g Fr(i)d Fn(h)p Fr(E)s(g)964 1085 y Fo(a)985 1078 y Fr(;)g(f)1031 1085 y Fo(a)1052 1078 y Fn(i)k(\000)1132 1038 y Fl(\012)1156 1078 y Fr(K)1201 1058 y Fo(a)1197 1091 y Fp(+)1227 1078 y Fr(g)1250 1085 y Fo(a)1282 1078 y Fq(+)f Fr(K)1376 1058 y Fo(b)1372 1091 y Fp(+)1402 1078 y Fr(g)1425 1085 y Fo(b)1443 1078 y Fr(;)d(f)1489 1085 y Fp(+)1518 1078 y Fq(\(0\))1580 1038 y Fl(\013)1612 1078 y Fr(:)257 1188 y Fw(Hence)15 b(w)o(e)h(get)626 1297 y Fq(0)e(=)716 1257 y Fl(\012)739 1297 y Fr(g)762 1304 y Fo(a)784 1297 y Fr(;)806 1257 y Fl(\010)835 1297 y Fr(K)880 1277 y Fo(a)p Fm(\003)876 1309 y(\000)918 1297 y Fr(f)942 1304 y Fm(\000)972 1297 y Fq(\(0\))d Fn(\000)g Fr(K)1140 1277 y Fo(a)p Fm(\003)1136 1309 y Fp(+)1178 1297 y Fr(f)1202 1304 y Fp(+)1232 1297 y Fq(\(0\))g Fn(\000)g Fr(iE)s(f)1435 1304 y Fo(a)1456 1257 y Fl(\011\013)1517 1297 y Fq(+)197 b Fw(\(4.24\))709 1335 y Fl(\012)732 1376 y Fr(g)755 1383 y Fo(b)773 1376 y Fr(;)795 1335 y Fl(\010)824 1376 y Fr(K)869 1355 y Fo(b)p Fm(\003)865 1388 y(\000)904 1376 y Fr(f)928 1383 y Fm(\000)957 1376 y Fq(\(0\))12 b Fn(\000)f Fr(K)1126 1355 y Fo(b)p Fm(\003)1122 1388 y Fp(+)1160 1376 y Fr(f)1184 1383 y Fp(+)1214 1376 y Fq(\(0\))g(+)g Fr(iE)s(f)1416 1383 y Fo(b)1433 1335 y Fl(\011\013)1494 1376 y Fr(:)257 1485 y Fw(F)l(rom)k(\(4.24\))i(w)o(e)f(obtain)711 1595 y Fr(K)756 1574 y Fo(a)p Fm(\003)752 1607 y(\000)795 1595 y Fr(f)819 1602 y Fm(\000)848 1595 y Fq(\(0\))c Fn(\000)f Fr(K)1017 1574 y Fo(a)p Fm(\003)1013 1607 y Fp(+)1055 1595 y Fr(f)1079 1602 y Fp(+)1109 1595 y Fq(\(0\))g Fn(\000)g Fr(iE)s(f)1312 1602 y Fo(a)1346 1595 y Fq(=)j(0)330 b Fw(\(4.25\))257 1704 y(and)717 1770 y Fr(K)762 1750 y Fo(b)p Fm(\003)758 1783 y(\000)797 1770 y Fr(f)821 1777 y Fm(\000)851 1770 y Fq(\(0\))11 b Fn(\000)g Fr(K)1019 1750 y Fo(b)p Fm(\003)1015 1783 y Fp(+)1054 1770 y Fr(f)1078 1777 y Fp(+)1107 1770 y Fq(\(0\))h(+)f Fr(iE)s(f)1310 1777 y Fo(b)1340 1770 y Fq(=)j(0)336 b Fw(\(4.26\))257 1861 y(F)l(rom)15 b(\(4.25\))i(and)g(\(4.26\))g(w)o(e)f(deduce)728 1970 y Fr(f)752 1977 y Fo(a)787 1970 y Fq(=)d Fr(iE)s(K)939 1950 y Fo(a)p Fm(\003)935 1982 y(\000)978 1970 y Fr(f)1002 1977 y Fm(\000)1031 1970 y Fq(\(0\))f Fn(\000)e Fr(iE)s(K)1255 1950 y Fo(b)p Fm(\003)1251 1982 y Fp(+)1290 1970 y Fr(f)1314 1977 y Fp(+)1343 1970 y Fq(\(0\))347 b Fw(\(4.27\))257 2080 y(and)713 2146 y Fr(f)737 2153 y Fo(b)768 2146 y Fq(=)13 b Fn(\000)p Fr(iE)s(K)959 2125 y Fo(b)p Fm(\003)955 2158 y(\000)994 2146 y Fr(f)1018 2153 y Fm(\000)1047 2146 y Fq(\(0\))f(+)f Fr(iE)s(K)1271 2125 y Fo(b)p Fm(\003)1267 2158 y Fp(+)1305 2146 y Fr(f)1329 2153 y Fp(+)1359 2146 y Fq(\(0\))331 b Fw(\(4.28\))257 2236 y(No)o(w)22 b(b)o(y)f(a)h (straigh)o(tforw)o(ard)g(computation)e(one)i(v)o(eri\034es)e(that)i Fr(K)1549 2218 y Fo(a)1545 2249 y Fm(\000)1575 2236 y Fr(f)1599 2243 y Fo(a)1643 2236 y Fq(=)g Fr(K)1748 2218 y Fo(b)1744 2249 y Fm(\000)1774 2236 y Fr(f)1798 2243 y Fo(b)1838 2236 y Fq(=)257 2302 y Fr(f)281 2309 y Fm(\000)311 2302 y Fq(\(0\))g Fw(and)g Fr(K)540 2284 y Fo(a)536 2315 y Fp(+)566 2302 y Fr(f)590 2309 y Fo(a)626 2302 y Fq(+)14 b Fr(K)723 2284 y Fo(b)719 2315 y Fp(+)749 2302 y Fr(f)773 2309 y Fo(b)813 2302 y Fq(=)23 b Fr(f)898 2309 y Fp(+)928 2302 y Fq(\(0\))f Fw(whic)o(h)f(sho)o(ws)h(that)g Fr(f)28 b Fn(2)23 b Fw(dom)o Fq(\()p Fr(K)t Fq(\))p Fw(.)38 b(Hence)257 2369 y Fr(K)302 2351 y Fm(\003)336 2369 y Fq(=)14 b Fr(K)t Fw(.)1385 b Fn(4)331 2435 y Fw(In)18 b(the)g(picture)g(represen)o (tation)g(the)h(b)q(oundary)h(conditions)e(can)h(b)q(e)g(expressed)257 2501 y(as)e(follo)o(ws:)1042 2626 y(16)p eop %%Page: 17 17 17 16 bop 179 413 416 4 v 594 419 a Fq(\))179 336 y Fr(\013)210 343 y Fo(b)227 336 y Fr(g)252 318 y Fo(b)250 349 y Fm(\000)280 336 y Fq(\(0\))14 b(=)450 317 y Fp(1)p 413 325 92 2 v 413 353 a(2)p Fo(m)p Fp(\()p Fo(b)p Fp(\))510 336 y Fr(g)535 318 y Fm(0)546 336 y Fq(\()p Fr(b)p Fq(\))d Fn(\000)g Fr(\024)694 343 y Fo(b)711 336 y Fr(g)r Fq(\()p Fr(b)p Fq(\))262 473 y Fn(\000)p Fr(i)333 454 y Fo(d)p 323 462 39 2 v 323 490 a(dx)366 473 y Fr(g)391 455 y Fo(b)389 486 y Fm(\000)p 1051 413 416 4 v 1043 419 a Fw(\()968 317 y Fp(1)p 931 325 92 2 v 931 353 a(2)p Fo(m)p Fp(\()p Fo(b)p Fp(\))1028 336 y Fr(g)1053 318 y Fm(0)1065 336 y Fq(\()p Fr(b)p Fq(\))f Fn(\000)p 1184 309 46 2 v 11 w Fr(\024)1212 343 y Fo(b)1229 336 y Fr(g)r Fq(\()p Fr(b)p Fq(\))k(=)g Fr(\013)1410 343 y Fo(b)1427 336 y Fr(g)1452 318 y Fo(b)1450 349 y Fp(+)1480 336 y Fq(\(0\))1259 473 y Fn(\000)p Fr(i)1329 454 y Fo(d)p 1320 462 39 2 v 1320 490 a(dx)1362 473 y Fr(g)1387 455 y Fo(b)1385 486 y Fp(+)p 179 662 416 4 v 594 668 a Fq(\))159 751 y Fr(\013)190 758 y Fo(a)210 751 y Fr(g)235 733 y Fo(a)233 764 y Fm(\000)263 751 y Fq(\(0\))g(=)435 732 y Fp(1)p 396 740 96 2 v 396 769 a(2)p Fo(m)p Fp(\()p Fo(a)p 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Fo(y)q Fp(\))p Fo(z)522 1479 y Fr(f)546 1486 y Fm(\000)575 1479 y Fq(\()p Fr(y)r Fq(\))25 b Fn(\010)g Fq(\()p Fr(H)15 b Fn(\000)c Fr(z)r Fq(\))896 1458 y Fm(\000)p Fp(1)943 1479 y Fr(f)16 b Fq(+)11 b Fr(iT)1078 1486 y Fm(\003)1097 1479 y Fq(\()p 1116 1451 25 2 v Fr(z)r Fq(\))1160 1458 y Fm(\003)1188 1411 y Fl(Z)1238 1424 y Fp(0)1216 1524 y Fm(\0001)1289 1479 y Fr(dy)k(e)1376 1458 y Fm(\000)p Fo(iy)q(z)1454 1479 y Fr(f)1478 1486 y Fm(\000)1508 1479 y Fq(\()p Fr(y)r Fq(\))24 b Fn(\010)184 1620 y Fr(i)209 1552 y Fl(Z)258 1565 y Fo(x)236 1665 y Fp(0)289 1620 y Fr(dy)15 b(e)376 1599 y Fo(i)p Fp(\()p Fo(x)p Fm(\000)p Fo(y)q Fp(\))p Fo(z)501 1620 y Fr(f)525 1627 y Fp(+)555 1620 y Fq(\()p Fr(y)r Fq(\))10 b(+)h Fr(ie)718 1599 y Fo(iz)q(x)769 1620 y Fr(T)c Fq(\()p Fr(z)r Fq(\))p Fr(f)16 b Fq(+)11 b Fr(i)p Fq(\002)1012 1627 y Fo(H)1045 1620 y Fq(\()p 1064 1592 V Fr(z)r Fq(\))1108 1599 y Fm(\003)1136 1552 y Fl(Z)1186 1565 y Fp(0)1164 1665 y Fm(\0001)1237 1620 y Fr(dy)k(e)1324 1599 y Fo(i)p Fp(\()p Fo(x)p Fm(\000)p Fo(y)q Fp(\))p Fo(z)1449 1620 y Fr(f)1473 1627 y Fm(\000)1503 1620 y Fq(\()p Fr(y)r Fq(\))71 1757 y Fj(for)h Fn(=)p Fw(m)o Fq(\()p Fr(z)r Fq(\))e Fr(>)f Fq(0)18 b Fj(and)218 1870 y Fq(\()p Fr(K)d Fn(\000)c Fr(z)r Fq(\))387 1850 y Fm(\000)p Fp(1)442 1870 y Fq(\()p Fr(f)485 1877 y Fm(\000)526 1870 y Fn(\010)g Fr(f)16 b Fn(\010)11 b Fr(f)690 1877 y Fp(+)719 1870 y Fq(\))j(=)775 b Fw(\(4.30\))201 1975 y Fn(\000)p Fr(i)265 1907 y Fl(Z)315 1921 y Fp(0)293 2020 y Fo(x)343 1975 y Fr(dy)16 b(e)431 1955 y Fo(i)p Fp(\()p Fo(x)p Fm(\000)p Fo(y)q Fp(\))p Fo(z)555 1975 y Fr(f)579 1982 y Fm(\000)609 1975 y Fq(\()p Fr(y)r Fq(\))11 b Fn(\000)f Fr(ie)773 1955 y Fo(iz)q(x)824 1975 y Fr(T)853 1982 y Fm(\003)873 1975 y Fq(\()p Fr(z)r Fq(\))h Fn(\000)f Fr(i)p Fq(\002)1051 1982 y Fo(H)1085 1975 y Fq(\()p Fr(z)r Fq(\))1156 1907 y Fl(Z)1206 1921 y Fm(1)1183 2020 y Fp(0)1251 1975 y Fr(dy)16 b(e)1339 1955 y Fo(i)p Fp(\()p Fo(x)p Fm(\000)p Fo(y)q Fp(\))p Fo(z)1464 1975 y Fr(f)1488 1982 y Fp(+)1517 1975 y Fq(\()p Fr(y)r Fq(\))11 b Fn(\010)201 2106 y Fq(\()p Fr(H)264 2085 y Fm(\003)295 2106 y Fn(\000)g Fr(z)r Fq(\))389 2085 y Fm(\000)p Fp(1)436 2106 y Fr(f)17 b Fn(\000)10 b Fr(iT)d Fq(\()p 598 2078 V Fr(z)q Fq(\))641 2085 y Fm(\003)669 2038 y Fl(Z)719 2051 y Fm(1)697 2151 y Fp(0)765 2106 y Fr(dy)15 b(e)852 2085 y Fm(\000)p Fo(iy)q(z)930 2106 y Fr(f)954 2113 y Fp(+)984 2106 y Fq(\()p Fr(y)r Fq(\))24 b Fn(\010)h(\000)p Fr(i)1200 