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y(oscillator,)g(it)h(is)h(easily)f(seen)i(that)e(\(1.8\))g(in)g (fact)h(fails)e(for)i(these)h(t)o(w)o(o)e(p)q(oten)o(tials;)g(th)o(us,) 249 1014 y(that)k(it)g(con)o(tin)o(ues)g(to)g(b)q(e)g(true)h(for)e(the) i(Thomas{F)m(ermi)c(p)q(oten)o(tial)i(m)o(ust)g(b)q(e)h(a)g(deep)249 1064 y(fact.)k(This)15 b(is)f(the)i(ap)q(erio)q(dicit)o(y)e(describ)q (ed)j(at)e(the)g(b)q(eginning)f(of)h(this)g(review,)g(whic)o(h)249 1113 y(is)f(the)g(essen)o(tial)h(con)o(ten)o(t)f(of)g(Theorem)f(0.1.) 249 1213 y(Ap)q(erio)q(dicit)o(y)h(of)f(zero{energy)j(Hamiltonian)11 b(paths)j(is)g(w)o(ell-kno)o(wn)f(to)h(pla)o(y)f(a)h(crucial)249 1263 y(role)d(in)g(the)h(study)g(of)f(eigen)o(v)n(alues)g(and)h (eigenfunctions.)17 b(In)12 b(our)f(setting,)h(Theorem)f(0.1)249 1313 y(en)o(ters)j(b)q(ecause)h(our)e(form)o(ulas)e(for)h(the)i(eigen)o (v)n(alue)e(sum)g(and)h(densit)o(y)g(in)o(v)o(olv)o(e)e(expres-)249 1362 y(sions)j(of)f(the)i(form)655 1436 y Fq(S)f Fr(=)799 1396 y Fk(X)738 1494 y Fp(1)p Fn(\024)p Fj(l)12 b Fr(0,)h(whic)o(h)h(can)g(b)q(e)h(view)o(ed)f(as)g(the)g (\014xed)g(p)q(oin)o(t)g(of)576 2109 y Fq(T)6 b Fr(\()p Fq(u)p Fr(\))11 b(=)h Fq(u)741 2115 y Fp(0)769 2109 y Fr(+)810 2052 y Fk(Z)852 2063 y Fj(x)833 2147 y(x)852 2151 y Fm(0)879 2038 y Fk( )912 2109 y Fq(u)936 2115 y Fp(1)964 2109 y Fr(+)1005 2052 y Fk(Z)1047 2063 y Fj(t)1028 2147 y(x)1047 2151 y Fm(0)1077 2081 y Fq(u)1101 2053 y Fm(3)1110 2062 y Fl(=)1120 2070 y Fm(2)1138 2081 y Fr(\()p Fq(s)p Fr(\))p 1077 2099 113 2 v 1105 2138 a Fq(s)1124 2118 y Fm(1)1134 2122 y Fl(=)1144 2130 y Fm(2)1204 2109 y Fq(ds)1245 2038 y Fk(!)1294 2109 y Fq(dt:)249 2284 y Fr(If)i(w)o(e)g(regard)g Fq(u)g Fr(as)689 2334 y Fq(u)p Fr(\()p Fq(x)p Fr(\))d(=)h Fq(u)848 2340 y Fp(0)875 2334 y Fr(+)e Fq(u)941 2340 y Fp(1)969 2334 y Fo(\001)e Fq(r)i Fo(\001)f Fq(z)j Fr(+)d Fq(z)1132 2317 y Fp(2)1151 2334 y Fq(f)t Fr(\()p Fq(z)r Fr(\))249 2408 y(with)j Fq(z)h Fr(=)f(\()p Fq(x)5 b Fo(\000)g Fq(x)524 2414 y Fp(0)543 2408 y Fr(\))p Fq(=r)q Fr(,)12 b Fq(f)k Fo(2)c Fq(H)738 2393 y Fp(1)768 2408 y Fr(and)g Fq(r)h Fr(small,)c(w)o(e)k (see)g(that)f Fq(T)18 b Fr(induces)13 b(in)e(a)h(trivial)e(w)o(a)o(y) 249 2458 y(an)k(op)q(erator)483 2448 y(~)475 2458 y Fq(T)20 b Fr(of)14 b(whic)o(h)g Fq(f)19 b Fr(is)14 b(its)g(\014xed)h(p)q(oin)o (t.)k(Th)o(us,)14 b(the)h(follo)o(wing)c(algorithm)h(is)i(a)249 2508 y(simple)d(consequence)16 b(of)c(the)h(\014xed)h(p)q(oin)o(t)e (theorem)h(on)f(Banac)o(h)h(spaces.)19 b(The)14 b(relev)n(an)o(t)249 2558 y(norm)f({)g(denoted)i(b)o(y)f Fo(j)-7 b(j)13 b(j)-7 b(j)13 b Fr({)g(throughout)h(this)g(section)h(w)o(e)f(will)e(b)q(e)j Fo(j)-7 b(j)13 b(j)-7 b(j)1381 2564 y Fp(1)1398 2558 y Fr(.)p eop 11 10 bop 249 275 a Fc(Interval)19 b(A)o(rithmetic)e(in)h(Quantum)h(Me) n(chanics)448 b Fi(11)249 516 y Fg(Algorithm)13 b(3.1.)18 b Fs(We)e(de)n(duc)n(e)f(c)n(onditions)g(on)h Fq(u)1062 522 y Fp(0)1080 516 y Fs(,)f Fq(u)1132 522 y Fp(1)1150 516 y Fs(,)g Fq(x)1202 522 y Fp(0)1220 516 y Fs(,)f Fq(r)i Fs(and)g Fq(\013)e Fs(under)h(which)1647 505 y Fr(~)1639 516 y Fq(T)249 565 y Fs(is)e(a)h(wel)r(l{de\014ne)n(d)g(c)n(ontr)n (action)f(in)h Fq(B)r Fr(\(0)p Fq(;)7 b(\013)p Fr(\))k Fo(\032)h Fq(H)1047 550 y Fp(1)1065 565 y Fs(,)i(and)g(we)f(c)n(ompute) h(an)g(upp)n(er)g(b)n(ound)249 615 y(for)g Fo(j)-7 b(j)340 605 y Fr(~)332 615 y Fq(T)6 b Fo(j)-7 b(j)379 621 y Fp(Lip)428 615 y Fs(.)249 715 y Fr(This)14 b(algorithm)d(is)j(then)g(applied)g(to) f(the)i(follo)o(wing)249 814 y Fg(Algorithm)d(3.2.)18 b Fs(Given)d(appr)n(opriate)e(intervals)h Fq(x)1104 799 y Fn(\003)1122 814 y Fs(,)g Fq(u)1173 799 y Fn(\003)1173 825 y Fp(0)1206 814 y Fs(and)h Fq(u)1310 799 y Fn(\003)1310 825 y Fp(1)1329 814 y Fs(,)f(and)g(r)n(epr)n(esentable)249 864 y Fq(r)q Fs(,)e(we)g(c)n(onstruct)g(a)g(neighb)n(orho)n(o)n(d)h Fo(U)t Fr(\()p Fq(I)871 870 y Fp(0)890 864 y Fq(;)7 b(:)g(:)g(:)e(;)i (I)1001 870 y Fj(N)1032 864 y Fr(;)g(0)p Fq(;)g(C)1121 870 y Fj(g)1139 864 y Fr(;)g(0\))k Fs(such)i(that)f(for)f(any)i Fq(x)1544 870 y Fp(0)1574 864 y Fo(2)e Fq(x)1637 849 y Fn(\003)1656 864 y Fs(,)249 914 y Fq(u)273 920 y Fp(0)310 914 y Fo(2)18 b Fq(u)380 899 y Fn(\003)380 924 y Fp(0)399 914 y Fs(,)h(and)g Fq(u)539 920 y Fp(1)576 914 y Fo(2)f Fq(u)646 899 y Fn(\003)646 924 y Fp(1)665 914 y Fs(,)h(and)g(any)h (solution)f Fq(u)f Fs(of)g Fr(\(3.3\))f Fs(with)h(any)h(of)g(these)g (initial)249 964 y(c)n(onditions,)c(we)g(have)491 1070 y Fq(u)p Fr(\()p Fq(x)p Fr(\))c(=)h Fq(u)650 1076 y Fp(0)677 1070 y Fr(+)e Fq(u)743 1076 y Fp(1)771 1070 y Fo(\001)e Fr(\()p Fq(x)h Fo(\000)h Fq(x)906 1076 y Fp(0)924 1070 y Fr(\))g(+)f Fq(z)1012 1053 y Fp(2)1031 1070 y Fq(f)t Fr(\()p Fq(z)r Fr(\))p Fq(;)92 b(z)14 b Fr(=)1294 1042 y Fq(x)9 b Fo(\000)g Fq(x)1392 1048 y Fp(0)p 1294 1060 117 2 v 1342 1098 a Fq(r)1416 1070 y(;)249 1173 y Fs(for)14 b(some)h Fq(f)i Fo(2)11 b(U)t Fs(.)249 1273 y Fg(Description)o(.)16 b Fr(First,)e(w)o(e)g(construct,)h(in)e(a)h(heuristic)h(w)o(a)o(y)m(,)d (a)h(p)q(olynomial)817 1402 y Fq(p)p Fr(\()p Fq(z)r Fr(\))e(=)962 1350 y Fj(N)946 1363 y Fk(X)949 1451 y Fj(i)p Fp(=0)1013 1402 y Fq(p)1034 1408 y Fj(i)1055 1402 y Fq(z)1076 1385 y Fj(i)1090 1402 y Fq(;)249 1539 y Fr(whic)o(h)17 b(appro)o(ximately)d (solv)o(es)777 1529 y(~)769 1539 y Fq(T)6 b(p)16 b Fr(=)h Fq(p)p Fr(,)g(and)g(w)o(e)g(set)h Fq(\013)f Fr(suc)o(h)h(that)f Fo(j)-7 b(j)p Fq(p)p Fo(j)g(j)15 b(\024)i Fq(\013)p Fr(.)27 b(Next,)249 1589 y(w)o(e)15 b(lo)q(ok)f(for)h Fq(\013)494 1595 y Fp(0)526 1589 y Fo(\025)f Fq(\013)g Fr(suc)o(h)i(that)f(the)h (conditions)f(on)g Fq(x)1155 1595 y Fp(0)1173 1589 y Fr(,)g Fq(u)1224 1595 y Fp(0)1242 1589 