2038 y Fl(Z)1249 2051 y Fm(1)1227 2151 y Fo(x)1295 2106 y Fr(dy)16 b(e)1383 2085 y Fo(i)p Fp(\()p Fo(x)p Fm(\000)p Fo(y)q Fp(\))p Fo(z)1507 2106 y Fr(f)1531 2113 y Fp(+)1561 2106 y Fq(\()p Fr(y)r Fq(\))71 2239 y Fj(for)g Fn(=)p Fw(m)o Fq(\()p Fr(z)r Fq(\))e Fr(<)f Fq(0)p Fj(.)71 2345 y(Pr)n(o)n(of:)20 b Fw(Let)c Fn(=)p Fw(m)n Fq(\()p Fr(z)r Fq(\))e Fr(>)g Fq(0)p Fw(.)22 b(W)l(e)15 b(set)339 2478 y Fr(g)362 2485 y Fm(\000)392 2478 y Fq(\()p Fr(x)p Fq(\))41 b(:=)g Fr(i)617 2410 y Fl(Z)666 2423 y Fo(x)644 2522 y Fm(\0001)717 2478 y Fr(dy)16 b(e)805 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2429 y Fn(\000)p Fr(\013)1003 2436 y Fo(b)1061 2409 y Fp(1)p 1025 2417 92 2 v 1025 2446 a(2)p Fo(m)p Fp(\()p Fo(b)p Fp(\))1121 2429 y Fr(v)1147 2410 y Fm(0)1145 2441 y Fo(a)1165 2429 y Fq(\()p Fr(b;)h(z)r Fq(\))41 b Fr(\013)1343 2436 y Fo(a)1364 2429 y Fr(\024)1392 2436 y Fo(b)983 2500 y Fn(\000)p Fr(\013)1053 2507 y Fo(b)1070 2500 y Fr(v)1094 2507 y Fo(a)1115 2500 y Fq(\()p Fr(b;)8 b(z)r Fq(\))114 b Fr(\013)1366 2507 y Fo(a)1430 2378 y Fl(!)1478 2464 y Fr(g)1501 2471 y Fm(\000)1531 2464 y Fq(\(0\))p Fr(:)145 b Fw(\(4.41\))1042 2626 y(18)p eop %%Page: 19 19 19 18 bop 71 274 a Fw(Since)15 b Fr(K)243 256 y Fo(a)239 286 y Fm(\000)269 274 y Fr(h)297 281 y Fo(a)329 274 y Fq(+)c Fr(K)423 256 y Fo(b)419 286 y Fm(\000)449 274 y Fr(h)477 281 y Fo(b)508 274 y Fq(=)j(0)i Fw(and)443 371 y Fq(1)p 397 393 116 2 v 397 438 a Fr(W)7 b Fq(\()p Fr(z)r Fq(\))518 404 y Fr(K)563 384 y Fo(a)559 417 y Fm(\000)597 319 y Fl( )657 369 y Fr(\013)688 376 y Fo(b)705 369 y Fr(\024)733 376 y Fo(a)796 369 y Fr(\013)827 376 y Fo(a)891 350 y Fp(1)p 853 358 96 2 v 853 386 a(2)p Fo(m)p Fp(\()p Fo(a)p Fp(\))953 369 y Fr(v)979 351 y Fm(0)977 382 y Fo(b)993 369 y Fq(\()p Fr(a;)h(z)r Fq(\))662 441 y Fn(\000)p Fr(\013)732 448 y Fo(b)848 441 y Fr(\013)879 448 y Fo(a)900 441 y Fr(v)924 448 y Fo(b)941 441 y Fq(\()p Fr(a;)g(z)r Fq(\))1124 319 y Fl(!)1172 404 y Fq(+)355 b Fw(\(4.42\))526 537 y Fq(1)p 480 559 116 2 v 480 605 a Fr(W)7 b Fq(\()p Fr(z)r Fq(\))601 570 y Fr(K)646 550 y Fo(b)642 583 y Fm(\000)680 485 y Fl( )740 535 y Fn(\000)p Fr(\013)810 542 y Fo(b)869 516 y Fp(1)p 832 524 92 2 v 832 552 a(2)p Fo(m)p Fp(\()p Fo(b)p Fp(\))928 535 y Fr(v)954 517 y Fm(0)952 548 y Fo(a)973 535 y Fq(\()p Fr(b;)h(z)r Fq(\))41 b Fr(\013)1151 542 y Fo(a)1172 535 y Fr(\024)1200 542 y Fo(b)791 607 y Fn(\000)p Fr(\013)861 614 y Fo(b)878 607 y Fr(v)902 614 y Fo(a)922 607 y Fq(\()p Fr(b;)8 b(z)r Fq(\))114 b Fr(\013)1173 614 y Fo(a)1238 485 y Fl(!)1291 570 y Fq(=)14 b Fr(I)71 701 y Fw(w)o(e)h(immedi)o (ately)e(\034nd)k(that)f Fr(K)668 683 y Fo(a)664 713 y Fm(\000)694 701 y Fr(g)717 708 y Fo(a)749 701 y Fq(+)11 b Fr(K)843 683 y Fo(b)839 713 y Fm(\000)870 701 y Fr(g)893 708 y Fo(b)924 701 y Fq(=)j Fr(g)999 708 y Fm(\000)1029 701 y Fq(\(0\))p Fw(.)21 b(W)l(e)16 b(note)g(that)546 791 y Fr(g)569 798 y Fp(+)599 791 y Fq(\(0\))e(=)g Fr(iT)7 b Fq(\()p Fr(z)r Fq(\))p Fr(f)15 b Fq(+)c(\002)969 798 y Fo(H)1003 791 y Fq(\()p 1022 763 25 2 v Fr(z)r Fq(\))1066 770 y Fm(\003)1085 791 y Fr(g)1108 798 y Fm(\000)1138 791 y Fq(\(0\))p Fr(:)351 b Fw(\(4.43\))71 880 y(Using)16 b(\(2.17\))h(w)o(e)e(compute)g(that)446 1007 y Fr(K)491 987 y Fo(a)487 1019 y Fp(+)517 1007 y Fr(h)545 1014 y Fo(a)580 1007 y Fq(=)686 973 y Fr(i)p 637 996 116 2 v 637 1041 a(W)7 b Fq(\()p Fr(z)r Fq(\))766 922 y Fl( )1028 973 y Fq(0)826 1043 y Fr(\013)857 1050 y Fo(a)886 1003 y Fl(R)919 1016 y Fo(b)910 1060 y(a)945 1043 y Fr(dy)16 b(v)1034 1050 y Fo(b)1050 1043 y Fq(\()p Fr(y)r(;)8 b(z)r Fq(\))p Fr(f)d Fq(\()p Fr(y)r Fq(\))1274 922 y Fl(!)1565 1007 y Fw(\(4.44\))71 1131 y(and)418 1222 y Fr(K)463 1202 y Fo(b)459 1235 y Fp(+)489 1222 y Fr(h)517 1229 y Fo(b)548 1222 y Fq(=)654 1189 y Fr(i)p 605 1211 V 605 1257 a(W)i Fq(\()p Fr(z)r Fq(\))734 1137 y Fl( )794 1192 y Fn(\000)p Fr(\013)864 1199 y Fo(b)889 1152 y Fl(R)922 1165 y Fo(b)913 1209 y(a)948 1192 y Fr(dy)16 b(v)1037 1199 y Fo(a)1057 1192 y Fq(\()p Fr(y)r(;)8 b(z)r Fq(\))p Fr(f)d Fq(\()p Fr(y)r Fq(\))1015 1258 y(0)1281 1137 y Fl(!)1329 1222 y Fr(:)222 b Fw(\(4.45\))71 1342 y(By)15 b(de\034nition)h(\(3.3\))g(w)o(e)g(\034nally)g(obtain)622 1431 y Fr(K)667 1411 y Fo(a)663 1444 y Fp(+)693 1431 y Fr(h)721 1438 y Fo(a)753 1431 y Fq(+)11 b Fr(K)847 1411 y Fo(b)843 1444 y Fp(+)873 1431 y Fr(h)901 1438 y Fo(b)932 1431 y Fq(=)i Fr(iT)7 b Fq(\()p Fr(z)r Fq(\))p Fr(f)s(:)425 b Fw(\(4.46\))71 1521 y(F)l(urthermore,)14 b(from)h(\(3.29\),)h(\(4.40\))h(and)g(\(4.41\))g(w)o(e)e(\034nd)i(that) 252 1614 y Fq(1)p 207 1636 V 207 1682 a Fr(W)7 b Fq(\()p Fr(z)r Fq(\))327 1648 y Fr(K)372 1627 y Fo(a)368 1660 y Fp(+)406 1563 y Fl( )467 1613 y Fr(\013)498 1620 y Fo(b)515 1613 y Fr(\024)543 1620 y Fo(a)605 1613 y Fr(\013)636 1620 y Fo(a)701 1593 y Fp(1)p 662 1601 96 2 v 662 1630 a(2)p Fo(m)p Fp(\()p Fo(a)p Fp(\))762 1613 y Fr(v)788 1595 y Fm(0)786 1626 y Fo(b)803 1613 y Fq(\()p Fr(a;)h(z)r Fq(\))472 1684 y Fn(\000)p Fr(\013)542 1691 y Fo(b)658 1684 y Fr(\013)689 1691 y Fo(a)710 1684 y Fr(v)734 1691 y Fo(b)750 1684 y Fq(\()p Fr(a;)g(z)r Fq(\))934 1563 y Fl(!)982 1648 y Fr(g)1005 1655 y Fm(\000)1034 1648 y Fq(\(0\)+)431 b Fw(\(4.47\))335 1780 y Fq(1)p 290 1802 116 2 v 290 1848 a Fr(W)7 b Fq(\()p Fr(z)r Fq(\))410 1814 y Fr(K)455 1793 y Fo(b)451 1826 y Fp(+)489 1729 y Fl( )550 1779 y Fn(\000)p Fr(\013)620 1786 y Fo(b)678 1759 y Fp(1)p 642 1767 92 2 v 642 1796 a(2)p Fo(m)p Fp(\()p Fo(b)p Fp(\))738 1779 y Fr(v)764 1761 y Fm(0)762 1791 y Fo(a)782 1779 y Fq(\()p Fr(b;)h(z)r Fq(\))41 b Fr(\013)960 1786 y Fo(a)981 1779 y Fr(\024)1009 1786 y Fo(b)600 1850 y Fn(\000)p Fr(\013)670 1857 y Fo(b)687 1850 y Fr(v)711 1857 y Fo(a)732 1850 y Fq(\()p Fr(b;)8 b(z)r Fq(\))114 b Fr(\013)983 1857 y Fo(a)1047 1729 y Fl(!)1095 1814 y Fr(g)1118 1821 y Fm(\000)1148 1814 y Fq(\(0\))14 b(=)g(\002)1314 1821 y Fo(H)1347 1814 y Fq(\()p 1366 1786 25 2 v Fr(z)r Fq(\))1410 1793 y Fm(\003)1430 1814 y Fr(g)1453 1821 y Fm(\000)1483 1814 y Fq(\(0\))p Fr(:)71 1945 y Fw(By)22 b(\(4.43\),)k(\(4.46\))e(and)h(\(4.47\))f(w)o(e)f(\034nally)g(get)h Fr(K)1085 1926 y Fo(a)1081 1957 y Fp(+)1111 1945 y Fr(g)1134 1952 y Fo(a)1171 1945 y Fq(+)16 b Fr(K)1270 1926 y Fo(b)1266 1957 y Fp(+)1296 1945 y Fr(g)1319 1952 y Fo(b)1363 1945 y Fq(=)26 b Fr(g)1450 1952 y Fp(+)1480 1945 y Fq(\(0\))e Fw(whic)o(h)71 2011 y(completes)14 b(the)i(pro)q(of.)1141 b Fn(4)144 2077 y Fw(F)l(rom)15 b(\(4.29\))i(w)o(e)e(obtain)403 2166 y Fr(P)441 2146 y Fe(K)434 2179 y(H)466 2166 y Fq(\()p Fr(K)g Fn(\000)c Fr(z)r Fq(\))635 2146 y Fm(\000)p Fp(1)682 2166 y Fq(\(0)p Fr(;)d(f)s(;)g Fq(0\))14 b(=)g(\()p Fr(H)h Fn(\000)c Fr(z)r Fq(\))1073 2146 y Fm(\000)p Fp(1)1120 2166 y Fr(f)s(;)56 b(f)20 b Fn(2)14 b Fk(H)p Fr(;)207 b Fw(\(4.48\))71 2256 y(for)16 b Fr(z)g Fn(2)e Fv(C)264 2263 y Fp(+)313 2256 y Fw(and)393 2345 y Fr(P)431 2325 y Fe(K)424 2358 y(H)457 2345 y Fq(\()p Fr(K)h Fn(\000)10 b Fr(z)r Fq(\))625 2325 y Fm(\000)p Fp(1)672 2345 y Fq(\(0)p Fr(;)e(f)s(;)g Fq(0\))14 b(=)g(\()p Fr(H)958 2325 y Fm(\003)989 2345 y Fn(\000)d Fr(z)r Fq(\))1083 2325 y Fm(\000)p Fp(1)1130 2345 y Fr(f)s(;)56 b(f)19 b Fn(2)14 b Fk(H)p Fr(;)198 b Fw(\(4.49\))71 2435 y(for)20 b Fr(z)j Fn(2)f Fv(C)283 2442 y Fm(\000)315 2435 y Fw(.)34 b(Hence)19 b(the)i(op)q(erator)g Fr(K)k Fw(is)20 