y Fr(,)g Fq(u)1293 1595 y Fp(1)1311 1589 y Fr(,)g Fq(r)h Fr(and)f Fq(\013)1482 1595 y Fp(0)1515 1589 y Fr(giv)o(en)g(b)o(y)249 1639 y(Algorithm)c(3.1)h(hold)g(uniformly)f(for)h(all)g Fq(x)937 1645 y Fp(0)967 1639 y Fo(2)f Fq(x)1030 1624 y Fn(\003)1049 1639 y Fr(,)i Fq(u)1098 1645 y Fp(0)1128 1639 y Fo(2)e Fq(u)1191 1624 y Fn(\003)1191 1649 y Fp(0)1223 1639 y Fr(and)i Fq(u)1327 1645 y Fp(1)1357 1639 y Fo(2)e Fq(u)1420 1624 y Fn(\003)1420 1649 y Fp(1)1439 1639 y Fr(.)18 b(Next,)13 b(since)249 1689 y Fq(f)19 b Fr(is)13 b(the)i(\014xed)f(p)q(oin)o(t)g (of)665 1678 y(~)657 1689 y Fq(T)6 b Fr(,)13 b(w)o(e)h(ha)o(v)o(e)770 1809 y Fo(j)-7 b(j)p Fq(p)8 b Fo(\000)i Fq(f)t Fo(j)-7 b(j)11 b(\024)974 1781 y(j)-7 b(j)p Fq(p)8 b Fo(\000)1069 1770 y Fr(~)1061 1781 y Fq(T)e(p)p Fo(j)-7 b(j)p 959 1799 185 2 v 959 1842 a Fr(1)9 b Fo(\000)g(j)-7 b(j)1055 1832 y Fr(~)1047 1842 y Fq(T)6 b Fo(j)-7 b(j)1094 1848 y Fp(Lip)249 1931 y Fr(W)m(e)13 b(get)h(the)h(required)f Fo(U)t Fr(,)g(b)o(y)f(putting)h Fq(I)902 1937 y Fj(i)927 1931 y Fr(=)e([)t Fq(p)1008 1937 y Fj(i)1026 1931 y Fq(;)s(p)1062 1937 y Fj(i)1080 1931 y Fr(])h(for)h Fq(i)e Fr(=)f(0)p Fq(;)c(:)g(:)g(:)e(;)i(N)18 b Fr(and)c(setting)g Fq(C)1650 1937 y Fj(g)249 1981 y Fr(equal)g(to)f(the)i(ratio)e(ab)q(o)o(v)o(e.)18 b(Q.E.D.)13 b(This)g(has)h(the)h(follo)o(wing)c(easy)j(consequence)j(:) 249 2080 y Fg(Algorithm)c(3.3.)20 b Fs(Given)15 b(disjoint)g(intervals) f Fq(x)1038 2065 y Fn(\003)1038 2091 y Fp(0)1072 2080 y Fs(and)i Fq(x)1177 2065 y Fn(\003)1177 2091 y Fp(1)1196 2080 y Fs(,)f(and)h(intervals)e Fq(u)1498 2065 y Fn(\003)1498 2091 y Fp(0)1532 2080 y Fs(and)i Fq(u)1637 2065 y Fn(\003)1637 2091 y Fp(1)1656 2080 y Fs(,)249 2130 y(we)g(c)n(onstruct)f(intervals)h Fq(y)684 2115 y Fn(\003)683 2141 y Fp(0)720 2130 y Fs(and)g Fq(y)822 2115 y Fn(\003)821 2141 y Fp(1)858 2130 y Fs(such)g(that)g(al) r(l)g(solutions)g Fq(u)f Fs(to)h(\(3.3\))g(with)g(initial)249 2180 y(values)f(e)n(qual)g(to)f(any)i Fq(u)631 2186 y Fp(0)661 2180 y Fo(2)11 b Fq(u)724 2165 y Fn(\003)724 2190 y Fp(0)757 2180 y Fs(and)16 b(any)f Fq(u)941 2186 y Fp(1)971 2180 y Fo(2)c Fq(u)1034 2165 y Fn(\003)1034 2190 y Fp(1)1053 2180 y Fs(,)j(for)g(any)i Fq(x)11 b Fo(2)g Fq(x)1324 2165 y Fn(\003)1324 2190 y Fp(0)1357 2180 y Fs(ar)n(e)k(guar)n(ante)n(e)n(d)g(to)249 2230 y(exist)g(as)g(p)n(ositive)f(solutions)h(on)h Fr([)t Fq(x)t(;)s(x)865 2215 y Fn(0)881 2230 y Fr(])p Fs(,)e(and)h(furthermor) n(e)f(satisfy)713 2321 y Fq(u)p Fr(\()p Fq(x)777 2304 y Fn(0)788 2321 y Fr(\))e Fo(2)f Fq(y)876 2304 y Fn(\003)875 2331 y Fp(0)896 2321 y Fq(;)91 b(u)1023 2304 y Fn(0)1034 2321 y Fr(\()p Fq(x)1074 2304 y Fn(0)1086 2321 y Fr(\))12 b Fo(2)f Fq(y)1174 2304 y Fn(\003)1173 2331 y Fp(1)1194 2321 y Fq(;)249 2412 y Fs(for)j(al)r(l)h Fq(x)399 2397 y Fn(0)422 2412 y Fo(2)c Fq(x)485 2397 y Fn(\003)485 2423 y Fp(1)504 2412 y Fs(.)p eop 12 11 bop 249 275 a Fi(12)1110 b Fh(Chapter)18 b(1)249 516 y Ft(The)k(Initial)f(V)-6 b(alue)22 b(Problem)g(at)h(0)249 615 y Fr(It)14 b(concerns)i(the)e(solution)f(to)h(the)h(Initial)d(V)m (alue)h(Problem)572 707 y Fq(u)596 689 y Fn(00)617 707 y Fr(\()p Fq(x)p Fr(\))e(=)h Fq(x)752 689 y Fn(\000)778 677 y Fm(1)787 685 y Fl(=)796 693 y Fm(2)814 707 y Fq(u)838 677 y Fm(3)847 685 y Fl(=)857 694 y Fm(2)875 707 y Fr(\()p Fq(x)p Fr(\))28 b Fq(u)p Fr(\(0\))11 b(=)h(1)27 b Fq(u)1163 689 y Fn(0)1175 707 y Fr(\(0\))11 b(=)h Fo(\000)p Fq(w)238 b Fr(\(3)p Fq(:)p Fr(4\))249 798 y(for)14 b Fq(w)e(>)g Fr(0,)h(whic)o(h)h(can)g(b)q(e)g(view)o(ed)g(as)g(the)h(\014xed)f(p)q (oin)o(t)g(of)578 924 y Fq(T)6 b Fr(\()p Fq(u)p Fr(\))12 b(=)f(1)e(+)791 868 y Fk(Z)832 878 y Fj(x)814 962 y Fp(0)860 853 y Fk( )893 924 y Fo(\000)p Fq(w)h Fr(+)1007 868 y Fk(Z)1048 878 y Fj(t)1030 962 y Fp(0)1075 896 y Fq(u)1099 869 y Fm(3)1108 877 y Fl(=)1117 885 y Fm(2)1136 896 y Fr(\()p Fq(s)p Fr(\))p 1075 915 113 2 v 1103 953 a Fq(s)1122 933 y Fm(1)1131 937 y Fl(=)1141 945 y Fm(2)1202 924 y Fq(ds)1243 853 y Fk(!)1292 924 y Fq(dt:)249 1051 y Fr(If)k(w)o(e)g(put) 690 1100 y Fq(u)p Fr(\()p Fq(x)p Fr(\))e(=)f(1)e Fo(\000)h Fq(w)g Fo(\001)f Fq(r)h Fo(\001)f Fq(z)1029 1083 y Fp(2)1057 1100 y Fr(+)g Fq(z)1119 1083 y Fp(3)1138 1100 y Fq(f)t Fr(\()p Fq(z)r Fr(\))p Fq(;)356 b Fr(\(3)p Fq(:)p Fr(5\))249 1181 y(with)16 b Fq(z)i Fr(=)e(\()p Fq(x=r)q Fr(\))528 1153 y Fm(1)537 1162 y Fl(=)546 1170 y Fm(2)565 1181 y Fr(,)g Fq(f)21 b Fo(2)15 b Fq(H)715 1166 y Fp(1)733 1181 y Fr(,)i(and)f Fq(r)h Fr(is)g(small,)d Fq(T)22 b Fr(induces)c(again)d(an)h(op)q(erator)1597 1170 y(~)1589 1181 y Fq(T)22 b Fr(of)249 1231 y(whic)o(h)16 b Fq(f)k Fr(is)c(its)g(\014xed)g(p)q(oin)o(t.)24 b(After)16 b(this,)g(w)o(e)g (just)g(pro)q(ceed)i(in)d(a)h(similar)d(manner)i(to)249 1280 y(the)g(previous)f(case.)249 1451 y Ft(The)22 b(Initial)f(V)-6 b(alue)22 b(Problem)g(at)h(In\014nit)n(y)249 1550 y Fr(This)14 b(is)610 1600 y Fq(u)634 1583 y Fn(00)655 1600 y Fr(\()p Fq(x)p Fr(\))d(=)h Fq(x)790 1583 y Fn(\000)816 1570 y Fm(1)825 1579 y Fl(=)834 1587 y Fm(2)853 1600 y Fq(u)877 1570 y Fm(3)885 1579 y Fl(=)895 1587 y Fm(2)913 1600 y Fr(\()p Fq(x)p Fr(\))28 b Fq(u)p Fr(\()p Fo(1)p Fr(\))11 b(=)h(0)27 b Fq(b)1216 1606 y Fp(1)1246 1600 y Fr(=)12 b Fq(b)275 b Fr(\(3)p Fq(:)p Fr(6\))249 1675 y(where)16 b(the)g(last)e(condition)h(is)g(in)o(terpreted)h(in)f(the)g(sense)i(of) d(\(3.2\).)21 b(The)15 b(solution)f(will)249 1725 y(b)q(e)h(written)f (as)654 1784 y Fq(u)p Fr(\()p Fq(x)p Fr(\))e(=)794 1756 y(144)p 794 1774 63 2 v 804 1812 a Fq(x)828 1800 y Fp(3)869 1750 y Fk(\000)888 1784 y Fr(1)d(+)g Fq(b)e(x)1008 1766 y Fn(\000)p Fj(\013)1066 1784 y Fr(+)j Fq(z)1129 1766 y Fp(2)1148 1784 y Fq(f)t Fr(\()p Fq(z)r