b(indeed)g(a)h(self-adjoin)o(t)f(dilation)g(of)h(the)71 2501 y(maximal)13 b(dissipativ)o(e)i(op)q(erator)i Fr(H)t Fw(.)22 b(It)15 b(can)i(b)q(e)f(sho)o(wn)h(that)g Fr(K)j Fw(is)c(minim)o(al.)856 2626 y(19)p eop %%Page: 20 20 20 19 bop 257 274 a Fu(5)77 b(Eigenfunction)25 b(expansion)257 392 y Fw(The)c(self-adjoin)o(t)g(op)q(erator)h Fr(K)j Fw(is)c(absolutely)f(con)o(tin)o(uous)h(and)h(its)e(sp)q(ectrum)g(co-) 257 458 y(incides)g(with)g(the)h(real)f(axis,)h(i.e.)33 b Fr(\033)r Fq(\()p Fr(K)t Fq(\))21 b(=)h Fv(R)p Fw(.)31 b(Its)21 b(m)o(ultipli)o(ci)o(t)o(y)c(is)k(t)o(w)o(o.)34 b(Let)21 b(us)257 524 y(compute)15 b(the)h(generalized)f (eigenfunctions)1112 511 y Fr(~)1110 524 y(\036)p Fq(\()p Fn(\001)p Fr(;)8 b(\025)p Fq(\))p Fw(,)15 b Fr(\025)g Fn(2)f Fv(R)p Fw(,)e(of)17 b Fr(K)t Fw(.)k(W)l(e)16 b(set)656 630 y Fr(~)654 643 y(\036)p Fq(\()p Fr(x;)8 b(\025)p Fq(\))14 b(=)f Fr(\036)893 650 y Fm(\000)923 643 y Fq(\()p Fr(x;)8 b(\025)p Fq(\))j Fn(\010)g Fr(\036)p Fq(\()p Fr(x;)d(\025)p Fq(\))j Fn(\010)f Fr(\036)1334 650 y Fp(+)1364 643 y Fq(\()p Fr(x;)e(\025)p Fq(\))296 b Fw(\(5.1\))257 762 y(for)17 b Fr(x)c Fn(2)426 750 y Fq(^)420 762 y(\012)k Fw(where)655 900 y Fr(\036)684 907 y Fm(\000)713 900 y Fq(\()p Fr(x;)8 b(\025)p Fq(\))14 b(=)895 830 y Fl(\022)952 868 y Fr(\036)981 850 y Fo(b)981 880 y Fm(\000)1010 868 y Fq(\()p Fr(x;)8 b(\025)p Fq(\))952 934 y Fr(\036)981 916 y Fo(a)981 946 y Fm(\000)1010 934 y Fq(\()p Fr(x;)g(\025)p Fq(\))1147 830 y Fl(\023)1192 900 y Fr(;)105 b(x)14 b Fn(2)g Fv(R)1439 907 y Fm(\000)1466 900 y Fr(;)296 b Fw(\(5.2\))257 1045 y(and)648 1131 y Fr(\036)677 1138 y Fp(+)706 1131 y Fq(\()p Fr(x;)8 b(\025)p Fq(\))14 b(=)888 1061 y Fl(\022)952 1099 y Fr(\036)981 1081 y Fo(b)981 1111 y Fp(+)1010 1099 y Fq(\()p Fr(x;)8 b(\025)p Fq(\))945 1165 y Fr(\036)974 1147 y Fo(a)974 1177 y Fp(+)1004 1165 y Fq(\()p Fr(x;)g(\025)p Fq(\))p Fr(;)1154 1061 y Fl(\023)1199 1131 y Fr(;)105 b(x)14 b Fn(2)g Fv(R)1446 1138 y Fp(+)1472 1131 y Fr(:)290 b Fw(\(5.3\))257 1254 y(F)l(rom)15 b(the)h(equation)440 1389 y Fq(\()p Fr(K)506 1376 y(~)504 1389 y(\036)p Fq(\)\()p Fr(x;)8 b(\025)p Fq(\))41 b(=)g Fn(\000)p Fr(i)863 1355 y(d)p 849 1377 54 2 v 849 1423 a(dx)907 1389 y(\036)936 1396 y Fm(\000)965 1389 y Fq(\()p Fr(x;)8 b(\025)p Fq(\))j Fn(\010)g Fr(l)q Fq(\()p Fr(\036)p Fq(\()p Fr(x;)d(\025)p Fq(\)\))i Fn(\010)h(\000)p Fr(i)1476 1355 y(d)p 1462 1377 V 1462 1423 a(dx)1520 1389 y(\036)1549 1396 y Fp(+)1578 1389 y Fq(\()p Fr(x;)d(\025)p Fq(\))82 b Fw(\(5.4\))709 1477 y Fq(=)41 b Fr(\025)8 b Fq(\()q Fr(\036)873 1484 y Fm(\000)903 1477 y Fq(\()p Fr(x;)g(\025)p Fq(\))j Fn(\010)f Fr(\036)p Fq(\()p Fr(x;)e(\025)p Fq(\))j Fn(\010)g Fr(\036)1314 1484 y Fp(+)1343 1477 y Fq(\()p Fr(x;)d(\025)p Fq(\)\))h Fr(;)257 1596 y(x)14 b Fn(2)351 1584 y Fq(^)346 1596 y(\012)p Fw(,)i(w)o(e)g(\034nd)g(the)g(equations)509 1742 y Fn(\000)p Fr(i)583 1708 y(d)p 570 1730 V 570 1776 a(dx)635 1671 y Fl(\022)693 1709 y Fr(\036)722 1691 y Fo(b)722 1722 y Fm(\000)751 1709 y Fq(\()p Fr(x;)8 b(\025)p Fq(\))693 1776 y Fr(\036)722 1757 y Fo(a)722 1788 y Fm(\000)751 1776 y Fq(\()p Fr(x;)g(\025)p Fq(\))888 1671 y Fl(\023)938 1742 y Fq(=)14 b Fr(\025)1026 1671 y Fl(\022)1091 1709 y Fr(\036)1120 1691 y Fo(b)1120 1722 y Fm(\000)1150 1709 y Fq(\()p Fr(x;)8 b(\025)p Fq(\))1084 1776 y Fr(\036)1113 1757 y Fo(a)1113 1788 y Fm(\000)1143 1776 y Fq(\()p Fr(x;)g(\025)p Fq(\))p Fr(;)1293 1671 y Fl(\023)1338 1742 y Fr(;)105 b(x)14 b Fn(2)g Fv(R)1585 1749 y Fm(\000)1611 1742 y Fr(;)151 b Fw(\(5.5\))720 1893 y Fr(l)q Fq(\()p Fr(\036)p Fq(\()p Fr(x;)8 b(\025)p Fq(\)\))13 b(=)h Fr(\025\036)p Fq(\()p Fr(x;)8 b(\025)p Fq(\))p Fr(;)105 b(x)14 b Fn(2)g Fq(\012)p Fr(;)362 b Fw(\(5.6\))257 1990 y(and)509 2076 y Fn(\000)p Fr(i)583 2042 y(d)p 570 2064 V 570 2110 a(dx)635 2005 y Fl(\022)693 2043 y Fr(\036)722 2025 y Fo(b)722 2056 y Fp(+)751 2043 y Fq(\()p Fr(x;)8 b(\025)p Fq(\))693 2110 y Fr(\036)722 2091 y Fo(a)722 2122 y Fp(+)751 2110 y Fq(\()p Fr(x;)g(\025)p Fq(\))888 2005 y Fl(\023)938 2076 y Fq(=)14 b Fr(\025)1026 2005 y Fl(\022)1091 2043 y Fr(\036)1120 2025 y Fo(b)1120 2056 y Fp(+)1150 2043 y Fq(\()p Fr(x;)8 b(\025)p Fq(\))1084 2110 y Fr(\036)1113 2091 y Fo(a)1113 2122 y Fp(+)1143 2110 y Fq(\()p Fr(x;)g(\025)p Fq(\))p Fr(;)1293 2005 y Fl(\023)1338 2076 y Fr(;)105 b(x)14 b Fn(2)g Fv(R)1585 2083 y Fp(+)1611 2076 y Fr(:)151 b Fw(\(5.7\))257 2202 y(The)17 b(equations)f(\(5.5\))g(-)h(\(5.7\))f (ha)o(v)o(e)g(the)g(solutions)671 2350 y Fr(\036)700 2357 y Fm(\000)729 2350 y Fq(\()p Fr(x;)8 b(\025)p Fq(\))14 b(=)911 2279 y Fl(\022)968 2317 y Fr(C)1007 2299 y Fo(b)1003 2330 y Fm(\000)968 2384 y Fr(C)1007 2366 y Fo(a)1003 2396 y Fm(\000)1054 2279 y Fl(\023)1099 2350 y Fr(e)1122 2329 y Fo(ix\025)1176 2350 y Fr(;)105 b(x)14 b Fn(2)g Fv(R)1423 2357 y Fm(\000)1449 2350 y Fr(;)313 b Fw(\(5.8\))591 2501 y Fr(\036)p Fq(\()p Fr(x;)8 b(\025)p Fq(\))13 b(=)h Fr(C)836 2508 y Fo(a)857 2501 y Fr(v)881 2508 y Fo(a)901 2501 y Fq(\()p Fr(x;)8 b(\025)p Fq(\))j(+)g Fr(C)1112 2508 y Fo(b)1129 2501 y Fr(v)1153 2508 y Fo(b)1170 2501 y Fq(\()p Fr(x;)d(\025)p Fq(\))p Fr(;)106 b(x)13 b Fn(2)h Fq(\012)p Fr(;)233 b Fw(\(5.9\))1042 2626 y(20)p eop %%Page: 21 21 21 20 bop 71 274 a Fw(and)484 357 y Fr(\036)513 364 y Fp(+)543 357 y Fq(\()p Fr(x;)8 b(\025)p Fq(\))13 b(=)724 287 y Fl(\022)782 324 y Fr(C)821 306 y Fo(b)817 337 y Fp(+)782 391 y Fr(C)821 373 y Fo(a)817 403 y Fp(+)867 287 y Fl(\023)912 357 y Fr(e)935 336 y Fo(ix\025)989 357 y Fr(;)105 b(x)14 b Fn(2)g Fv(R)1236 364 y Fp(+)1262 357 y Fr(:)289 b Fw(\(5.10\))71 476 y(The)13 b(eigenfunctions)f(ha)o(v) o(e)h(to)g(satisfy)g(the)g(b)q(oundary)i(conditions,)e(cf.)20 b(Theorem)12 b(4.1,)546 591 y Fr(K)591 570 y Fo(a)587 603 y Fm(\000)617 591 y Fr(\036)646 598 y Fo(a)667 591 y Fq(\()p Fr(\025)p Fq(\))g(+)f Fr(K)839 570 y Fo(b)835 603 y Fm(\000)865 591 y Fr(\036)894 598 y Fo(b)911 591 y Fq(\()p Fr(\025)p Fq(\))j(=)g Fr(\036)1072 598 y Fm(\000)1101 591 y Fq(\(0)p Fr(;)8 b(\025)p Fq(\))352 b Fw(\(5.11\))71 706 y(and)540 772 y Fr(K)585 752 y Fo(a)581 785 y Fp(+)611 772 y Fr(\036)640 779 y Fo(a)660 772 y Fq(\()p Fr(\025)p Fq(\))12 b(+)f Fr(K)832 752 y Fo(b)828 785 y Fp(+)858 772 y Fr(\036)887 779 y Fo(b)904 772 y Fq(\()p Fr(\025)p Fq(\))j(=)g Fr(\036)1065 779 y Fp(+)1094 772 y Fq(\(0)p Fr(;)8 b(\025)p Fq(\))p Fr(;)345 b Fw(\(5.12\))71 865 y(where)159 1006 y Fr(\036)188 1013 y Fo(a)209 1006 y Fq(\()p Fr(\025)p Fq(\))14 b(=)341 920 y Fl( )445 951 y Fp(1)p 406 959 96 2 v 406 988 a(2)p Fo(m)p Fp(\()p Fo(a)p Fp(\))506 971 y Fr(\036)535 952 y Fm(0)547 971 y Fq(\()p Fr(a;)8 b(\025)p Fq(\))459 1042 y Fr(\036)p Fq(\()p Fr(a;)g(\025)p Fq(\))681 920 y Fl(!)778 1006 y Fw(and)49 b Fr(\036)934 1013 y Fo(b)951 1006 y Fq(\()p Fr(\025)p Fq(\))14 b(=)1083 920 y Fl( )1185 951 y Fp(1)p 1148 959 92 2 v 1148 988 a(2)p Fo(m)p Fp(\()p Fo(b)p Fp(\))1245 971 y Fr(\036)1274 952 y Fm(0)1285 971 y Fq(\()p Fr(b;)8 b(\025)p Fq(\))1200 1042 y Fr(\036)p Fq(\()p Fr(b;)g(\025)p Fq(\))1415 920 y Fl(!)1463 1006 y Fr(:)88 b Fw(\(5.13\))71 1155 y(A)15 b(straigh)o(tforw)o(ard)i(computation)f (sho)o(ws)h(that)404 1271 y Fr(C)439 1278 y Fo(a)474 1271 y Fq(=)c Fn(\000)605 1237 y Fr(\013)636 1244 y Fo(b)p 569 1259 120 2 v 569 1305 a Fr(W)7 b Fq(\()p Fr(\025)p Fq(\))693 1271 y Fr(C)732 1250 y Fo(b)728 1283 y Fm(\000)855 1271 y Fw(and)98 b Fr(C)1066 1278 y Fo(b)1097 1271 y Fq(=)1188 1237 y Fr(\013)1219 1244 y Fo(a)p 1154 1259 V 1154 1305 a Fr(W)7 b Fq(\()p Fr(\025)p Fq(\))1278 1271 y Fr(C)1317 1250 y Fo(a)1313 1283 y Fm(\000)1343 1271 y Fr(:)208 b Fw(\(5.14\))71 1405 y(This)16 b(yields)189 1520 y Fr(\036)p Fq(\()p Fr(x;)8 b(\025)p Fq(\))14 b(=)g Fn(\000)479 1486 y Fr(\013)510 1493 y Fo(b)p 444 1509 V 444 1554 a Fr(W)7 b Fq(\()p Fr(\025)p Fq(\))568 1520 y Fr(C)607 1500 y Fo(b)603 1532 y Fm(\000)632 1520 y Fr(v)656 1527 y Fo(a)677 1520 y Fq(\()p Fr(x;)h(\025)p Fq(\))j(+)891 1486 y Fr(\013)922 1493 y Fo(a)p 858 1509 V 858 1554 a Fr(W)c Fq(\()p Fr(\025)p Fq(\))982 1520 y Fr(C)1021 1500 y Fo(a)1017 1532 y Fm(\000)1046 1520 y Fr(v)1070 1527 y Fo(b)1087 1520 y Fq(\()p Fr(x;)h(\025)p Fq(\))p Fr(;)105 b(x)14 b Fn(2)g Fq(\012)119 b Fw(\(5.15\))71 1657 y(where)18 b Fr(v)238 1664 y Fo(a)258 1657 y Fq(\()p Fr(x;)8 b(\025)p Fq(\))19 b Fw(and)g Fr(v)514 1664 y Fo(b)531 1657 y Fq(\()p Fr(x;)8 b(\025)p Fq(\))19 b Fw(the)f(elemen)o (tary)e(solutions)j(\(2.6\))g(and)g(\(2.7\).)29 b(Using)71 1724 y(the)16 b(adjoin)o(t)g(op)q(erator)h Fr(T)545 1731 y Fm(\003)565 1724 y Fq(\()p Fr(\025)p Fq(\))631 1705 y Fm(\003)651 1724 y Fw(,)e(cf.)21 b(\(3.8\),)16 b(w)o(e)g(\034nd)373 1839 y Fr(\036)p Fq(\()p Fr(x;)8 b(\025)p Fq(\))14 b(=)f(\()p Fr(T)631 1846 y Fm(\003)651 1839 y Fq(\()p Fr(\025)p Fq(\))717 1818 y Fm(\003)737 1839 y Fr(C)772 1846 y Fm(\000)801 1839 y Fq(\)\()p Fr(x)p Fq(\))p Fr(;)105 b(x)14 b Fn(2)g Fq(\012)p Fr(;)105 b(\025)15 b Fn(2)f Fv(R)p Fr(;)174 b Fw(\(5.16\))71 1954 y(where)719 2037 y Fr(C)754 2044 y Fm(\000)797 2037 y Fq(:=)862 1967 y Fl(\022)920 2005 y Fr(C)959 1987 y Fo(b)955 2017 y Fm(\000)920 2071 y Fr(C)959 2053 y Fo(a)955 2083 y Fm(\000)1005 1967 y Fl(\023)1565 2037 y Fw(\(5.17\))71 2159 y(Inserting)15 b(\(5.16\))i(in)o(to)f (\(5.12\))h(w)o(e)f(\034nd)575 2274 y Fr(C)610 2281 y Fp(+)653 2274 y Fq(=)e(\002)743 2281 y Fo(H)776 2274 y Fq(\()p Fr(\025)p Fq(\))842 2254 y Fm(\003)863 2274 y Fr(C)898 2281 y Fm(\000)927 2274 y Fr(;)105 b(\025)15 b Fn(2)f 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y Fm(0)1225 1306 y Fq(\))1244 1251 y Fl(E)1288 1306 y Fq(=)426 b Fw(\(5.23\))616 1409 y Fq(1)p 601 1431 V 601 1477 a(2)p Fr(\031)668 1375 y Fl(Z)718 1388 y Fp(+)p Fm(1)696 1488 y Fp(0)791 1443 y Fr(dx)14 b(e)881 1422 y Fo(ix)p Fp(\()p Fo(\025)p Fm(\000)p Fo(\025)996 1410 y Fi(0)1006 1422 y Fp(\))1030 1402 y Fl(\012)1054 1443 y Fq(\002)1092 1450 y Fo(H)1125 1443 y Fq(\()p Fr(\025)p Fq(\))1191 1422 y Fm(\003)1211 1443 y Fr(C)1246 1450 y Fm(\000)1276 1443 y Fr(;)8 b Fq(\002)1336 1450 y Fo(H)1369 1443 y Fq(\()p Fr(\025)1416 1422 y Fm(0)1428 1443 y Fq(\))1447 1422 y Fm(\003)1467 1443 y Fr(C)1506 1422 y Fm(0)1502 1455 y(\000)1531 1402 y Fl(\013)1569 1443 y Fq(=)601 1536 y(1)p 601 1559 25 2 v 601 1604 a(2)631 1570 y Fr(\016)r Fq(\()p Fr(\025)i Fn(\000)h Fr(\025)790 1549 y Fm(0)802 1570 y Fq(\))829 1530 y Fl(\012)853 1570 y Fq(\002)891 1577 y Fo(H)925 1570 y Fq(\()p Fr(\025)p Fq(\))991 1549 y Fm(\003)1011 1570 y Fr(C)1046 1577 y Fm(\000)1075 1570 y Fr(;)d Fq(\002)1135 1577 y Fo(H)1169 1570 y Fq(\()p Fr(\025)1216 1549 y Fm(0)1228 1570 y Fq(\))1247 1549 y Fm(\003)1266 1570 y Fr(C)1305 1549 y Fm(0)1301 1582 y(\000)1331 1530 y Fl(\013)1365 1570 y Fq(+)620 1653 y Fr(i)p 601 1676 54 2 v 601 1721 a Fq(2)p Fr(\031)660 1687 y Fn(P)756 1653 y Fq(1)p 704 1676 130 2 v 704 1721 a Fr(\025)j Fn(\000)g Fr(\025)821 1707 y Fm(0)846 1647 y Fl(\012)870 1687 y Fq(\002)908 1694 y Fo(H)942 1687 y Fq(\()p Fr(\025)p Fq(\))1008 1667 y Fm(\003)1028 1687 y Fr(C)1063 1694 y Fm(\000)1092 1687 y Fr(;)d Fq(\002)1152 1694 y Fo(H)1185 1687 y Fq(\()p Fr(\025)1232 1667 y Fm(0)1244 1687 y Fq(\))1263 1667 y Fm(\003)1283 1687 y Fr(C)1322 1667 y Fm(0)1318 1699 y(\000)1348 1647 y Fl(\013)257 1818 y Fw(where)16 b Fn(h)q(\001)p Fr(;)8 b Fn(\001i)16 b Fw(is)g(the)g(scalar)g(pro)q(duct)h(in)f Fv(C)1047 1800 y Fp(2)1070 1818 y Fw(.)21 b(Since)403 1927 y Fq(1)p 388 1949 54 2 v 388 1995 a(2)p Fr(\031)456 1893 y Fl(Z)505 1906 y Fo(b)483 2006 y(a)531 1961 y Fr(dx)600 1948 y(~)598 1961 y(\036)627 1940 y Fo(C)652 1944 y Fi(\000)680 1961 y Fq(\()p Fr(x;)8 b(\025)p Fq(\))p 804 1906 210 2 v 806 1948 a Fr(~)804 1961 y(\036)833 1942 y Fo(C)860 1930 y Fi(0)858 1952 y(\000)886 1961 y Fq(\()p Fr(x;)g(\025)983 1946 y Fm(0)995 1961 y Fq(\))14 b(=)686 b Fw(\(5.24\))486 2066 y Fq(1)p 471 2089 54 2 v 471 2134 a(2)p Fr(\031)539 2032 y Fl(Z)588 2045 y Fo(b)566 2145 y(a)628 2100 y Fr(dx)p Fq(\()p Fr(T)729 2107 y Fm(\003)748 2100 y Fq(\()p Fr(\025)p Fq(\))814 2080 y Fm(\003)834 2100 y Fr(C)869 2107 y Fm(\000)899 2100 y Fq(\)\()p Fr(x)p Fq(\))p 992 2057 296 2 v 8 w Fr(T)1021 2107 y Fm(\003)1040 2100 y Fq(\()p Fr(\025)1087 2086 y Fm(0)1099 2100 y Fq(\))1118 2086 y Fm(\003)1138 2100 y Fr(C)1177 2083 y Fm(0)1173 2111 y(\000)1202 2100 y Fq(\)\()p Fr(x)p Fq(\))13 b(=)490 2194 y Fr(i)p 471 2217 54 2 v 471 2262 a Fq(2)p Fr(\031)530 2228 y Fn(P)626 2194 y Fq(1)p 574 2217 130 2 v 574 2262 a Fr(\025)e Fn(\000)g Fr(\025)691 2248 y Fm(0)717 2188 y Fl(\012)740 2228 y Fr(C)775 2235 y Fm(\000)805 2228 y Fr(;)d(C)866 2207 y Fm(0)862 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2479 y Fq(\()p Fn(\001)p Fr(;)g(\025)1038 2458 y Fm(0)1050 2479 y Fq(\))1069 2408 y Fl(\023)1119 2479 y Fq(=)14 b Fr(\016)r Fq(\()p Fr(\025)d Fn(\000)g Fr(\025)1331 2458 y Fm(0)1343 2479 y Fq(\))1370 2438 y Fl(\012)1394 2479 y Fr(C)1429 2486 y Fm(\000)1458 2479 y Fr(;)d(C)1519 2458 y Fm(0)1515 2491 y(\000)1544 2438 y Fl(\013)1576 2479 y Fr(:)162 b Fw(\(5.25\))1042 2626 y(22)p eop %%Page: 23 23 23 22 bop 71 274 a Fw(Setting)367 353 y Fr(~)361 366 y( )395 346 y Fm(\000)424 366 y Fq(\()p Fn(\001)p Fr(;)8 b(\025;)g(b)p Fq(\))14 b(:=)689 333 y(1)p 653 355 96 2 v 653 365 a Fn(p)p 695 365 54 2 v 41 x Fq(2)p Fr(\031)756 353 y(~)753 366 y(\036)782 346 y Fo(e)798 352 y Fh(b)816 366 y Fq(\()p Fn(\001)p Fr(;)8 b(\025)p Fq(\))49 b Fw(where)g Fr(e)1164 373 y Fo(b)1194 366 y Fq(:=)1260 296 y Fl(\022)1317 334 y Fq(1)1317 400 y(0)1362 296 y Fl(\023)1565 366 y Fw(\(5.26\))71 480 y(and)362 550 y Fr(~)355 563 y( )389 542 y Fm(\000)418 563 y Fq(\()p Fn(\001)p Fr(;)8 b(\025;)g(a)p Fq(\))14 b(:=)688 529 y(1)p 652 551 96 2 v 652 561 a Fn(p)p 694 561 54 2 v 41 x Fq(2)p Fr(\031)755 550 y(~)752 563 y(\036)781 542 y Fo(e)797 546 y Fh(a)819 563 y Fq(\()p Fn(\001)p Fr(;)8 b(\025)p Fq(\))48 b Fw(where)h Fr(e)1166 570 y Fo(a)1200 563 y Fq(:=)1265 492 y Fl(\022)1323 530 y Fq(0)1323 597 y(1)1368 492 y Fl(\023)1565 563 y Fw(\(5.27\))71 676 y(one)16 b(gets)g(the)g(follo)o(wing)g(theorem:)71 772 y Fx(Theorem)h(5.1)24 b Fj(L)n(et)c Fr(V)31 b Fn(2)19 b Fr(L)635 754 y Fp(2)655 772 y Fq(\([)p Fr(a;)8 b(b)p Fq(]\))p Fj(,)19 b Fn(=)p Fw(m)o Fq(\()p Fr(V)11 b Fq(\))19 b(=)g(0)i Fj(and)g Fr(\024)1225 779 y Fo(a)1246 772 y Fr(;)8 b(\024)1296 779 y Fo(b)1332 772 y Fn(2)19 b Fv(C)1417 779 y Fp(+)1450 772 y Fj(.)31 b(Then)21 b(the)71 838 y(functions)15 b Fn(f)p Fr( )338 820 y Fm(\000)367 838 y Fq(\()p Fn(\001)p Fr(;)8 b(\025;)g(b)p Fq(\))p Fr(;)g( )568 820 y Fm(\000)597 838 y Fq(\()p Fn(\001)p Fr(;)g(\025;)g(a)p Fq(\))p Fn(g)772 845 y Fo(\025)p Fm(2)p Fg(R)855 838 y Fj(p)n(erform)13 b(a)i(c)n(omplete)g(orthonormal)f(system)71 905 