Fr(\))1225 1750 y Fk(\001)1252 1784 y Fq(;)319 b Fr(\(3)p Fq(:)p Fr(7\))249 1875 y(where)15 b Fq(f)h Fo(2)c Fq(H)483 1860 y Fp(1)501 1875 y Fr(,)h Fq(z)h Fr(=)e Fq(R)635 1860 y Fj(\013)658 1875 y Fq(x)682 1860 y Fn(\000)p Fj(\013)732 1875 y Fr(,)h(for)h(some)f Fq(R)g Fr(large.)18 b(If)c(w)o(e)g(set)737 1966 y Fq(T)761 1972 y Fp(1)780 1966 y Fr(\()p Fq(f)t Fr(\))e(=)g Fq(bR)942 1949 y Fn(\000)p Fj(\013)991 1966 y Fq(z)g Fr(+)d Fq(z)1084 1949 y Fp(2)1103 1966 y Fq(f)t Fr(\()p Fq(z)r Fr(\))802 2065 y Fq(T)826 2071 y Fp(2)845 2065 y Fr(\()p Fq(g)q Fr(\))j(=)g(\(1)d(+)g Fq(g)q Fr(\))1078 2036 y Fm(3)1088 2044 y Fl(=)1098 2052 y Fm(2)484 2173 y Fq(T)508 2179 y Fp(3)527 2173 y Fr(\()p Fq(g)q Fr(\))j(=)g(12)699 2121 y Fn(1)685 2134 y Fk(X)685 2222 y Fj(n)p Fp(=2)849 2145 y Fq(a)871 2151 y Fj(n)893 2145 y Fq(z)914 2130 y Fj(n)p Fn(\000)p Fp(2)p 758 2164 311 2 v 758 2202 a Fr(\()p Fq(n\013)d Fr(+)h(3\)\()p Fq(n\013)f Fr(+)g(4\))1074 2173 y Fq(;)48 b(g)q Fr(\()p Fq(z)r Fr(\))12 b(=)1266 2134 y Fk(X)1264 2223 y Fj(n)p Fn(\025)p Fp(0)1334 2173 y Fq(a)1356 2179 y Fj(n)1379 2173 y Fq(z)1400 2156 y Fj(n)1422 2173 y Fq(:)249 2300 y Fr(A)j(solution)e(of)h(\(3.6\))g (comes)g(from)e(an)i Fq(f)20 b Fr(satisfying)1112 2289 y(~)1104 2300 y Fq(T)6 b(f)17 b Fr(=)c Fq(f)t Fr(,)i(with)1370 2289 y(~)1362 2300 y Fq(T)j Fr(=)13 b Fq(T)1473 2306 y Fp(3)1501 2300 y Fo(\016)d Fq(T)1556 2306 y Fp(2)1584 2300 y Fo(\016)f Fq(T)1638 2306 y Fp(1)1657 2300 y Fr(.)249 2350 y(As)14 b(b)q(efore,)f(w)o(e)g(w)o(ould)g(lik)o(e)f(to)h(go)f (ahead)i(and)e(c)o(hec)o(k)i(that)g(indeed)1343 2339 y(~)1335 2350 y Fq(T)19 b Fr(is)13 b(a)g(con)o(traction,)249 2399 y(but)k(the)g(problem)e(here)i(is)f(considerably)h(more)e(subtle)i (than)f(in)g(the)h(previous)g(cases,)249 2449 y(due)f(to)f(the)h(fact)f (that)g Fq(T)652 2455 y Fp(3)686 2449 y Fr(do)q(es)h(not)f(scale)h (with)f Fq(R)p Fr(.)21 b(As)16 b(a)f(consequence,)j(con)o(traction)249 2499 y(prop)q(erties)e(of)501 2489 y(~)493 2499 y Fq(T)k Fr(either)15 b(hold)f(or)g(don't,)g(and)g(taking)f(large)i Fq(R)f Fr(w)o(on't)f(help)i(m)o(uc)o(h.)j(W)m(e)p eop 13 12 bop 249 275 a Fc(Interval)19 b(A)o(rithmetic)e(in)h(Quantum)h(Me) n(chanics)448 b Fi(13)249 516 y Fr(are)15 b(luc)o(ky)m(,)f(ho)o(w)o(ev) o(er,)g(that)h(the)g(norm)e(of)h Fq(T)953 522 y Fp(2)987 516 y Fr(is)g(essen)o(tially)1234 499 y Fp(3)p 1234 506 17 2 v 1234 530 a(2)1255 516 y Fr(,)h(and)f(that)h(the)g(norm)e(of)249 565 y Fq(T)273 571 y Fp(3)306 565 y Fr(is)h(essen)o(tially)897 604 y(12)p 767 623 303 2 v 767 661 a(\(2)p Fq(\013)9 b Fr(+)g(3\)\(2)p Fq(\013)g Fr(+)g(4\))1086 632 y Fq(<)1135 616 y Fp(1)p 1135 623 17 2 v 1135 647 a(2)249 739 y Fr(whic)o(h)i(sa)o (ys)g(that)g(the)g(Lipsc)o(hitz)h(norm)d(of)937 728 y(~)929 739 y Fq(T)17 b Fr(will)10 b(appro)o(ximately)e(b)q(e)1374 722 y Fp(3)p 1374 729 V 1374 753 a(4)1396 739 y Fr(.)17 b(The)11 b(follo)o(wing)249 788 y(result)k(is)e(th)o(us)i(crucial.)249 888 y Fg(Lemma)k(3.4.)27 b Fs(Put)17 b Fq(\014)i Fr(=)d(0)p Fq(:)p Fr(3)p Fs(.)26 b(Assume)18 b(that)1032 853 y Fk(\014)1032 878 y(\014)1044 877 y Fr(\026)1046 888 y Fq(b)1064 853 y Fk(\014)1064 878 y(\014)1094 888 y Fr(=)e Fq(R)1174 873 y Fn(\000)p Fj(\013)1223 888 y Fo(j)p Fq(b)p Fo(j)g(\024)g Fr(0)p Fq(:)p Fr(23)p Fs(.)25 b(Then)1561 878 y Fr(~)1553 888 y Fq(T)e Fs(is)18 b(a)249 945 y(c)n(ontr)n(action)d(in)g Fq(B)r Fr(\(0)p Fq(;)7 b(\014)r Fr(\))p Fs(,)15 b(and)h Fo(j)-7 b(j)780 934 y Fr(~)773 945 y Fq(T)5 b Fo(j)-7 b(j)819 951 y Fp(Lip)880 945 y Fo(\024)12 b Fr(0)p Fq(:)p Fr(8652)p Fs(.)249 1045 y Fr(Once)j(this)e(is)h(settled,)g(ev)o (erything)g(pro)q(ceeds)h(as)e(b)q(efore,)h(and)g(w)o(e)f(can)h(easily) f(conclude)249 1094 y(the)i(follo)o(wing:)249 1194 y Fg(Algorithm)k(3.5.)33 b Fs(Given)20 b Fq(b)737 1179 y Fn(\003)775 1194 y Fs(\(interval\))f(and)h Fq(R)g Fs(\(r)n(epr)n (esentable\),)f(we)g(pr)n(o)n(duc)n(e)h Fo(U)1650 1200 y Fp(1)249 1244 y Fs(such)15 b(that,)g(for)f(any)i Fq(b)11 b Fo(2)g Fq(b)673 1229 y Fn(\003)692 1244 y Fs(,)k(the)g(solution)g Fq(u)g Fs(of)f(\(3.6\))h(is)g(given)g(by)505 1355 y Fq(y)q Fr(\()p Fq(x)p Fr(\))d(=)643 1327 y(144)p 643 1345 63 2 v 653 1383 a Fq(x)677 1371 y Fp(3)717 1321 y Fk(\000)736 1355 y Fr(1)d(+)g Fq(bx)849 1338 y Fn(\000)p Fj(\013)908 1355 y Fr(+)g Fq(z)970 1338 y Fp(2)989 1355 y Fq(f)t Fr(\()p Fq(z)r Fr(\))1066 1321 y Fk(\001)1093 1355 y Fq(;)91 b(z)14 b Fr(=)e Fq(R)1305 1338 y Fj(\013)1328 1355 y Fq(x)1352 1338 y Fn(\000)p Fj(\013)1402 1355 y Fq(;)249 1458 y Fs(with)19 b Fq(f)24 b Fo(2)19 b(U)461 1464 y Fp(1)480 1458 y Fs(.)32 b(Her)n(e,)19 b Fo(U)668 1464 y Fp(1)706 1458 y Fs(dep)n(ends)i(only)e(on)h Fq(b)1042 1443 y Fn(\003)1061 1458 y Fs(,)g(i.e.,)f(it)g(is)g(indep)n(endent)i (of)e(which)249 1507 y(p)n(articular)10 b Fq(b)g Fs(in)h Fq(b)528 1492 y Fn(\003)558 1507 y Fs(we)f(ar)n(e)h(c)n(onsidering.)17 b(In)11 b(p)n(articular,)g(we)f(c)n(an)h(pr)n(o)n(duc)n(e)g(two)f (intervals)249 1557 y Fq(u)273 1542 y Fn(\003)273 1568 y Fp(0)312 1557 y Fs(and)20 b Fq(u)421 1542 y Fn(\003)421 1568 y Fp(1)460 1557 y Fs(such)h(that,)g(if)e Fq(u)h Fs(is)f(the)h(solution)g(to)g(\(3.6\),)h(we)f(have)g Fq(u)p Fr(\()p Fq(R)p Fr(\))h Fo(2)f Fq(u)1564 1542 y Fn(\003)1564 1568 y Fp(0)1603 1557 y Fs(and)249 1607 y Fq(u)273 1592 y Fn(0)284 1607 y Fr(\()p Fq(R)p Fr(\))12 b Fo(2)f Fq(u)423 1592 y Fn(\003)423 1617 y Fp(1)442 1607 y Fs(.)249 1777 y Ft(The)22 b(Boundary)j(V)-6 b(alue)23 b(Problem)249 1877 y Fr(Next)15 b(w)o(e)f(discuss)h(ho)o(w)e(to)h(solv) o(e)582 1968 y Fq(u)606 1951 y Fn(00)627 1968 y Fr(\()p Fq(x)p Fr(\))e(=)g Fq(x)763 1951 y Fn(\000)789 1939 y Fm(1)797 1947 y Fl(=)807 1955 y Fm(2)825 1968 y Fq(u)849 1939 y Fm(3)858 1947 y Fl(=)868 1955 y Fm(2)886 1968 y Fr(\()p Fq(x)p Fr(\))28 b Fq(u)p Fr(\(0\))11 b(=)h(1)27 b Fq(u)p Fr(\()p Fo(1)p Fr(\))11 b(=)h(0)p Fq(:)249 2060 y Fr(Unlik)o(e)i(the)g(previous)h(section,)f(w)o(e)g(will)f(a)o(v)o (oid)g(a)h(serious)g(form)o(ulation)d(of)j(the)h(metho)q(d)249 2109 y(\(whic)o(h)k(b)q(ecomes)g(a)g(little)f(tec)o(hnical\),)i(and)f (instead)h(w)o(e)f(concen)o(trate)h(on)f(the)h(main)249 2159 y(simple)12 b(ideas.)249 2259 y(What)k(w)o(e)h(will)e(b)q(e)h (using)h(here)g(is)f(a)g(rigorous)h(v)n(arian)o(t)e(of)h(the)h Fs(sho)n(oting)f Fr(metho)q(d:)22 b(w)o(e)249 2309 y(will)13 b(b)q(egin)i(making)d(a)j(heuristic)g(guess)h(for,)e(sa)o(y)m(,)g Fq(y)1081 2294 y Fn(0)1093 2309 y Fr(\(0\),)g(and)h(w)o(e)g(will)e (then)i(follo)o(w)e(the)249 2358 y(solution)h(forw)o(ard)h(un)o(til)f (it)g(violates)h(one)g(of)f(the)i(prop)q(erties)g(w)o(e)f(kno)o(w)g(to) g(b)q(e)g(true)h(for)249 2408 y(the)h(true)f(Thomas{F)m(ermi)d (function:)21 b(this)16 b(will)f(determine)g(whether)j(our)d(c)o(hoice) i(w)o(as)249 2458 y(an)g(upp)q(er)h(b)q(ound)f(or)g(a)f(lo)o(w)o(er)h (b)q(ound)g(for)f(the)i(actual)f(v)n(alue)f(for)g Fq(y)1363 2443 y Fn(0)1376 2458 y Fr(\(0\).)27 b(This)17 b(yields)249 2508 y(b)q(ounds)f(for)g Fo(\000)p Fq(w)523 2514 y Fp(0)556 2508 y Fr(=)f Fq(y)624 2493 y Fn(0)637 2508 y Fr(\(0\).)24 b(W)m(e)15 b(then)i(carry)f(out)g(a)f(similar)f(pro)q(cedure)j(at)f Fo(1)p Fr(.)23 b(Let's)249 2558 y(get)14 b(in)o(to)f(a)h(little)f(more) g(detail:)p eop 14 13 bop 249 275 a Fi(14)1110 b Fh(Chapter)18 b(1)249 516 y Fr(Pic)o(k)h(a)f(v)n(alue)g Fq(w)h Fr(and,)g(with)g(the)g(aid)f (of)g(the)h(algorithms)d(in)i(the)i(previous)e(section,)249 565 y(obtain)13 b(b)q(ounds)i(for)e(the)i(solution)e(of)566 650 y Fq(u)590 633 y Fn(00)611 650 y Fr(\()p Fq(x)p Fr(\))e(=)h Fq(x)746 633 y Fn(\000)772 621 y Fm(1)781 629 y Fl(=)790 637 y Fm(2)809 650 y Fq(u)833 621 y Fm(3)841 629 y Fl(=)851 637 y Fm(2)869 650 y Fr(\()p Fq(x)p Fr(\))28 b Fq(u)p Fr(\(0\))11 b(=)h(1)28 b Fq(u)1158 633 y Fn(0)1169 650 y Fr(\(0\))12 b(=)f Fo(\000)p Fq(w)q(;)249 735 y Fr(around)k(small)d (in)o(terv)n(als)j(co)o(v)o(ering,)f(sa)o(y)m(,)g([)t(0)t Fq(;)s(M)9 b Fr(].)20 b(If,)15 b(at)f(some)g(p)q(oin)o(t,)g(w)o(e)h (can)g(deduce)249 785 y(using)e(these)h(b)q(ounds)g(that)f Fq(u)g Fr(is)f(increasing,)h(a)g(trivial)f(ODE)h(analysis)f(sho)o(ws)i (that)f(this)249 835 y(c)o(hoice)i Fq(w)h Fr(is)e(a)g Fs(lower)g Fr(b)q(ound)h(for)f Fq(w)830 841 y Fp(0)861 835 y Fr(=)f Fo(\000)p Fq(y)959 820 y Fn(0)972 835 y Fr(\(0\).)20 b(If)15 b(ho)o(w)o(ev)o(er,)f(the)i(solution)d(v)n (anished,)249 885 y(the)j(same)f(trivial)g(ODE)h(analysis)f(yields)g (that)h Fq(w)1060 891 y Fp(0)1093 885 y Fq(<)f(w)q Fr(.)23 b(The)17 b(problem)d(here)j(is)f(that)249 935 y(the)j(computer)f(alone) f(w)o(on't)g(ev)o(er)i(b)q(e)g(able)f(to)g(tell)g(us)g(that)g(the)h (solution)e(v)n(anishes,)249 984 y(since)c(it)f(needs)i(it)e(to)h(b)q (e)g(p)q(ositiv)o(e)f(to)g(p)q(erform)g(the)h(di\013eren)o(t)h (algorithms)c(ab)q(o)o(v)o(e:)17 b(so,)12 b(in)249 1034 y(practice,)g(ev)o(en)g(if)e(the)i(solution)e(w)o(ould)g(v)n(anish)g (at)h(some)f(p)q(oin)o(t,)h(the)g(computer)g(will)e(just)249 1084 y(require,)18 b(at)e(eac)o(h)h(step,)h(smaller)d(and)i(smaller)e (in)o(terv)n(als)h(to)h(propagate)f(the)i(solution)249 1134 y(forw)o(ard,)f(without)g(ev)o(er)h(giving)e(negativ)o(e)h(b)q (ounds)g(for)g Fq(u)p Fr(\()p Fq(x)p Fr(\).)28 b(Hence,)19 b(the)e(follo)o(wing)249 1184 y(simple)12 b(lemma)f(is)j(helpful:)249 1283 y Fg(Lemma)h(3.6.)k Fs(L)n(et)c Fq(u)f Fs(b)n(e)h(the)g(solution)g (of)f Fr(\(3.3\),)f Fs(with)h Fq(u)1171 1289 y Fp(1)1201 1283 y Fq(<)e Fr(0)p Fs(.)18 b(If)864 1377 y Fr(2)p Fq(u)909 1346 y Fm(5)917 1355 y Fl(=)927 1363 y Fm(2)909 1388 y Fp(0)p 864 1396 82 2 v 874 1441 a Fq(x)898 1414 y Fm(1)907 1418 y Fl(=)916 1426 y Fm(2)898 1452 y Fp(0)961 1405 y Fo(\024)12 b Fq(u)1029 1388 y Fp(2)1029 1416 y(1)1048 1405 y Fq(;)523 b Fr(\(3)p Fq(:)p Fr(8\))249 1520 y Fs(then,)16 b(ther)n(e)f(exists)g(a)g(p)n(oint)h Fq(t)d(>)f(x)811 1526 y Fp(0)845 1520 y Fs(such)k(that)f Fq(u)g Fs(c)n(an)h(b)n(e)g (extende)n(d)g(as)g(a)f(wel)r(l{de\014ne)n(d)249 1570 y(p)n(ositive)g(solution)g(of)f(the)h(ODE)h(to)e Fr([)t Fq(x)869 1576 y Fp(0)892 1570 y Fq(;)s(t)t Fr(\))h Fs(and,)h (furthermor)n(e,)d Fr(inf)1343 1577 y Fj(x)p Fn(2)p Fp(\()t Fj(x)1420 1581 y Fm(0)1440 1577 y Fj(;)t(t)t Fp(\))1493 1570 y Fq(u)p Fr(\()p Fq(x)p Fr(\))e(=)h(0)p Fs(.)249 1670 y Fr(As)g(a)f(consequence,)j(the)e(w)o(ord)g Fs(vanishes)g Fr(used)g(ab)q(o)o(v)o(e)f(is)g(simply)f(replaced)i(b)o(y)g(the)g(more) 249 1719 y(feasible)i(prop)q(ert)o(y)g(\(3.8\),)f(and)h(w)o(e)g(obtain) f(the)i(follo)o(wing)249 1819 y Fg(Algorithm)g(3.7.)24 b Fs(Given)17 b(a)f(r)n(epr)n(esentable)g Fq(w)q Fs(,)g(we)h(c)n (onstruct)f(an)h(algorithm)f(that,)g(if)249 1869 y(suc)n(c)n(essful,)f (wil)r(l)f(indic)n(ate)i(whether)f Fq(w)e(<)g(w)958 1875 y Fp(0)992 1869 y Fs(or)i Fq(w)f(>)e(w)1164 1875 y Fp(0)1183 1869 y Fs(.)20 b(Henc)n(e,)c(given)g(appr)n(opriate)249 1919 y Fq(x)273 1925 y Fj(i)298 1919 y Fo(2)11 b(R)k Fs(,)g(we)f(c)n(an)i(pr)n(o)n(duc)n(e)f Fq(y)724 1904 y Fn(\003)723 1929 y Fj(i)759 1919 y Fs(and)g Fq(y)860 1904 y Fn(0)873 1898 y(\003)873 1929 y Fj(i)903 1919 y Fo(2)d(I)s Fs(,)i Fq(i)e Fr(=)g(0)p Fq(;)7 b(:)g(:)g(:)t(;)g(m)p Fs(,)15 b(such)g(that)545 2004 y Fq(y)q Fr(\()p Fq(x)606 2010 y Fj(i)621 2004 y Fr(\))c Fo(2)h Fq(y)709 1986 y Fn(\003)708 2014 y Fj(i)729 2004 y Fq(;)91 b(y)853 1986 y Fn(0)865 2004 y Fr(\()p Fq(x)905 2010 y Fj(i)919 2004 y Fr(\))12 b Fo(2)f Fq(y)1007 1986 y Fn(0)1019 1981 y(\003)1019 2014 y Fj(i)1039 2004 y Fq(;)91 b(i)12 b Fr(=)g(0)p