y(of)j(gener)n(alize)n(d)h(eigenfunctions)i(of)d Fr(K)t Fj(,)h(i.e)105 957 y Fl(\020)141 999 y Fr(~)135 1012 y( )169 991 y Fm(\000)198 1012 y Fq(\()p Fn(\001)p Fr(;)8 b(\025;)g(\034)e Fq(\))p Fr(;)377 999 y(~)371 1012 y( )405 991 y Fm(\000)434 1012 y Fq(\()p Fn(\001)p Fr(;)i(\025)517 991 y Fm(0)529 1012 y Fr(;)g(\034)578 991 y Fm(0)589 1012 y Fq(\))608 957 y Fl(\021)651 1012 y Fq(=)14 b Fr(\016)r Fq(\()p Fr(\025)d Fn(\000)g Fr(\025)863 991 y Fm(0)875 1012 y Fq(\))p Fr(\016)916 1019 y Fo(\034)t(\034)956 1010 y Fi(0)968 1012 y Fr(;)57 b(\025;)8 b(\025)1117 991 y Fm(0)1144 1012 y Fn(2)14 b Fv(R)p Fr(;)54 b(\034)s(;)8 b(\034)1371 991 y Fm(0)1396 1012 y Fq(=)14 b Fr(a;)8 b(b;)34 b Fw(\(5.28\))71 1119 y Fj(and)17 b(their)h(line)n(ar)f(sp)n (an)g(is)h(dense)g(in)g Fk(K)p Fj(.)71 1215 y Fw(Since)i(the)h (eigenfunctions)g(of)h(the)f(system)f(b)q(eha)o(v)o(es)h(on)h Fv(R)1232 1222 y Fm(\000)1280 1215 y Fw(lik)o(e)d(free)i(w)o(a)o(v)o (es)f(one)71 1282 y(calls)15 b(it)h(the)g(complete)e(system)h(of)h (incoming)f(eigenfunctions.)20 b(Let)559 1410 y Fr(C)t Fq(\()p Fr(\025)p Fq(\))14 b(:=)743 1340 y Fl(\022)802 1378 y Fr(C)841 1360 y Fo(b)858 1378 y Fq(\()p Fr(\025)p Fq(\))800 1444 y Fr(C)839 1426 y Fo(a)860 1444 y Fq(\()p Fr(\025)p Fq(\))947 1340 y Fl(\023)992 1410 y Fr(;)56 b(\025)15 b Fn(2)f Fv(R)p Fr(:)360 b Fw(\(5.29\))71 1542 y(In)16 b(accordance)g(with)g(\(5.20\))h(w)o(e)e(set)169 1631 y Fr(~)167 1644 y(\036)196 1624 y Fo(C)r Fp(\()p Fo(\025)p Fp(\))273 1644 y Fq(\()p Fr(x;)8 b(\025)p Fq(\))14 b(:=)g Fr(C)t Fq(\()p Fr(\025)p Fq(\))p Fr(e)597 1624 y Fo(ix\025)662 1644 y Fn(\010)c Fq(\()p Fr(T)759 1651 y Fm(\003)779 1644 y Fq(\()p Fr(\025)p Fq(\))845 1624 y Fm(\003)865 1644 y Fr(C)t Fq(\()p Fr(\025)p Fq(\)\)\()p Fr(x)p Fq(\))g Fn(\010)h Fq(\002)1153 1651 y Fo(H)1187 1644 y Fq(\()p Fr(\025)p Fq(\))1253 1624 y Fm(\003)1273 1644 y Fr(C)t Fq(\()p Fr(\025)p Fq(\))p Fr(e)1401 1624 y Fo(ix\025)1455 1644 y Fr(;)96 b Fw(\(5.30\))71 1747 y Fr(x)16 b Fn(2)170 1734 y Fq(^)164 1747 y(\012)i Fw(and)g Fr(\025)f Fn(2)g Fv(R)p Fw(.)22 b(Ob)o(viously)l(,)17 b(the)g(functions)1024 1734 y Fr(~)1021 1747 y(\036)1050 1729 y Fo(C)r Fp(\()p Fo(\025)p Fp(\))1128 1747 y Fq(\()p Fn(\001)p Fr(;)8 b(\025)p Fq(\))17 b Fw(are)h(eigenfunctions)f(of)71 1813 y Fr(K)t Fw(,)e(i.e.)20 b Fr(K)276 1800 y(~)274 1813 y(\036)303 1795 y Fo(C)r Fp(\()p Fo(\025)p Fp(\))381 1813 y Fq(\()p Fn(\001)p Fr(;)8 b(\025)p Fq(\))14 b(=)g Fr(\025)579 1800 y(~)577 1813 y(\036)606 1795 y Fo(C)r Fp(\()p Fo(\025)p Fp(\))684 1813 y Fq(\()p Fn(\001)p Fr(;)8 b(\025)p Fq(\))p Fw(.)21 b(Moreo)o(v)o(er,)14 b(one)j(gets)f(that)200 1898 y Fq(1)p 164 1920 96 2 v 164 1930 a Fn(p)p 206 1930 54 2 v 41 x Fq(2)p Fr(\031)267 1919 y(~)265 1932 y(\036)294 1911 y Fo(C)r Fp(\()p Fo(\025)p Fp(\))371 1932 y Fq(\()p Fn(\001)p Fr(;)8 b(\025)p Fq(\))14 b(=)g Fr(C)578 1911 y Fo(b)594 1932 y Fq(\()p Fr(\025)p Fq(\))666 1919 y Fr(~)660 1932 y( )694 1911 y Fm(\000)724 1932 y Fq(\()p Fn(\001)p Fr(;)8 b(\025;)g(b)p Fq(\))i(+)i Fr(C)968 1911 y Fo(a)988 1932 y Fq(\()p Fr(\025)p Fq(\))1060 1919 y Fr(~)1054 1932 y( )1088 1911 y Fm(\000)1117 1932 y Fq(\()p Fn(\001)p Fr(;)c(\025;)g(a)p Fq(\))p Fr(;)56 b(\025)15 b Fn(2)f Fv(R)p Fr(;)85 b Fw(\(5.31\))71 2053 y(whic)o(h)15 b(yields)161 2105 y Fl(\022)238 2141 y Fq(1)p 202 2164 96 2 v 202 2174 a Fn(p)p 244 2174 54 2 v 41 x Fq(2)p Fr(\031)305 2162 y(~)303 2175 y(\036)332 2155 y Fo(C)r Fp(\()p Fo(\025)p Fp(\))409 2175 y Fq(\()p Fn(\001)p Fr(;)8 b(\025)p Fq(\))p Fr(;)573 2141 y Fq(1)p 538 2164 96 2 v 538 2174 a Fn(p)p 579 2174 54 2 v 579 2215 a Fq(2)p Fr(\031)641 2162 y(~)638 2175 y(\036)667 2155 y Fo(C)r Fp(\()p Fo(\025)729 2143 y Fi(0)740 2155 y Fp(\))756 2175 y Fq(\()p Fn(\001)p Fr(;)g(\025)839 2155 y Fm(0)851 2175 y Fq(\))870 2105 y Fl(\023)920 2175 y Fq(=)14 b Fr(\016)r Fq(\()p Fr(\025)d Fn(\000)g Fr(\025)1132 2155 y Fm(0)1144 2175 y Fq(\))d Fn(h)p Fr(C)t Fq(\()p Fr(\025)p Fq(\))p Fr(;)g(C)t Fq(\()p Fr(\025)1403 2155 y Fm(0)1415 2175 y Fq(\))p Fn(i)g Fr(:)90 b Fw(\(5.32\))71 2302 y(By)16 b(a)i(family)c(of)k(orthonormal)f(bases)h Fn(f)p Fr(e)853 2309 y Fp(1)872 2302 y Fq(\()p Fr(\025)p Fq(\))p Fr(;)8 b(e)983 2309 y Fp(2)1002 2302 y Fq(\()p Fr(\025)p Fq(\))p Fn(g)1093 2309 y Fo(\025)p Fm(2)p Fg(R)1181 2302 y Fw(in)17 b Fv(C)1272 2284 y Fp(2)1312 2302 y Fw(w)o(e)f(mean)g (that)i(the)71 2369 y(comp)q(onen)o(ts)11 b(of)h(the)g(v)o(ectors)g Fr(e)651 2376 y Fp(1)670 2369 y Fq(\()p Fr(\025)p Fq(\))h Fw(and)g Fr(e)863 2376 y Fp(2)882 2369 y Fq(\()p Fr(\025)p Fq(\))g Fw(are)f(Leb)q(esgue)h(measurable)e(functions)71 2435 y(suc)o(h)19 b(that)h Fn(h)p Fr(e)335 2442 y Fo(\034)356 2435 y Fq(\()p Fr(\025)p Fq(\))p Fr(;)8 b(e)467 2442 y Fo(\034)487 2433 y Fi(0)500 2435 y Fq(\()p Fr(\025)p Fq(\))p Fn(i)20 b Fq(=)f Fr(\016)684 2442 y Fo(\034)t(\034)724 2433 y Fi(0)756 2435 y Fw(for)g(a.e.)31 b Fr(\025)20 b Fn(2)f Fv(R)p Fw(.)28 b(Using)19 b(this)h(notion)g(one)f(gets)71 2501 y(follo)o(wing)c(corollary:)856 2626 y(23)p eop %%Page: 24 24 24 23 bop 257 274 a Fx(Corollary)20 b(5.2)k Fj(L)n(et)17 b Fr(V)25 b Fn(2)14 b Fr(L)819 256 y Fp(2)839 274 y Fq(\([)p Fr(a;)8 b(b)p Fq(]\))p Fj(,)16 b Fn(=)p Fw(m)o Fq(\()p Fr(V)11 b Fq(\))j(=)f(0)18 b Fj(and)g Fr(\024)1389 281 y Fo(a)1410 274 y Fr(;)8 b(\024)1460 281 y Fo(b)1491 274 y Fn(2)14 b Fv(C)1571 281 y Fp(+)1603 274 y Fj(.)23 b(If)257 340 y Fn(f)p Fr(e)305 347 y Fp(1)325 340 y Fq(\()p Fr(\025)p Fq(\))p Fr(;)8 b(e)436 347 y Fp(2)455 340 y Fq(\()p Fr(\025)p Fq(\))p Fn(g)546 347 y Fo(\025)p Fm(2)p Fg(R)635 340 y Fj(is)18 b(a)g(me)n(asur)n(able)g(family)g(of)g (orthonormal)f(b)n(ases)i(in)f(the)h(Hilb)n(ert)257 406 y(sp)n(ac)n(e)e Fv(C)417 388 y Fp(2)440 406 y Fj(,)g(then)i(the)f (system)f(of)g(eigenfunctions)k Fn(f)1231 393 y Fr(~)1225 406 y( )1259 388 y Fp(\(1\))1305 406 y Fq(\()p Fn(\001)p Fr(;)8 b(\025)p Fq(\))p Fr(;)1435 393 y(~)1429 406 y( )1463 388 y Fp(\(2\))1510 406 y Fq(\()p Fn(\001)p Fr(;)g(\025)p Fq(\))p Fn(g)1637 413 y Fo(\025)p Fm(2)p Fg(R)1707 406 y Fj(,)557 529 y Fr(~)551 542 y( )585 521 y Fp(\()p Fo(\034)t Fp(\))634 542 y Fq(\()p Fn(\001)p Fr(;)g(\025)p Fq(\))14 b(:=)856 508 y(1)p 820 530 96 2 v 820 540 a Fn(p)p 862 540 54 2 v 41 x Fq(2)p Fr(\031)923 529 y(~)920 542 y(\036)949 521 y Fo(e)965 525 y Fh(\034)985 521 y Fp(\()p Fo(\025)p Fp(\))1035 542 y Fq(\()p Fn(\001)p Fr(;)8 b(\025)p Fq(\))p Fr(;)58 b(\025)14 b Fn(2)g Fv(R)p Fr(;)55 b(\034)19 b Fq(=)14 b(1)p Fr(;)8 b Fq(2)p Fr(;)170 b Fw(\(5.33\))257 679 y Fj(p)n(erforms)15 b(a)i(c)n(omplete)g(orthonormal)f(system)h(of)f (gener)n(alize)n(d)i(eigenfunctions)h(of)d Fr(K)t Fj(.)257 792 y Fw(In)g(particular,)g(setting)615 912 y Fr(~)609 925 y( )643 904 y Fp(+)672 925 y Fq(\()p Fn(\001)p Fr(;)8 b(\025;)g(\034)e Fq(\))13 b(:=)942 891 y(1)p 907 913 96 2 v 907 923 a Fn(p)p 948 923 54 2 v 948 964 a Fq(2)p Fr(\031)1009 912 y(~)1007 925 y(\036)1036 904 y Fp(\002)1063 910 y Fh(H)1092 904 y Fp(\()p Fo(\025)p Fp(\))p Fo(e)1157 908 y Fh(\034)1178 925 y Fq(\()p Fn(\001)p Fr(;)8 b(\025)p Fq(\))p Fr(;)57 b(\034)19 b Fq(=)14 b Fr(a;)8 b(b;)226 b Fw(\(5.34\))257 1062 y(where)400 1168 y Fr(~)398 1182 y(\036)427 1161 y Fp(\002)454 1167 y Fh(H)483 1161 y Fp(\()p Fo(\025)p Fp(\))p Fo(C)557 1165 y Ff(+)584 1182 y Fq(\()p Fr(x;)8 b(\025)p Fq(\))14 b(=)f(\002)803 1189 y Fo(H)837 1182 y Fq(\()p Fr(\025)p