Fq(;)7 b(:)g(:)g(:)t(;)g(m:)249 2188 y Fr(W)m(e)14 b(no)o(w)f(do)h(the)g(same) f(at)h(in\014nit)o(y)m(.)249 2288 y Fg(Algorithm)k(3.8.)32 b Fs(Given)19 b(a)h(r)n(epr)n(esentable)e Fq(b)p Fs(,)h(and)h(assuming) g(b)n(ounds)g(for)e Fq(w)1571 2294 y Fp(0)1589 2288 y Fs(,)i(we)249 2338 y(c)n(onstruct)12 b(an)g(algorithm)f(that,)i(if)e (suc)n(c)n(essful,)h(wil)r(l)e(indic)n(ate)i(whether)f Fq(b)h(<)g(b)1466 2344 y Fp(1)1496 2338 y Fs(or)f Fq(b)h(>)g(b)1638 2344 y Fp(1)1656 2338 y Fs(.)249 2387 y(Next,)17 b(we)g(use)g(these)g (b)n(ounds)g(to)g(pr)n(o)n(duc)n(e)g Fq(x)974 2393 y Fj(i)1003 2387 y Fo(2)d(R)p Fs(,)k(and)f Fq(y)1214 2372 y Fn(\003)1213 2398 y Fj(i)1234 2387 y Fs(,)g Fq(y)1285 2372 y Fn(0)1298 2366 y(\003)1298 2398 y Fj(i)1332 2387 y Fo(2)d(I)s Fs(,)j Fq(i)e Fr(=)h(1)p Fq(;)7 b(:)g(:)g(:)t(;)g(m)p Fs(,)249 2437 y(such)15 b(that)545 2487 y Fq(y)q Fr(\()p Fq(x)606 2493 y Fj(i)621 2487 y Fr(\))c Fo(2)h Fq(y)709 2470 y Fn(\003)708 2497 y Fj(i)729 2487 y Fq(;)91 b(y)853 2470 y Fn(0)865 2487 y Fr(\()p Fq(x)905 2493 y Fj(i)919 2487 y Fr(\))12 b Fo(2)f Fq(y)1007 2470 y Fn(0)1019 2464 y(\003)1019 2497 y Fj(i)1039 2487 y Fq(;)91 b(i)12 b Fr(=)g(1)p Fq(;)7 b(:)g(:)g(:)t(;)g(m:)p eop 15 14 bop 249 275 a Fc(Interval)19 b(A)o(rithmetic)e(in)h(Quantum)h(Me) n(chanics)448 b Fi(15)249 516 y Fr(The)16 b(details)e(are)i(easy)m(.)21 b(In)15 b(fact)h(w)o(e)f(can)g(use)h(the)g(extra)f(information)d(giv)o (en)j(b)o(y)g(Algo-)249 565 y(rithm)h(3.3)g(to)h(detect)i(easily)e (when)h(the)g(test)g(solution)e(go)q(es)i(b)q(ey)o(ond)g(b)q(ounds.)28 b(Note)249 615 y(that)16 b(afterw)o(ards)g(w)o(e)f(can)h(tak)o(e)g(the) g(b)q(est)g(of)f(the)h(t)o(w)o(o)f(sources)i(of)e(information)e(ab)q (out)249 665 y(the)i(Thomas{F)m(ermi)10 b(function,)j(coming)f(from)g (0)i(or)g(from)e(in\014nit)o(y)m(.)249 765 y(A)i(delicate)h(p)q(oin)o (t)f(here)h(is)g(the)g(c)o(hoice)f(of)g(the)h Fq(x)1028 771 y Fj(i)1056 765 y Fr(ab)q(o)o(v)o(e,)f(but)g(w)o(e)h(refer)g(the)g (in)o(terested)249 814 y(reader)g(to)f([21)o(])f(for)h(details.)249 914 y(Note)k(also)f(that)g(the)i(re\014ned)f(b)q(ounds)g(for)f Fq(u)h Fr(giv)o(en)f(b)o(y)g(Algorithm)e(3.8,)i(allo)o(w)f(us)i(to)249 964 y(sharp)q(en)13 b(our)e(b)q(ounds)h(for)f Fq(w)707 970 y Fp(0)725 964 y Fr(.)17 b(These)c(impro)o(v)o(ed)c(b)q(ounds)j (for)f Fq(w)1283 970 y Fp(0)1313 964 y Fr(will)f(in)h(turn)g(impro)o(v) o(e)249 1014 y(on)17 b(the)h(older)f(b)q(ounds)h(for)f Fq(b)725 1020 y Fp(1)744 1014 y Fr(,)g(and)g(w)o(e)h(are)g(all)e(set)i (for)f(a)g(recursiv)o(e)i(algorithm,)c(the)249 1064 y(result)g(of)e (whic)o(h)h(is)g(quite)f(precise)j(b)q(ounds)e(for)g(the)g(Thomas{F)m (ermi)d(function:)249 1163 y Fg(Algorithm)i(3.9.)19 b Fs(We)c(c)n(an)g(pr)n(o)n(duc)n(e)g Fq(x)898 1169 y Fj(i)912 1163 y Fs(,)g Fq(r)959 1169 y Fj(i)984 1163 y Fo(2)c(R)p Fs(,)k(and)557 1254 y Fo(U)583 1260 y Fj(i)597 1254 y Fr(\()p Fq(I)634 1237 y Fj(i)631 1265 y Fp(0)650 1254 y Fq(;)7 b(:)g(:)g(:)t(;)g(I)763 1237 y Fj(i)760 1265 y(N)792 1254 y Fr(;)g Fq(C)841 1260 y Fj(h;i)883 1254 y Fq(;)g(C)932 1260 y Fj(g)q(;i)972 1254 y Fq(;)g Fr(2\))p Fq(;)90 b(i)12 b Fr(=)g(1)p Fq(;)7 b(:)g(:)g(:)e(;)i(m;)249 1346 y Fs(such)15 b(that)418 1437 y Fq(y)q Fr(\()p Fq(x)479 1443 y Fj(i)503 1437 y Fr(+)10 b Fq(z)h Fo(\001)e Fq(r)615 1443 y Fj(i)628 1437 y Fr(\))j Fo(2)f(U)721 1443 y Fj(i)735 1437 y Fr(\()p Fq(I)772 1420 y Fj(i)769 1447 y Fp(0)788 1437 y Fq(;)c(:)g(:)g(:)e(;)i(I)902 1420 y Fj(i)899 1447 y(N)930 1437 y Fr(;)g Fq(C)979 1443 y Fj(h;i)1021 1437 y Fq(;)g(C)1070 1443 y Fj(g)q(;i)1110 1437 y Fq(;)g Fr(2\))p Fq(;)91 b(i)12 b Fr(=)f(1)p Fq(;)c(:)g(:)g(:)e(;)i(m;)249 1528 y Fs(and)503 1549 y Fj(m)495 1562 y Fk([)491 1650 y Fj(i)p Fp(=1)545 1601 y Fr(\()t Fq(x)589 1607 y Fj(i)612 1601 y Fo(\000)i Fq(r)672 1607 y Fj(i)690 1601 y Fq(;)s(x)729 1607 y Fj(i)752 1601 y Fr(+)h Fq(r)813 1607 y Fj(i)830 1601 y Fr(\))i(=)g(\()t Fq(x)946 1607 y Fp(1)974 1601 y Fo(\000)d Fq(r)1034 1607 y Fp(1)1057 1601 y Fq(;)s(x)1096 1607 y Fj(m)1137 1601 y Fr(+)g Fq(r)1197 1607 y Fj(m)1233 1601 y Fr(\))j Fo(\032)f Fr(\()t(0)t Fq(;)t Fo(1)t Fr(\))249 1783 y(What)e(follo)o(ws)f(is)h(some)f(rigorous)h(n)o(umerical)f(v)n (alues)h(w)o(e)h(obtained)f(for)g(sev)o(eral)g(Thomas{)249 1833 y(F)m(ermi)j(parameters.)249 1932 y Fg(Lemma)j(3.10.)k Fs(The)c(fol)r(lowing)f(ine)n(qualities)g(hold:)536 2024 y Fr(1)p Fq(:)p Fr(58807102261)o(1278)8 b Fo(\024)k Fq(w)965 2030 y Fp(0)995 2024 y Fo(\024)f Fr(1)p Fq(:)p Fr(58807102261147)o(1) 592 2115 y Fo(\000)p Fr(13)p Fq(:)p Fr(2709738479)e Fo(\025)j Fq(b)959 2121 y Fp(1)989 2115 y Fo(\025)f(\000)p Fr(13)p Fq(:)p Fr(2709738482)536 2190 y(0)p Fq(:)p Fr(4863485380)o(4359)o(4)e Fo(\024)j Fr(\012)965 2173 y Fp(2)965 2200 y Fj(c)995 2190 y Fo(\024)g Fr(0)p Fq(:)p Fr(4863485380)o(4686)o(9)604 2264 y(2)p Fq(:)p Fr(104025280219)c Fo(\024)k Fq(r)960 2270 y Fj(c)988 2264 y Fo(\024)g Fr(2)p Fq(:)p Fr(104025280274)p eop 16 15 bop 249 275 a Fi(16)1110 b Fh(Chapter)18 b(1)249 516 y Ft(4)69 b(THE)23 b(APERIODICITY)e(INEQUALITY)249 615 y Fr(Once)e(w)o(e)g(ha)o(v)o(e)f(appropriate)g(b)q(ounds)h(for)e (the)i(Thomas{F)m(ermi)c(function,)j(the)h(next)249 665 y(step)c(is)f(to)f(use)i(them)e(to)h(pro)o(v)o(e)g(Theorem)f(0.1.)249 765 y(There)20 b(is)f(the)g(straigh)o(tforw)o(ard)f(approac)o(h)h(of)f (simply)f(computing)g(expression)j(\(2.1\))249 814 y(for)d(a)f (\014nite)h(collection)g(of)f(small)f(in)o(terv)n(als)h(\012)1011 799 y Fn(\003)1011 825 y Fj(i)1047 814 y Fr(whic)o(h)h(co)o(v)o(er)g (all)f(of)g(\()t(0)t Fq(;)t Fr(\012)t(\))1501 820 y Fj(c)1518 814 y Fr(.)27 b(As)17 b(w)o(e)249 864 y(will)g(see)k(here,)f(this)f (will)f(w)o(ork)g(with)h(more)f(or)h(less)g(trivial)f(mo)q (di\014cations)f(only)h(for)249 914 y(for)e(in)o(terv)n(als)g(whic)o(h) h(are)g(not)f(near)h(0)g(or)f(\012)970 920 y Fj(c)987 914 y Fr(;)h(in)o(terv)n(als)g(near)g(0)f(will)f(require)i(a)g(more)249 964 y(sophisticated)f(approac)o(h)f(to)g(computing)f(\(2.1\),)g(but)h (near)h(\012)1252 970 y Fj(c)1284 964 y Fr(w)o(e)f(will)f(ha)o(v)o(e)h (to)g(use)h(a)249 1014 y(completely)g(di\013eren)o(t)i(approac)o(h.)27 b(This)17 b(section)g(explains)g(the)g(analysis)g(in)o(v)o(olv)o(ed)e (in)249 1064 y(the)g(computation)d(of)h Fq(F)643 1048 y Fn(00)677 1064 y Fr(for)h(the)h(di\013eren)o(t)f(\012)1006 1048 y Fn(\003)1006 1074 y Fj(i)1025 1064 y Fr(.)249 1234 y Ft(Case)23 b(1)249 1333 y Fr(W)m(e)16 b(b)q(egin)h(with)f(the)h (computation)e(of)h Fq(F)937 1318 y Fn(00)958 1333 y Fr(\(\012)1004 1318 y Fn(\003)1023 1333 y Fr(\))g(for)h(\012)1152 1318 y Fn(\003)1187 1333 y Fr(su\016cien)o(tly)g(far)f(from)f(0)h(and) 249 1383 y(\012)279 1389 y Fj(c)296 1383 y Fr(.)i(Use)d(\(2.1\))e(to)h (write)g Fq(F)6 b Fr(\(\012\))11 b(=)h Fq(I)830 1389 y Fp(1)849 1383 y Fr(\(\012\))d(+)h Fq(I)980 1389 y Fp(2)999 1383 y Fr(\(\012\))f(+)h Fq(I)1130 1389 y Fp(3)1149 1383 y Fr(\(\012\))k(with)677 1506 y Fq(I)695 1512 y Fp(1)726 1506 y Fr(=)770 1449 y Fk(Z)811 1459 y Fj(b)793 1543 y(a)835 1472 y Fk(\000)854 1506 y Fq(u)p Fr(\()p Fq(r)q Fr(\))9 b Fo(\000)h Fr(\012)1011 1488 y Fp(2)1029 1472 y Fk(\001)1048 1480 y Fn(\000)1074 1467 y Fm(3)1083 1476 y Fl(=)1093 1484 y Fm(2)1118 1506 y Fq(y)q Fr(\()p Fq(r)q Fr(\))d Fq(dr)322 b Fr(\(4)p Fq(:)p Fr(1)p Fq(a)p Fr(\))388 1662 y Fq(I)406 1668 y Fp(2)437 1662 y Fr(=)16 b(lim)480 1689 y Fj(\016)q Fn(!)p Fp(0)553 1591 y Fk( )586 1605 y(Z)628 1616 y Fj(a)609 1700 y(r)624 1704 y Fm(1)641 1700 y Fp(\(\012\)+)p Fj(\016)741 1628 y Fk(\000)760 1662 y Fq(u)p Fr(\()p Fq(r)q Fr(\))9 b Fo(\000)h Fr(\012)917 1645 y Fp(2)935 1628 y Fk(\001)954 1636 y Fn(\000)980 1624 y Fm(3)989 1632 y Fl(=)999 1640 y Fm(2)1024 1662 y Fq(y)q Fr(\()p Fq(r)q Fr(\))d Fq(dr)k Fo(\000)e Fq(G)1230 1668 y Fp(1)1248 1662 y Fr(\(\012\))p Fq(\016)1330 1645 y Fn(\000)1356 1632 y Fm(1)1366 1641 y Fl(=)1375 1649 y Fm(2)1394 1591 y Fk(!)1566 1662 y Fr(\(4)p Fq(:)p Fr(1)p Fq(b)p Fr(\))379 1815 y Fq(I)397 1821 y Fp(3)427 1815 y Fr(=)16 b(lim)471 1842 y Fj(\016)q Fn(!)p Fp(0)544 1744 y Fk( )576 1759 y(Z)618 1769 y Fj(r)633 1773 y Fm(2)650 1769 y Fp(\(\012\))p Fn(\000)p Fj(\016)600 1853 y(b)750 1782 y Fk(\000)769 1815 y Fq(u)p Fr(\()p Fq(r)q Fr(\))10 b Fo(\000)f Fr(\012)926 1798 y Fp(2)945 1782 y Fk(\001)964 1790 y Fn(\000)990 1777 y Fm(3)999 1786 y Fl(=)1008 1794 y Fm(2)1033 1815 y Fq(y)q Fr(\()p Fq(r)q Fr(\))e Fq(dr)k Fo(\000)f Fq(G)1240 1821 y Fp(2)1258 1815 y Fr(\(\012\))p Fq(\016)1340 1798 y Fn(\000)1366 1786 y Fm(1)1375 1794 y Fl(=)1385 1802 y Fm(2)1403 1744 y Fk(!)1565 1815 y Fr(\(4)p Fq(:)p Fr(1)p Fq(c)p Fr(\))249 1925 y(with)k(the)g Fq(G)448 1931 y Fj(i)475 1925 y Fr(suc)o(h)h(that)f(the)g(limit)e(is)h (\014nite,)h(and)g Fq(a)d(<)h(b)i Fr(carefully)f(c)o(hosen.)249 2025 y(The)h(computation)e(of)h Fq(I)641 2031 y Fp(1)673 2025 y Fr(seems)h(to)g(b)q(e)g(rather)g(easy)h(at)e(\014rst)h(sigh)o (t:)k(w)o(e)c(just)g(break)g(up)616 2147 y Fq(I)634 2153 y Fp(1)665 2147 y Fr(=)728 2095 y Fj(n)708 2108 y Fk(X)711 2196 y Fj(i)p Fp(=1)775 2091 y Fk(Z)817 2101 y Fj(t)830 2105 y Fl(i)p Fm(+1)798 2185 y Fj(t)811 2189 y Fl(i)887 2113 y Fk(\000)906 2147 y Fq(u)p Fr(\()p Fq(r)q Fr(\))9 b Fo(\000)h Fr(\012)1063 2130 y Fp(2)1081 2113 y Fk(\001)1100 2121 y Fn(\000)1126 2109 y Fm(3)1135 2117 y Fl(=)1145 2125 y Fm(2)1170 2147 y Fq(y)q Fr(\()p Fq(r)q Fr(\))d Fq(dr)o(;)249 2275 y Fr(with)18 b Fq(t)363 2281 y Fp(1)401 2275 y Fr(=)h Fq(a)g Fr(and)f Fq(t)593 2281 y Fj(n)p Fp(+1)677 2275 y Fr(=)h Fq(b)p Fr(.)32 b(This)18 b(in)o(v)o(olv)o(es)g (only)f(basic)i(op)q(erations)g(with)f Fq(u)g Fr(and)249 2324 y(th)o(us)g(feasible)f(with)g(the)h(standard)g(in)o(terv)n(al)e (arithmetic)g(pac)o(k)n(ages.)28 b(A)18 b(deep)q(er)h(issue)249 2374 y(is)e(that)g(of)f(sp)q(eed:)25 b(it)16 b(turns)i(out)f(that)g(w)o (e)g(will)e(b)q(e)i(forced)h(to)e(tak)o(e)h(the)g(in)o(terv)n(als)g (\012)1650 2359 y Fn(\003)1650 2385 y Fj(i)249 2424 y Fr(v)o(ery)h(small,)d(and)i(th)o(us)h(need)h(to)e(compute)g Fq(I)992 2430 y Fp(1)1028 2424 y Fr(man)o(y)e(times;)j(in)f(fact,)h(so) f(man)o(y)f(that)249 2474 y(the)g(direct)h(computation)d(approac)o(h)h (fails)g(due)h(to)f(time)g(considerations,)h(and)f(w)o(e)h(are)249 2524 y(forced)f(to)e(use)i(a)f(certain)g(amoun)o(t)e(of)h(tric)o(k)o (ery)i(whic)o(h)f(w)o(e)g(will)e(omit)g(here.)p eop 17 16 bop 249 275 a Fc(Interval)19 b(A)o(rithmetic)e(in)h(Quantum)h(Me) n(chanics)448 b Fi(17)249 516 y Fr(The)12 b(computation)f(of)g Fq(I)636 522 y Fp(2)667 516 y Fr(and)h Fq(I)764 522 y Fp(3)794 516 y Fr(is)g(m)o(uc)o(h)f(easier:)18 b(w)o(e)12 b(use)h(Algorithm)c(3.2)i(to)h(compute)249 565 y Fo(U)275 571 y Fp(1)308 565 y Fr(suc)o(h)i(that)494 670 y Fq(u)p Fr(\()p Fq(x)p Fr(\))e(=)g(\012)660 653 y Fp(2)687 670 y Fr(+)e Fq(z)r(f)t Fr(\()p Fq(z)r Fr(\))p Fq(;)91 b(z)13 b Fr(=)1011 642 y Fq(x)c Fo(\000)h Fq(r)1105 648 y Fp(1)1123 642 y Fr(\(\012\))p 1011 660 175 2 v 1088 698 a Fq(r)1190 670 y(;)90 b(f)16 b Fo(2)11 b(U)1393 676 y Fp(1)1412 670 y Fq(;)159 b Fr(\(4)p Fq(:)p Fr(2\))249 769 y(where)15 b Fq(r)e Fo(\025)e(j)p Fq(a)e Fo(\000)h Fq(r)548 775 y Fp(1)566 769 y Fr(\(\012\))p Fo(j)k Fr(and)f Fo(U)760 775 y Fp(1)793 769 y Fr(is)g(uniform)f(for)i(all)e(\012)g Fo(2)f Fr(\012)1222 754 y Fn(\003)1241 769 y Fr(.)18 b(Therefore,)455 821 y Fk(Z)496 831 y Fj(a)478 915 y(r)493 919 y Fm(1)509 915 y Fp(\(\012\)+)p Fj(\016)610 843 y Fk(\000)629 877 y Fq(u)p Fr(\()p