Fq(\))p Fr(C)938 1189 y Fp(+)968 1182 y Fr(e)991 1161 y Fo(ix\025)1056 1182 y Fn(\010)e Fq(\()p Fr(T)c Fq(\()p Fr(\025)p Fq(\))1227 1161 y Fm(\003)1246 1182 y Fr(C)1281 1189 y Fp(+)1311 1182 y Fq(\)\()p Fr(x)p Fq(\))j Fn(\010)h Fr(C)1491 1189 y Fp(+)1520 1182 y Fr(e)1543 1161 y Fo(ix\025)1598 1182 y Fr(;)140 b Fw(\(5.35\))257 1301 y Fr(x)21 b Fn(2)366 1288 y Fq(^)361 1301 y(\012)p Fw(,)h Fr(\034)27 b Fq(=)21 b Fr(a;)8 b(b)p Fw(,)20 b Fr(C)677 1308 y Fp(+)728 1301 y Fn(2)i Fv(C)816 1283 y Fp(2)838 1301 y Fw(,)g(one)f(de\034nes)f(a)h (complete)d(orthonormal)j(system)e(of)257 1367 y(eigenfunctions)d(of)g Fr(K)t Fw(.)21 b(Since)16 b(the)f(eigenfunctions)h(b)q(eha)o(v)o(es)f (on)i Fv(R)1529 1374 y Fp(+)1572 1367 y Fw(lik)o(e)d(free)h(w)o(a)o(v)o (es)257 1433 y(one)h(calls)e(them)g(the)g(complete)f(system)h(of)h (outgoing)i(eigenfunctions.)j(Using)15 b(\(5.31\))257 1499 y(one)i(gets)628 1518 y Fl( )697 1558 y Fr(~)691 1571 y( )725 1553 y Fp(+)754 1571 y Fq(\()p Fn(\001)p Fr(;)8 b(\025;)g(b)p Fq(\))694 1626 y Fr(~)688 1640 y( )722 1621 y Fp(+)751 1640 y Fq(\()p Fn(\001)p Fr(;)g(\025;)g(a)p Fq(\))921 1518 y Fl(!)975 1603 y Fq(=)14 b(\002)1065 1583 y Fo(t)1065 1616 y(H)1098 1603 y Fq(\()p Fr(\025)p Fq(\))1172 1518 y Fl( )1242 1558 y Fr(~)1235 1571 y( )1269 1553 y Fm(\000)1298 1571 y Fq(\()p Fn(\001)p Fr(;)8 b(\025;)g(b)p Fq(\))1239 1626 y Fr(~)1233 1640 y( )1267 1621 y Fm(\000)1296 1640 y Fq(\()p Fn(\001)p Fr(;)g(\025;)g(a)p Fq(\))1466 1518 y Fl(!)1752 1603 y Fw(\(5.36\))257 1739 y(where)16 b Fq(\002)436 1721 y Fo(t)436 1751 y(H)469 1739 y Fq(\()p Fr(\025)p Fq(\))g Fw(is)g(the)f(transp)q(osed)i(matrix)d(of)i Fq(\002)1179 1746 y Fo(H)1212 1739 y Fq(\()p Fr(\025)p Fq(\))p Fw(,)g(i.e.,)e(the)h(matrix)f(where)h(lines)257 1805 y(and)i(columns)e(are)h(in)o(terc)o(hanged.)k(Since)c Fq(\002)1095 1787 y Fo(t)1095 1818 y(H)1128 1805 y Fq(\()p Fr(\025)p Fq(\))f(=)e(\002)1298 1812 y Fo(H)1332 1805 y Fq(\()p Fr(\025)p Fq(\))p Fw(,)j Fr(\025)e Fn(2)g Fv(R)p Fw(,)f(w)o(e)j(\034nd)617 1877 y Fl( )686 1916 y Fr(~)680 1930 y( )714 1911 y Fp(+)743 1930 y Fq(\()p Fn(\001)p Fr(;)8 b(\025;)g(b)p Fq(\))683 1985 y Fr(~)677 1998 y( )711 1980 y Fp(+)740 1998 y Fq(\()p Fn(\001)p Fr(;)g(\025;)g(a)p Fq(\))911 1877 y Fl(!)964 1962 y Fq(=)14 b(\002)1054 1969 y Fo(H)1087 1962 y Fq(\()p Fr(\025)p Fq(\))1161 1877 y Fl( )1231 1916 y Fr(~)1224 1930 y( )1258 1911 y Fm(\000)1287 1930 y Fq(\()p Fn(\001)p Fr(;)8 b(\025;)g(b)p Fq(\))1228 1985 y Fr(~)1222 1998 y( )1256 1980 y Fm(\000)1285 1998 y Fq(\()p Fn(\001)p Fr(;)g(\025;)g(a)p Fq(\))1455 1877 y Fl(!)1503 1962 y Fr(:)235 b Fw(\(5.37\))257 2115 y(Using)16 b(the)g(incoming)f(eigenfunctions)h(w)o(e)f(in)o(tro)q(duce) h(the)g(transformations)728 2261 y Fq(\(\010)782 2268 y Fm(\000)810 2261 y Fr(~)-22 b(g)r Fq(\)\)\()p Fr(\025)p Fq(\))15 b(=:)g(^)-26 b Fr(g)r Fq(\()p Fr(\025)p Fq(\))14 b(=)1178 2191 y Fl(\022)1239 2229 y Fq(^)-26 b Fr(g)1262 2210 y Fo(b)1280 2229 y Fq(\()p Fr(\025)p Fq(\))1237 2295 y(^)g Fr(g)1260 2277 y Fo(a)1281 2295 y Fq(\()p Fr(\025)p Fq(\))1369 2191 y Fl(\023)1752 2261 y Fw(\(5.38\))257 2406 y(where)577 2481 y Fq(^)g Fr(g)600 2460 y Fo(\034)622 2481 y Fq(\()p Fr(\025)p Fq(\))14 b(:=)767 2413 y Fl(Z)799 2518 y Fp(^)795 2526 y(\012)831 2481 y Fr(dx)906 2426 y Fl(\020)934 2481 y Fr(~)-22 b(g)r Fq(\()p Fr(x)p Fq(\))p Fr(;)1055 2468 y(~)1049 2481 y( )1083 2460 y Fm(\000)1111 2481 y Fq(\()p Fr(x;)8 b(\025;)g(\034)e Fq(\))1276 2426 y Fl(\021)1314 2481 y Fr(;)57 b(\034)19 b Fq(=)14 b Fr(a;)8 b(b:)192 b Fw(\(5.39\))1042 2626 y(24)p eop %%Page: 25 25 25 24 bop 71 274 a Fw(The)13 b(op)q(erator)i Fq(\010)397 281 y Fm(\000)440 274 y Fq(:)f Fk(K)g Fn(\000)-9 b(!)614 261 y Fq(^)609 274 y Fk(K)14 b Fq(=)g Fr(L)741 256 y Fp(2)761 274 y Fq(\()p Fv(R)p Fr(;)7 b Fv(C)871 256 y Fp(2)893 274 y Fq(\))14 b Fw(is)f(unitary)h(and)g(is)f(called)g(the)g (incoming)71 340 y(F)l(ourier)i(transformation.)21 b(The)15 b(in)o(v)o(erse)g(incoming)f(F)l(ourier)h(transformation)h Fq(\010)1594 320 y Fm(\000)p Fp(1)1594 351 y Fm(\000)1657 340 y Fw(is)71 406 y(giv)o(en)f(b)o(y)324 548 y Fr(~)-22 b(g)r Fq(\()p Fr(x)p Fq(\))13 b(=)482 480 y Fl(Z)510 592 y Fg(R)544 548 y Fr(d\025)629 500 y Fl(X)620 606 y Fo(\034)t Fp(=)p Fo(a;b)725 534 y Fr(~)719 548 y( )753 527 y Fm(\000)782 548 y Fq(\()p Fr(x;)8 b(\025;)g(\034)e Fq(\))r(^)-26 b Fr(g)972 527 y Fo(\034)993 548 y Fq(\()p Fr(\025)p Fq(\))p Fr(;)59 b Fq(^)-26 b Fr(g)16 b Fn(2)e Fr(L)1249 527 y Fp(2)1269 548 y Fq(\()p Fv(R)p Fr(;)8 b Fv(C)1379 527 y Fp(2)1402 548 y Fq(\))p Fr(:)130 b Fw(\(5.40\))71 716 y(W)l(e)16 b(note)g(that)725 792 y Fq(\010)760 799 y Fm(\000)790 792 y Fr(K)t Fq(\010)870 771 y Fm(\000)p Fp(1)870 804 y Fm(\000)931 792 y Fq(=)e Fr(M)535 b Fw(\(5.41\))71 896 y(where)15 b Fr(M)22 b Fw(is)15 b(the)h(m)o(ultiplic)o(ation)e(op)q(erator)j(b)o(y)e(the)h (indep)q(enden)o(t)f(v)m(ariable)h Fr(\025)g Fw(on)1647 883 y Fq(^)1643 896 y Fk(K)p Fw(,)71 962 y(i.e.)362 1038 y(dom)o Fq(\()p Fr(M)5 b Fq(\))14 b(:=)623 998 y Fl(\010)654 1038 y Fq(^)-26 b Fr(g)16 b Fn(2)e Fr(L)771 1017 y Fp(2)791 1038 y Fq(\()p Fv(R)p Fr(;)8 b Fv(C)901 1017 y Fp(2)924 1038 y Fq(\))14 b(:)f Fr(\025)r Fq(^)-26 b Fr(g)s Fq(\()p Fr(\025)p Fq(\))14 b Fn(2)g Fr(L)1198 1017 y Fp(2)1218 1038 y Fq(\()p Fv(R)p Fr(;)7 b Fv(C)1328 1017 y Fp(2)1350 1038 y Fq(\))1369 998 y Fl(\011)1565 1038 y Fw(\(5.42\))71 1143 y(and)513 1218 y Fq(\()p Fr(M)g Fq(^)-26 b Fr(g)s Fq(\)\()p Fr(\025)p Fq(\))14 b(:=)f Fr(\025)r Fq(^)-26 b Fr(g)s Fq(\()p Fr(\025)p Fq(\))p Fr(;)59 b Fq(^)-26 b Fr(g)16 b Fn(2)e Fw(dom)o Fq(\()p Fr(M)5 b Fq(\))p Fr(:)318 b Fw(\(5.43\))71 1323 y(Using)24 b(the)g(outgoing)i (eigenfunctions)e(one)g(easily)g(de\034nes)g(an)h(outgoing)h(F)l (ourier)71 1389 y(transformation)16 b Fq(\010)438 1396 y Fp(+)481 1389 y Fq(:)e Fk(K)g Fn(\000)-9 b(!)654 1376 y Fq(^)650 1389 y Fk(K)14 b Fq(=)g Fr(L)782 1371 y Fp(2)802 1389 y Fq(\()p Fv(R)p Fr(;)7 b Fv(C)911 1371 y Fp(2)934 1389 y Fq(\))p Fw(.)71 1584 y Fu(6)77 b(Concluding)24 b(remarks)71 1707 y Fw(In)c(a)i(forthcoming)e(pap)q(er)i([19])f(w)o(e)g (in)o(tend)f(to)i(use)f(dissipativ)o(e)f(Sc)o(hr\366)q(dinger-t)o(yp)q (e)71 1773 y(op)q(erators)g(as)f(ingredien)o(ts)f(for)h(Sc)o (hr\366dinger-P)o(oisson)g(systems.)27 b(Suc)o(h)18 b(an)i(applica-)71 1839 y(tion)c(in)g(mind)g(let)f(us)i(mak)o(e)e(some)h(remarks.)21 b(Firstly)l(,)15 b(to)i(include)e(electro-magnetic)71 1905 y(radiation)20 b(e\033ects)g(in)f(the)h(mo)q(del)f(one)h(has)g(to) h(extend)e(the)h(considerations)g(to)g(dis-)71 1971 y(sipativ)o(e)e(p)q (oten)o(tials,)i(i.e.)29 b Fn(=)p Fw(m)o Fq(\()p Fr(V)11 b Fq(\()p Fr(x)p Fq(\)\))19 b Fn(\024)g Fq(0)p Fw(,)h Fr(x)f Fn(2)h Fq([)p Fr(a;)8 b(b)p Fq(])p Fw(.)29 b(Hence,)19 b(w)o(e)g(ha)o(v)o(e)f(to)i(con-)71 2038 y(struct)14 b(self-adjoin)o(t)g(dilations)g Fr(K)19 b Fw(for)c(Sc)o(hr\366)q (dinger-lik)o(e)e(op)q(erators)i(with)g(dissipativ)o(e)71 2104 y(b)q(oundary)j(conditions)g(and)g(dissipativ)o(e)e(p)q(oten)o (tials.)24 b(This)18 b(will)e(b)q(e)h(done)h(in)f(a)h(sub-)71 2170 y(sequen)o(t)h(pap)q(er.)35 b(Secondly)l(,)21 b(w)o(e)f(note)h (that)g(the)g(self-adjoin)o(t)f(dilation)g Fr(K)25 b Fw(of)c Fr(H)k Fw(is)71 2236 y(not)e(semi-b)q(ounded)f(from)g(b)q(elo)o (w.)41 b(This)23 b(observ)m(ation)h(has)f(the)g(unpleasan)o(t)g(con-)71 2302 