Fq(x)p Fr(\))9 b Fo(\000)g Fr(\012)789 860 y Fp(2)808 843 y Fk(\001)827 851 y Fn(\000)853 839 y Fm(3)862 847 y Fl(=)871 855 y Fm(2)897 877 y Fq(y)q Fr(\()p Fq(x)p Fr(\))e Fq(dx)k Fr(=)1082 821 y Fk(Z)1124 831 y Fj(a)1105 915 y(r)1120 919 y Fm(1)1137 915 y Fp(\(\012\)+)p Fj(\016)1237 877 y Fq(z)1258 860 y Fn(\000)1284 847 y Fm(3)1293 856 y Fl(=)1303 864 y Fm(2)1330 866 y Fr(~)1321 877 y Fq(f)5 b Fr(\()p Fq(z)r Fr(\))i Fq(dx;)249 1002 y Fr(for)17 b(a)g(new)h(function)618 991 y(~)609 1002 y Fq(f)5 b Fr(\()p Fq(z)r Fr(\))18 b(=)g Fq(y)q Fr(\()p Fq(x)p Fr(\))p Fq(f)856 987 y Fn(\000)882 975 y Fm(3)892 983 y Fl(=)902 991 y Fm(2)920 1002 y Fr(\()p Fq(z)r Fr(\),)g(that)g (can)g(also)e(b)q(e)i(enclosed)h(in)e(a)g(com-)249 1052 y(putable)d Fo(U)t Fr(.)k(Th)o(us,)c(if)577 1132 y(~)568 1143 y Fq(f)5 b Fr(\()p Fq(z)r Fr(\))12 b(=)703 1104 y Fk(X)702 1193 y Fj(n)p Fn(\025)p Fp(0)772 1143 y Fq(a)794 1149 y Fj(n)816 1143 y Fq(z)837 1126 y Fj(n)872 1143 y Fo(2)f(U)t Fr(\()p Fq(J)980 1149 y Fp(0)999 1143 y Fq(;)c(:)g(:)g(:)e(;)i(J)1115 1149 y Fj(N)1145 1143 y Fr(;)g Fq(C)1194 1149 y Fj(h)1215 1143 y Fq(;)g(C)1264 1149 y Fj(g)1283 1143 y Fr(;)g(1\))p Fq(;)249 1270 y Fr(w)o(e)14 b(see)h(that)387 1359 y Fk(Z)428 1369 y Fj(a)410 1454 y(r)425 1458 y Fm(1)441 1454 y Fp(\(\012\)+)p Fj(\016)542 1382 y Fk(\000)561 1416 y Fq(u)p Fr(\()p Fq(x)p Fr(\))9 b Fo(\000)g Fr(\012)721 1399 y Fp(2)740 1382 y Fk(\001)759 1390 y Fn(\000)785 1378 y Fm(3)794 1386 y Fl(=)803 1394 y Fm(2)829 1416 y Fq(y)q Fr(\()p Fq(x)p Fr(\))e Fq(dx)k Fr(=)h Fq(r)1047 1376 y Fk(X)1046 1465 y Fj(n)p Fn(\025)p Fp(0)1150 1388 y Fq(a)1172 1394 y Fj(n)p 1121 1406 103 2 v 1121 1446 a Fq(n)d Fo(\000)1201 1430 y Fp(1)p 1201 1437 17 2 v 1201 1461 a(2)1239 1416 y Fq(z)1260 1399 y Fj(n)p Fn(\000)1312 1389 y Fm(1)p 1312 1395 15 2 v 1312 1410 a(2)1333 1331 y Fk(\014)1333 1356 y(\014)1333 1380 y(\014)1333 1405 y(\014)1333 1430 y(\014)1333 1455 y(\014)1347 1341 y Fj(z)q Fp(=)1395 1325 y Fl(a)p Fa(\000)p Fl(r)1447 1331 y Fm(1)1463 1325 y(\(\012\))p 1395 1334 112 2 v 1443 1351 a Fl(r)1347 1482 y Fj(z)q Fp(=)1395 1471 y Fl(\016)p 1395 1476 15 2 v 1395 1492 a(r)1520 1416 y Fq(;)249 1560 y Fr(whic)o(h)14 b(implies)598 1648 y Fq(I)616 1654 y Fp(2)647 1648 y Fo(2)d Fq(r)f Fo(\001)752 1596 y Fj(N)737 1609 y Fk(X)736 1697 y Fj(n)p Fp(=0)839 1620 y Fq(J)862 1626 y Fj(n)p 810 1639 103 2 v 810 1679 a Fq(n)f Fo(\000)891 1662 y Fp(1)p 891 1669 17 2 v 891 1693 a(2)924 1590 y Fk(\022)960 1620 y Fq(a)g Fo(\000)h Fq(r)1052 1626 y Fp(1)1070 1620 y Fr(\(\012\))p 960 1639 173 2 v 1036 1677 a Fq(r)1137 1590 y Fk(\023)1168 1598 y Fj(n)p Fn(\000)1219 1588 y Fm(1)p 1219 1594 15 2 v 1219 1609 a(2)1250 1648 y Fo(\006)f Fq(\017;)249 1786 y Fr(with)14 b Fo(j)o Fq(\017)p Fo(j)d(\024)h Fq(r)466 1728 y Fk(\022)520 1769 y Fj(C)544 1773 y Fl(h)p 501 1777 82 2 v 501 1813 a Fj(N)s Fp(+)561 1797 y(1)p 561 1804 17 2 v 561 1828 a(2)597 1786 y Fr(+)d(2)p Fq(C)689 1792 y Fj(g)708 1728 y Fk(\023)739 1786 y Fr(.)18 b Fq(I)787 1792 y Fp(3)819 1786 y Fr(is)c(done)g(in)g(a)f(similar)f(manner.)249 1967 y Ft(Case)23 b(2)249 2067 y Fr(The)15 b(problem)f(no)o(w)g(is)h (that)g(when)g(\012)g(is)g(small,)d(\(4.2\))i(w)o(ould)g(ha)o(v)o(e)h (to)f(b)q(e)i(done)f(either)249 2117 y(v)o(ery)f(far)f(\(for)g Fq(I)500 2123 y Fp(3)519 2117 y Fr(\))g(of)g(v)o(ery)h(close)f(to)h(0)f (\(for)g Fq(I)966 2123 y Fp(2)985 2117 y Fr(\),)g(and)g(this)g(is)g (impractical.)j(The)e(answ)o(er)249 2166 y(to)g(this)g(degenerate)h (case)g(go)q(es)g(through)e(the)i(c)o(hange)f(of)f(v)n(ariables)h(giv)o (en)f(b)o(y)h Fq(r)q Fr(\()p Fq(t)p Fr(\),)f(the)249 2216 y(in)o(v)o(erse)f(of)f Fq(u)h Fr(around)f(0)h(for)f Fq(I)715 2222 y Fp(2)734 2216 y Fr(,)g(and)h(around)f(in\014nit)o(y)g (for)g Fq(I)1194 2222 y Fp(3)1213 2216 y Fr(;)h(w)o(e)g(will)e(concen)o (trate)k(here)249 2266 y(on)g Fq(I)325 2272 y Fp(2)344 2266 y Fr(.)k(This)13 b(c)o(hange)i(of)e(v)n(ariables)g(yields)249 2386 y Fq(I)267 2392 y Fp(2)297 2386 y Fr(=)361 2358 y Fq(d)p 346 2376 52 2 v 346 2414 a(d)p Fr(\012)410 2315 y Fk( )443 2386 y Fr(\012)480 2329 y Fk(Z)521 2340 y Fj(a)503 2424 y(r)518 2428 y Fm(1)534 2424 y Fp(\(\012\))593 2352 y Fk(\000)612 2386 y Fq(u)p Fr(\()p Fq(r)q Fr(\))c Fo(\000)h Fr(\012)769 2369 y Fp(2)787 2352 y Fk(\001)806 2360 y Fn(\000)832 2348 y Fm(1)841 2356 y Fl(=)851 2364 y Fm(2)890 2358 y Fq(dr)p 890 2376 42 2 v 901 2414 a(r)937 2315 y Fk(!)981 2386 y Fr(=)1045 2358 y Fq(d)p 1030 2376 52 2 v 1030 2414 a(d)p Fr(\012)1093 2315 y Fk( )1126 2386 y Fr(\012)1163 2329 y Fk(Z)1205 2340 y Fj(u)p Fp(\()p Fj(a)p Fp(\))1186 2424 y(\012)1210 2415 y Fm(2)1277 2352 y Fk(\000)1296 2386 y Fq(t)f Fo(\000)h Fr(\012)1392 2369 y Fp(2)1410 2352 y Fk(\001)1429 2360 y Fn(\000)1455 2348 y Fm(1)1464 2356 y Fl(=)1474 2364 y Fm(2)1499 2386 y Fq(w)q Fr(\()p Fq(t)p Fr(\))d Fq(dt)1621 2315 y Fk(!)1661 2386 y Fq(;)249 2508 y Fr(for)12 b Fq(w)q Fr(\()p Fq(t)p Fr(\))g(=)g Fq(r)465 2493 y Fn(0)476 2508 y Fr(\()p Fq(t)p Fr(\))p Fq(=r)q Fr(\()p Fq(t)p Fr(\))q(.)17 b(The)c(\014rst)h(step)f (is)g(therefore)h(to)e(compute)g Fq(w)q Fr(,)h(whic)o(h)f(hinges)h(on) 249 2558 y(the)i(follo)o(wing)c(in)o(v)o(ersion)i(algorithm.)p eop 18 17 bop 249 275 a Fi(18)1110 b Fh(Chapter)18 b(1)249 516 y Fg(Algorithm)9 b(4.1.)17 b Fs(Given)c Fq(N)j Fo(\025)c Fr(0)p Fs(,)g(we)f(pr)n(o)n(duc)n(e)h(intervals)f Fq(a)1218 501 y Fn(\003)1218 526 y Fp(2)1237 516 y Fq(;)c(:)g(:)g(:)e(;)i(a)1352 501 y Fn(\003)1352 527 y Fj(N)1383 516 y Fs(,)12 b(and)g(a)g(c)n (onstant)249 565 y Fq(C)279 571 y Fj(N)310 565 y Fs(,)j(such)g(that)455 622 y Fk(\014)455 647 y(\014)455 672 y(\014)455 697 y(\014)455 722 y(\014)468 695 y Fq(r)q Fr(\()p Fq(t)p Fr(\))10 b Fo(\000)f Fq(t)601 649 y Fk(\020)626 695 y Fr(1)g(+)714 643 y Fj(N)699 656 y Fk(X)698 743 y Fj(n)p Fp(=2)767 695 y Fq(a)789 701 y Fj(n)812 687 y Fr(\026)812 695 y Fq(t)836 678 y Fj(n)858 649 y Fk(\021)883 622 y(\014)883 647 y(\014)883 672 y(\014)883 697 y(\014)883 722 y(\014)909 695 y Fo(\024)i Fq(C)982 701 y Fj(N)1020 695 y Fq(t)f Fo(\001)1066 687 y Fr(\026)1065 695 y Fq(t)1090 678 y Fj(N)s Fp(+1)1163 695 y Fq(;)1225 687 y Fr(\026)1224 695 y Fq(t)i Fr(=)1294 636 y Fk(\022)1344 667 y Fq(t)p 1330 685 43 2 v 1330 723 a(x)1354 729 y Fp(0)1377 636 y Fk(\023)1408 633 y Fm(1)1417 641 y Fl(=)1426 649 y Fm(2)1452 695 y Fq(;)249 822 y Fs(for)i(c)n(onstants)i Fq(a)520 828 y Fj(i)545 822 y Fo(2)11 b Fq(a)606 807 y Fn(\003)606 833 y Fj(i)625 822 y Fs(,)k Fq(i)d Fr(=)g(2)p Fq(;)7 b(:)g(:)g(:)t(;)g(N)e Fs(,)14 b(and)i(for)e Fq(t)d Fo(\024)h Fq(\021)q Fs(.)249 922 y Fr(The)h(p)q(oin)o(t)f(to)g(realize) h(here)g(is)f(that)h(the)g(function)f Fq(u)p Fr(\()p Fq(x)p Fr(\))g(is)g(not)g(analytic)g(either)h(around)249 972 y(0)e(\(where)i(it)f(is)f(still)g(is)h(analytic)e(in)815 942 y Fo(p)p 849 942 24 2 v 849 972 a Fq(x)p Fr(\),)i(and)f(is)h(ev)o (en)g(w)o(orse)g(around)g(in\014nit)o(y)m(,)e(where)j(it)249 1028 y(is)g(a)f(com)o(bination)f(of)h Fq(x)629 1013 y Fn(\000)p Fp(2)686 1028 y Fr(and)h(p)q(o)o(w)o(ers)g(of)f Fq(x)972 1013 y Fp(\(7)p Fn(\000)1028 989 y(p)p 1055 989 34 2 v 24 x Fp(73)o(\))p Fj(=)p Fp(2)1136 1028 y Fr(.)18 b(In)13 b(spite)g(of)f(the)i(fact)f(that)g(this)249 1078 y(function)e(is)g(not)h(analytic,)e(w)o(e)i(can)f(in)g(principle)h (design)f(an)g(algorithm)e(for)i(this)h(case)g(b)o(y)249 1128 y(com)o(bining)i(di\013eren)o(t)j(manipulations)c(with)j(analytic) f(functions.)25 b(Ho)o(w)o(ev)o(er,)17 b(it)f(turns)249 1178 y(out)f(to)f(b)q(e)h(more)f(e\013ectiv)o(e)i(to)f(switc)o(h)g (here)g(to)g(gen)o(uine)g(real)f(v)n(ariable)g(theory)m(,)g(and)h(to) 249 1227 y(use)e(the)f(Banac)o(h)g(algebra)f Fq(C)710 1212 y Fp(0)740 1227 y Fr(instead)h(of)f(analytic)f(functions.)18 b(As)12 b(a)f(consequence,)j(the)249 1277 y(calculation)h(of)g(the)h (deriv)n(ativ)o(es)g(has)g(to)g(b)q(e)g(dealt)g(with)f(in)h(a)f (di\013eren)o(t)i(manner)e(than)249 1327 y(b)q(efore.)k(Of)14 b(course)h(w)o(e)f(omit)e(all)g(details,)i(and)f(simply)f(state)j(the)f (main)e(\014nal)h(result:)249 1427 y Fg(Algorithm)d(4.2.)17 b Fs(We)c(pr)n(o)n(duc)n(e)g(a)f(neighb)n(orho)n(o)n(d)h Fo(U)1096 1412 y Fp(0)1115 1427 y Fr(\()p Fq(I)1149 1433 y Fp(0)1168 1427 y Fq(;)7 b(:)g(:)g(:)e(;)i(I)1279 1433 y Fj(N)1310 1427 y Fr(;)g Fq(C)1359 1433 y Fj(h)1379 1427 y Fq(;)g Fr(0;)g Fo(1)p Fr(\))k Fs(such)i(that)582 1535 y Fq(h)p Fr(\()p Fq(t)p Fr(\))668 1507 y Fp(def)675 1535 y Fo(\021)22 b Fq(t)7 b(w)782 1518 y Fn(0)793 1535 y Fr(\()p Fq(t)p Fr(\))j(+)f Fq(w)q Fr(\()p Fq(t)p Fr(\))j(=)g Fq(f)t Fr(\()1066 1527 y(\026)1065 1535 y Fq(t)p Fr(\))p Fq(;)92 b(f)16 b Fo(2)11 b(U)1305 1518 y Fp(0)1324 1535 y Fq(;)249 1626 y Fs(for)j Fq(t)e Fo(\024)g Fq(\021)q Fs(.)249 1726 y Fr(Sa)o(y)h(then)i(that)423 1855 y Fq(f)443 1861 y Fj(h)465 1855 y Fr(\()482 1848 y(\026)481 1855 y Fq(t)p Fr(\))d(=)584 1803 y Fj(N)569 1816 y Fk(X)568 1904 y Fj(n)p Fp(=0)637 1855 y Fq(a)659 1861 y Fj(n)682 1848 y Fr(\026)682 1855 y Fq(t)706 1838 y Fj(n)738 1855 y Fr(+)d Fq(H)s Fr(\()834 1848 y(\026)833 1855 y Fq(t)p Fr(\))p Fq(;)49 b Fo(j)o Fq(H)s Fr(\()991 1848 y(\026)990 1855 y Fq(t)p Fr(\))p Fo(j)11 b(\024)h Fq(")1107 1861 y Fj(h)1136 1855 y Fo(j)1148 1848 y Fr(\026)1147 1855 y Fq(t)p Fo(j)1174 1834 y Fj(N)s Fp(+1)1254 1855 y Fq(;)48 b Fr(0)12 b Fq(<)f(t)h Fo(\024)g Fq(\021)q(;)249 1982 y Fr(for)f Fq(a)332 1988 y Fj(n)367 1982 y Fr(in)g(computable)f(in)o (terv)n(als)i(and)f Fq(")896 1988 y Fj(n)931 1982 y Fr(b)q(ounded)h(b)o (y)g(a)f(computable)g(n)o(um)o(b)q(er.)16 b(Then)280 2081 y Fq(d)p 265 2099 52 2 v 265 2137 a(d)p Fr(\012)329 2038 y Fk( )362 2109 y Fr(\012)399 2052 y Fk(Z)440 2062 y Fj(\016)422 2147 y(r)437 2151 y Fm(1)453 2147 y Fp(\(\012\))512 2075 y Fk(\000)531 2109 y Fq(u)p Fr(\()p Fq(r)q Fr(\))9 b Fo(\000)g Fr(\012)687 2092 y Fp(2)706 2075 y Fk(\001)725 2083 y Fn(\000)751 2071 y Fm(1)760 2079 y Fl(=)770 2087 y Fm(2)800 2081 y Fq(dr)p 800 2099 42 2 v 811 2137 a(r)846 2038 y Fk(!)891 2109 y Fr(=)954 2081 y Fq(d)p 939 2099 52 2 v 939 2137 a(d)p Fr(\012)1003 2038 y Fk( )1036 2109 y Fr(\012)1066 2092 y Fp(2)1091 2052 y Fk(Z)1133 2062 y Fp(\012)1157 2050 y Fa(\000)p Fm(2)1195 2062 y Fj(u)p Fp(\()p Fj(\016)q Fp(\))1114 2147 y(1)1266 2109 y Fr(\()p Fq(t)g Fo(\000)h Fr(1\))1385 2088 y Fn(\000)1411 2075 y Fm(1)1420 2084 y Fl(=)1429 2092 y Fm(2)1454 2109 y Fq(w)q Fr(\()p Fq(t)p Fr(\012)1546 2092 y Fp(2)1565 2109 y Fr(\))d Fq(dt)1625 2038 y Fk(!)357 2279 y Fr(=)k(2\012)458 2222 y Fk(Z)499 2233 y Fp(\012)523 2220 y Fa(\000)p Fm(2)562 2233 y Fj(u)p Fp(\()p Fj(\016)q Fp(\))481 2317 y(1)633 2279 y Fr(\()p Fq(t)e Fo(\000)g Fr(1\))751 2258 y Fn(\000)777 2245 y Fm(1)786 2254 y Fl(=)796 2262 y Fm(2)821 2279 y Fq(h)p Fr(\()p Fq(t)p Fr(\012)906 2262 y Fp(2)925 2279 y Fr(\))e Fq(dt)i Fo(\000)g Fr(2)1063 2245 y Fk(\000)1082 2279 y Fq(u)p Fr(\()p Fq(\016)r Fr(\))g Fo(\000)h Fr(\012)1239 2262 y Fp(2)1257 2245 y Fk(\001)1276 2253 y Fn(\000)1302 2241 y Fm(1)1311 2249 y Fl(=)1321 2257 y Fm(2)1346 2279 y Fq(w)q Fr(\()p Fq(u)p Fr(\()p Fq(\016)r Fr(\)\))p Fq(u)p Fr(\()p Fq(\016)r Fr(\))249 2422 y(=)i(2\012)367 2370 y Fj(N)352 2382 y Fk(X)351 2470 y Fj(n)p Fp(=0)420 2422 y Fq(a)442 2428 y Fj(n)465 2422 y Fq(x)489 2403 y Fn(\000)515 2391 y Fl(n)527 2399 y(=)537 2407 y Fm(2)489 2433 y Fp(0)555 2422 y Fr(\012)585 2405 y Fj(n)615 2365 y Fk(Z)656 2375 y Fp(\012)680 2363 y Fa(\000)p Fm(2)719 2375 y Fj(u)p Fp