y(sequence)d(that)i(w)o(e)g(can)g(regard)g(the)g(op)q(erator)h Fr(K)j Fw(only)21 b(v)o(ery)g(conditionally)g(as)h(a)71 2369 y(ph)o(ysical)17 b(Hamiltonian)f(since,)i(in)f(general,)h(suc)o(h) g(Hamiltonians)f(ha)o(v)o(e)g(to)i(b)q(e)f(semi-)71 2435 y(b)q(ounded)g(from)e(b)q(elo)o(w.)24 b(T)l(o)18 b(o)o(v)o(ercome)c (this)j(disadv)m(an)o(tage)h(one)g(can)f(use)g(b)q(oundary)71 2501 y(conditions)d(whic)o(h)g(dep)q(end)h(on)g(the)f(sp)q(ectral)h (parameter)e Fr(\025)p Fw(.)21 b(In)14 b(particular,)h(one)f(can)856 2626 y(25)p eop %%Page: 26 26 26 25 bop 257 274 a Fw(consider)16 b(instead)g(of)g(\(1.2\))g(and)g (\(1.3\))g(a)g(family)e(of)i(maximal)c(dissipativ)o(e)j(op)q(erators) 257 340 y Fn(f)p Fr(H)t Fq(\()p Fr(\025)p Fq(\))p Fn(g)417 353 y Fo(\025)p Fm(2)p 462 326 50 2 v Fg(C)483 357 y Ff(+)530 340 y Fw(de\034ned)h(b)o(y)299 497 y(dom)o Fq(\()p Fr(H)t Fq(\()p Fr(\025)p Fq(\)\))f(=)605 412 y Fl(\()666 459 y Fr(g)h Fn(2)e Fr(W)805 441 y Fp(1)p Fo(;)p Fp(2)852 459 y Fq(\([)p Fr(a;)8 b(b)p Fq(]\))k(:)1076 439 y Fp(1)p 1046 448 79 2 v 1046 476 a Fo(m)p Fp(\()p Fo(x)p Fp(\))1129 459 y Fr(g)1154 441 y Fm(0)1166 459 y Fq(\()p Fr(x)p Fq(\))h Fn(2)h Fr(W)1345 441 y Fp(1)p Fo(;)p Fp(2)1392 459 y Fq(\([)p Fr(a;)8 b(b)p Fq(]\))p Fr(;)710 511 y Fp(1)p 671 519 96 2 v 671 548 a(2)p Fo(m)p Fp(\()p Fo(a)p Fp(\))771 531 y Fr(g)796 513 y Fm(0)808 531 y Fq(\()p Fr(a)p Fq(\))13 b(=)h Fn(\000)p Fr(\024)1004 538 y Fo(a)1025 531 y Fq(\()p Fr(\025)p Fq(\))p Fr(g)r Fq(\()p Fr(a)p Fq(\))p Fr(;)1257 511 y Fp(1)p 1221 519 92 2 v 1221 548 a(2)p Fo(m)p Fp(\()p Fo(b)p Fp(\))1317 531 y Fr(g)1342 513 y Fm(0)1354 531 y Fq(\()p Fr(b)p Fq(\))f(=)h Fr(\024)1506 538 y Fo(b)1523 531 y Fq(\()p Fr(\025)p Fq(\))p Fr(g)r Fq(\()p Fr(b)p Fq(\))1694 412 y Fl(\))1776 497 y Fw(\(6.1\))257 642 y(and)662 708 y Fq(\()p Fr(H)t Fq(\()p Fr(\025)p Fq(\))p Fr(g)r Fq(\)\()p Fr(x)p Fq(\))g(=)g(\()p Fr(l)q Fq(\()p Fr(g)r Fq(\)\)\()p Fr(x)p Fq(\))48 b Fr(g)16 b Fn(2)e Fw(dom)o Fq(\()p Fr(H)t Fq(\))p Fr(:)304 b Fw(\(6.2\))257 799 y(If)16 b Fr(\024)334 806 y Fo(a)355 799 y Fq(\()p Fr(\025)p Fq(\))h Fw(and)g Fr(\024)561 806 y Fo(b)578 799 y Fq(\()p Fr(\025)p Fq(\))p Fw(,)g Fr(\025)d Fn(2)h Fv(C)798 806 y Fp(+)830 799 y Fw(,)h(are)h(suitable)f(Nev)m(anlinna)f (functions,)h(then)h(one)f(can)257 865 y(pro)o(v)o(e)21 b(that)h(there)f(is)g(a)h(larger)g(Hilb)q(ert)e Fk(K)j Fn(\023)g Fk(H)e Fw(and)h(a)g(self-adjoin)o(t)g(op)q(erator)g Fr(K)257 931 y Fw(de\034ned)16 b(on)h Fk(K)p Fw(,)f(suc)o(h)g(that)606 1042 y Fr(P)644 1021 y Fe(K)637 1054 y(H)669 1042 y Fq(\()p Fr(K)f Fn(\000)c Fr(\025)p Fq(\))841 1021 y Fm(\000)p Fp(1)889 1042 y Fn(j)p Fk(H)i Fq(=)h(\()p Fr(H)t Fq(\()p Fr(\025)p Fq(\))e Fn(\000)e Fr(\025)p Fq(\))1241 1021 y Fm(\000)p Fp(1)1289 1042 y Fr(;)57 b(\025)14 b Fn(2)g Fv(C)1482 1049 y Fp(+)1514 1042 y Fr(:)248 b Fw(\(6.3\))257 1152 y(With)15 b(resp)q(ect)g(of)g(\(4.1\))g(the)g(op)q(erator)h Fr(K)k Fw(can)15 b(b)q(e)g(called)f(the)h(dilation)f(of)i(the)f(family) 257 1219 y Fn(f)p Fr(H)t Fq(\()p Fr(\025)p Fq(\))p Fn(g)417 1232 y Fo(\025)p Fm(2)p 462 1204 50 2 v Fg(C)483 1236 y Ff(+)513 1219 y Fw(.)25 b(In)17 b(con)o(trast)h(to)g(the)f(presen)o (t)g(case)h(it)f(is)g(p)q(ossible)h(that)f(this)h(dilation)257 1285 y Fr(K)j Fw(is)16 b(semib)q(ounded)f(from)g(b)q(elo)o(w.)331 1351 y(Finally)l(,)h(w)o(e)h(note)g(that)h(in)g(application)f(to)h(Sc)o (hr\366)q(dinger-P)o(oisson)g(systems)f(and)257 1417 y(to)g(semiconductor)e(ph)o(ysics)g(the)h(necessit)o(y)f(arises)h(to)h (extend)f(all)f(considerations)i(to)257 1483 y(dimensions)j(t)o(w)o(o)h (and)h(three)e(whic)o(h)h(generates)g(new)g(di\036culties)e(with)i (resp)q(ect)g(to)257 1550 y(dilations)16 b(and)h(eigenfunction)f (expansions.)257 1724 y Fu(A)n(c)n(kno)n(wledgemen)n(ts)257 1840 y Fw(The)d(second)h(author)f(thanks)h(the)f(Deutsc)o(he)f(F)l (orsc)o(h)o(ungsgemeinsc)o(haft)f(for)i(\034nancial)257 1906 y(supp)q(ort.)257 2081 y Fu(References)282 2196 y Fw([1])24 b Fa(Ad)o(amy)l(an,)15 b(V.)9 b(M.)p Fw(;)14 b Fa(Neidhardt,)k(H.)p Fw(:)i(On)14 b(the)f(absolutely)g(con)o(tin)o (uous)h(sub-)358 2262 y(space)i(for)g(non-selfadjoin)o(t)h(op)q (erators;)g(Math.)f(Nac)o(hr.)f Fx(210)h Fw(\(2000\),)i(5-42.)282 2369 y([2])24 b Fa(Allakhverdiev,)h(B.)9 b(P.)p Fw(:)32 b(On)21 b(dissipativ)o(e)f(extensions)h(of)h(the)f(symmetric)358 2435 y(Sc)o(hr\366)q(dinger)11 b(op)q(erator)i(in)f(W)l(eyl's)f (limit-ci)o(rcl)o(e)e(case;)k(Dokl.)e(Ak)m(ad.)h(Nauk)f(SSSR)358 2501 y Fx(V)-5 b(ol)20 b(293)c Fw(\(1987\),)h(777-781.)1042 2626 y(26)p eop %%Page: 27 27 27 26 bop 95 274 a Fw([3])24 b Fa(Allakhverdiev,)17 b(B.)10 b(P.)p Fw(,)k(and)h Fa(Maksudo)o(v,)i(F.)9 b(G.)p Fw(:)21 b(On)14 b(the)h(theory)f(of)h(the)171 340 y(c)o(haracteristic)d (function)h(and)i(sp)q(ectral)e(analysis)h(of)g(a)g(dissipativ)o(e)f (Sc)o(hr\366)q(dinger)171 406 y(op)q(erator;)k(Dokl.)e(Ak)m(ad.)h(Nauk) g(SSSR)h Fx(303)f Fw(\(1988\),)h(No.6,)f(1307-1309.)95 521 y([4])24 b Fa(Allakhverdiev,)c(B.)9 b(P.)p Fw(:)24 b(Sc)o(hr\366)q(dinger)17 b(t)o(yp)q(e)g(dissipativ)o(e)f(op)q(erator)j (with)e(a)171 588 y(matrix)10 b(p)q(oten)o(tial;)j(Sp)q(ectral)e (theory)h(of)g(op)q(erators)i(and)e(its)g(applications)g Fx(No.9)p Fw(,)171 654 y(11-41,)17 b(\020\311lm\021)j(Baku)c(1989.)95 769 y([5])24 b Fa(Allakhverdiev,)h(B.)9 b(P.)p Fw(:)31 b(On)21 b(dilation)g(theory)f(and)i(sp)q(ectral)f(analysis)g(of)171 835 y(dissipativ)o(e)15 b(Sc)o(hr\366)q(dinger)i(op)q(erators)h(in)e(W) l(eyl's)f(limit-circl)o(e)e(case;)k(Izv.)e(Ak)m(ad.)171 901 y(Nauk)h(SSSR,)g(Ser.)f(Mat.)h Fx(54)g Fw(\(1990\),)i(No.2,)d (242-257.)95 1016 y([6])24 b Fa(Allakhverdiev,)19 b(B.)9 b(P.)p Fw(:)21 b(On)16 b(the)g(theory)h(of)f(non-selfadjoin)o(t)h(op)q (erators)g(of)171 1083 y(Sc)o(hr\366)q(dinger)h(t)o(yp)q(e)g(with)g(a)h (matrix)d(p)q(oten)o(tial;)j(Izv.)e(Ross.)h(Ak)m(ad.)g(Nauk,)g(Ser.)171 1149 y(Mat.)e Fx(56)g Fw(\(1993\),)h(No.)f(2,)g(193-205.)95 1264 y([7])24 b Fa(Allakhverdiev,)17 b(B.)9 b(P.)p Fw(,)15 b(and)f Fa(Canoglu,)j(A.)p Fw(:)k(Sp)q(ectral)14 b(analysis)g(of)h (dissi-)171 1330 y(pativ)o(e)h(Sc)o(hr\366)q(dinger)g(op)q(erators;)i (Pro)q(c.)f(R.)f(So)q(c.)h(Edin)o(b.,)e(Sect.)h(A,)g(Math)g Fx(127)171 1396 y Fw(\(1997\),)h(No.)f(6,)g(1113-1121.)95 1511 y([8])24 b Fa(Ben)f(Abd)o(allah,)h(N.)p Fw(,)e Fa(Degond,)j(P.)p Fw(,)d Fa(Mark)o(o)o(wich,)h(P.)9 b(A.)p Fw(:)31 b(On)21 b(a)g(one-)171 1577 y(dimensional)j(Sc)o(hr\366)q(dinger-P)o(oisson)i (scattering)g(mo)q(del,)g(Z.)g(Angew.)f(Math.)171 1644 y(Ph)o(ys.)15 b Fx(48)i Fw(\(1997\),)g(No.)f(1,)g(135-155.)95 1759 y([9])24 b Fa(Ben)19 b(Abd)o(allah,)h(N.)p Fw(:)25 b(On)17 b(a)h(m)o(ultidim)o(ensional)d(Sc)o(hr\366)q(dinger-P)o(oisson) j(sys-)171 1825 y(tem)c(for)j(semiconductors;)d(Preprin)o(t)i(MPI)f (96.04,)i(T)l(oulouse)g(1996.)71 1940 y([10])24 b Fa(Ben)e(Abd)o (allah,)g(N.)p Fw(:)29 b(A)19 b(h)o(ybrid)g(kinetic-quan)o(tum)e(mo)q (del)h(for)i(stationary)171 2006 y(electron)15 b(transp)q(ort;)i(J.)f (Stat.)g(Ph)o(ys.)g Fx(90)g Fw(\(1998\),)h(No.3-4,)g(627-662.)71 2121 y([11])24 b Fa(F)o(oias,)e(C.)p Fw(;)e Fa(Sz.-Na)o(gy,)i(B.)p Fw(:)28 b(Harmonic)17 b(analysis)i(of)g(op)q(erators)h(on)g(Hilb)q(ert) 171 2187 y(space;)29 b(Ak)m(ad\351miai)23 b(Kiad\363,)k(Budap)q(est,)g (North-Holland)e(Publishing)g(Com-)171 2254 y(pan)o(y)l(,)15 b(Amsterdam-London,)g(1970.)71 2369 y([12])24 b Fa(Frensley,)29 b(W.)9 b(R.)p Fw(:)37 b(Boundary)25 b(conditions)f(for)g(op)q(en)h (quan)o(tum)e(systems)171 2435 y(driv)o(en)14 b(far)i(from)f (equilibrium)o(;)e(Reviews)h(of)i(Mo)q(dern)g(Ph)o(ysics)f Fx(62)h Fw(\(1990\),)h(No.)171 2501 y(3,)f(745-791.)856 2626 y(27)p eop %%Page: 28 28 28 27 bop 257 274 a Fw([13])25 b Fa(Gajewski,)19 b(H.)p Fw(:)i(On)c(the)e(existence)g(of)h(steady-state)h(carier)f (distributions)g(in)358 340 y(semiconductors.)10 b(In)h Fj(Pr)n(obleme)j(und)g(Metho)n(den)g(der)f(Mathematischen)i(Physik)p Fw(,)358 406 y(T)l(eubner-T)l(exte)g(Math.)h Fx(63)p Fw(,)g(T)l(eubner-V)l(erlag)f(1984,)j(pp)e(76-82.)257 519 y([14])25 b Fa(Gajewski,)20 b(H.)p Fw(;)d Fa(Gr)o(\366ger,)j(K.)p Fw(:)j(On)17 b(the)g(basic)g(equations)g(for)g(carier)f(trans-)358 585 y(p)q(ort)h(in)f(semiconductors;)e(J.)i(Math.)g(Anal.)f(Appl.)g Fx(113)h Fw(\(1986\),)i(12-35.)257 698 y([15])25 b Fa(Kaiser,)31 b(H.-Ch.)p Fw(;)g Fa(Rehber)o(g,)h(J.)p Fw(:)41 b(Ab)q(out)27 b(a)g(one-dimensional)d(station-)358 765 y(ary)18 b(Sc)o(hr\366)q (dinger-P)o(oisson)g(system)f(with)g(K)o(ohn-Sham)h(p)q(oten)o(tial;)g (Z.)f(Angew.)358 831 y(Math.)f(Ph)o(ys.)f Fx(50)h Fw(\(1999\),)i(No.3,) d(423-458.)257 944 y([16])25 b Fa(Kaiser,)19 b(H.-Ch.)p Fw(;)e Fa(Rehber)o(g,)j(J.)p Fw(:)j(Ab)q(out)17 b(a)g(stationary)h(Sc)o (hr\366)q(dinger-P)o(ois-)358 1010 y(son)26 b(system)f(with)g(K)o (ohn-Sham)h(p)q(oten)o(tial)g(in)f(a)i(b)q(ounded)f(t)o(w)o(o-)g(or)h (three-)358 1076 y(dimensional)f(domain;)33 b(Nonlinear)27 b(Anal.,)j(Theory)e(Metho)q(ds)h(Appl.)e Fx(41A)358 1142 y Fw(\(2000\))17 b(,)f(No.)g(1-2,)h(33-72.)257 1255 y([17])25 b Fa(Kaiser,)17 b(H.-Ch.)p Fw(;)f Fa(Rehber)o(g,)h(J.)p Fw(:)k(Ab)q(out)16 b(some)d(some)h(mathematical)e(ques-)358 1321 y(tions)h(concerning)g(the)g(em)o(b)q(edding)f(of)i(Sc)o(hr\366)q (dinger-P)o(oisson)g(systems)e(in)o(to)i(the)358 1388 y(drift-di\033usion)f(mo)q(del)g(of)h(semiconductor)e(devices.)g(In)h (B.)g(Fiedler,)f(K.)h(Gr\366ger,)358 1454 y(and)21 b(J.)g(Sprek)o(els,) f(editors,)i(Equadi\033)f(99)h(Pro)q(ceedings)f(of)g(the)g(In)o (ternational)358 1520 y(Conference)12 b(on)i(Di\033eren)o(tial)e (Equations,)i(v)o(olume)d(2,)i(pages)h(1328-1333)q(.)h(W)l(orld)358 1586 y(Scien)o(ti\034c,)e(2000.)257 1699 y([18])25 b Fa(Kaiser,)18 b(H-Ch.)p Fw(;)f Fa(Neidhardt,)j(H.)p Fw(;)d Fa(Rehber)o(g,)i(J.)p Fw(:)j(Macroscopic)16 b(curren)o(t)358 1765 y(induced)24 b(b)q(oundary)i(conditions)f(for)h(Sc)o(hr\366)q (dinger-t)o(yp)q(e)e(op)q(erators,)29 b(WIAS-)358 1831 y(Preprin)o(t)19 b Fx(630)p Fw(,)h(W)l(eierstrass-Institute)f(for)h (Applied)f(Analysis)g(and)h(Sto)q(c)o(has-)358 1898 y(tics,)15 b(Berlin)f(2001.)257 2010 y([19])25 b Fa(Kaiser,)20 b(H-Ch.)p Fw(;)f Fa(Neidhardt,)j(H.)p Fw(;)d Fa(Rehber)o(g,)i(J.)p Fw(:)k(T)l(o)o(w)o(ards)19 b(dissipativ)o(e)358 2077 y(Sc)o(hr\366)q(dinger-P)o(oisson)e(systems:)k(carrier)16 b(and)i(curren)o(t)e(densities.)f(In)h(prepara-)358 2143 y(tion.)257 2256 y([20])25 b Fa(Ka)l(to,)e(T.)p Fw(:)31 b(P)o(erturbation)21 b(theory)g(for)h(linear)e(op)q(erators;)k (Springer-V)l(erlag,)358 2322 y(Berlin)14 b(1966.)257 2435 y([21])25 b Fa(Land)o(a)o(u,)36 b(L.)9 b(A.)p Fw(;)39 b Fa(Lifschitz,)f(E.M.)p Fw(:)51 b(Quan)o(tenmec)o(hanik,)31 b(Ak)m(ademie-)358 2501 y(V)l(erlag,)15 b(Berlin)f(1971.)1042 2626 y(28)p eop %%Page: 29 29 29 28 bop 71 274 a Fw([22])24 b Fa(Lange,)45 b(H.)p Fw(;)h Fa(Toomire,)f(B.)p Fw(;)i Fa(Zweifel,)e(P.)9 b(F.)p Fw(:)62 b(An)36 b(o)o(v)o(erview)f(of)171 340 y(Sc)o(hr\366)q(dinger)22 b(P)o(oisson)g(problems,)f(Rep.)g(Math.)h(Ph)o(ys.)f Fx(36)h Fw(\(1995\),)i(No.)d(2-3,)171 406 y(331-345.)71 517 y([23])j Fa(Mark)o(o)o(wich,)f(P.)8 b(A.)p Fw(:)30 b(The)20 b(stationary)h(semiconductor)e(device)g(equations;)171 583 y(Springer-V)l(erlag,)c(Wien-New)g(Y)l(ork)h(1986.)71 693 y([24])24 b Fa(Mark)o(o)o(wich,)h(P.)9 b(A.)p Fw(;)25 b Fa(Ringhofer,)h(Ch.)9 b(A.)p Fw(;)26 b Fa(Schmeiser,)g(Ch.)p Fw(:)33 b(Semi-)171 759 y(conductor)16 b(Equations;)h(Springer-V)l (erlag,)e(Wien-New)g(Y)l(ork)h(1990.)71 869 y([25])24 b Fa(Nier,)j(F.)p Fw(:)34 b(A)22 b(stationary)h(Sc)o(hr\366)q(dinger-P) o(oisson)g(system)e(arising)i(from)e(the)171 935 y(mo)q(delling)14 b(of)j(electron)e(devices,)f(F)l(orum)i(Math.)f Fx(2)i Fw(\(1990\),)g(No.)f(5,)g(489-510.)71 1046 y([26])24 b Fa(Nier,)j(F.)p Fw(:)34 b(F)l(orm)o(ulation)20 b(v)m(ariationelle)h (des)i(systems)e(Sc)o(hr\366)q(dinger-P)o(oisson)171 1112 y(en)c(dimension)f Fr(d)g Fn(\024)f Fq(3)p Fw(,)i(Journal)h (Equations)g(Deriv.)e(P)o(artielles,)g(St.-Jeans-De-)171 1178 y(Mon)o(ts)g(1992,)h(No.)f(XVI)q(I)q(I)f(\(1992\).)71 1288 y([27])24 b Fa(Nier,)c(F.)p Fw(:)k(A)16 b(v)m(ariational)i(form)o (ulation)d(of)j(Sc)o(hr\366)q(dinger-P)o(oisson)g(systems)e(in)171 1354 y(dimension)11 b Fr(d)j Fn(\024)g Fq(3)p Fw(,)f(Comm)o(un.)d(P)o (artial)j(Di\033er.)f(Equations)h Fx(18)g Fw(\(1993\),)i(No.)d(7-8,)171 1421 y(1125-1147.)71 1531 y([28])24 b Fa(Nier,)19 b(F.)p Fw(:)j(Sc)o(hr\366)q(dinger-P)o(oisson)17 b(systems)f(in)g(dimension)e Fr(d)h Fn(\024)e Fq(3)p Fw(:)22 b(The)17 b(whole)171 1597 y(space,)f(Pro)q(c.)g(R.)g(So)q(c.)g(Edin)o(b.,)f(Sect.)g(A)h Fx(123)g Fw(\(1993\),)i(No.6,)d(1179-1201)q(.)71 1707 y([29])24 b Fa(P)l(a)-6 b(vlo)o(v,)20 b(B.)9 b(S.)p Fw(:)25 b(Self-adjoin)o(t)18 b(dilation)g(of)g(the)g(dissipativ)o(e)f(Sc)o (hr\366)q(dinger)h(op-)171 1773 y(erator)i(and)g(its)f(resolution)g(in) g(terms)f(of)i(eigenfuctions,)f(Mat.)g(S.,)h Fx(102\(144\))171 1840 y Fw(\(1977\),)d(511-536.)71 1950 y([30])24 b Fa(P)l(a)-6 b(vlo)o(v,)17 b(B.)9 b(S.)p Fw(:)22 b(Dilation)15 b(theory)h(and)g(sp)q (ectral)g(analysis)g(of)g(non-selfadjoin)o(t)171 2016 y(di\033eren)o(tial)c(op)q(erators,)k(T)l(ransl.,)e(I)q(I.)f(Ser.,)g (Am.)f(Math.)i(So)q(c.)g Fx(115)g Fw(\(1981\),)h(103-)171 2082 y(142)j(;)g(translation)g(from)e(Pro)q(c.)i(7th.)g(Win)o(ter)e(Sc) o(ho)q(ol,)i(Drogob)o(yc)o(h)f(1974,)i(3-69)171 2148 y(\(1976\).)71 2258 y([31])24 b Fa(P)l(a)-6 b(vlo)o(v,)23 b(B.S.)p Fw(:)32 b(Sp)q(ectral)21 b(theory)g(of)g(non-selfadjoin)o(t)g (op)q(erators,)j(In)c(Pro)q(c.)171 2325 y(In)o(t.)15 b(Congr.)i(Math.,)e(W)l(arsza)o(w)o(a)i(1983,)g(V)l(ol.2,)e(1011-1025) 20 b(\(1984\).)71 2435 y([32])k Fa(P)l(a)-6 b(vlo)o(v,)20 b(B.)10 b(S.)p Fw(:)27 b(Sp)q(ectral)18 b(analysis)h(of)g(a)g (dissipativ)o(e)f(singular)h(Sc)o(hr\366)q(dinger)171 2501 y(op)q(erator)j(in)e(terms)f(of)i(a)g(functional)f(mo)q(del.)f Fj(Partial)j(di\033er)n(ential)g(e)n(quations)856 2626 y Fw(29)p eop %%Page: 30 30 30 29 bop 358 274 a Fx(8)p Fw(,)15 b(95-163,)i(in)d(P)o(artial)h (Di\033eren)o(tial)f(Equations)i(VI)q(I)q(I.)d(Encycl.)h(Math.)h(Sci.)f Fx(65)p Fw(,)358 340 y(87-153.)k(\(1996\))257 448 y([33])25 b Fa(P)l(a)-6 b(vlo)o(v,)16 b(B.)9 b(S.)p Fw(:)21 b(Irrev)o(ersibilit)n (y)l(,)12 b(Lax-Phillips)i(approac)o(h)i(to)f(resonance)g(scat-)358 514 y(tering)c(and)i(sp)q(ectral)f(analysis)h(of)f(non-selfadjoin)o(t)g (op)q(erators)i(in)e(Hilb)q(ert)f(space,)358 580 y(In)o(t.)k(J.)h (Theor.)g(Ph)o(ys.)f Fx(38)i Fw(\(1999\),)g(No.1,)f(21-45.)257 688 y([34])25 b Fa(Vladimir)o(o)o(v,)i(V.)9 b(S.)p Fw(:)36 b(Generalized)22 b(functions)h(in)g(mathematical)e(ph)o(ysics.)358 754 y(Mir,)15 b(Mosco)o(w)h(1979.)1042 2626 y(30)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF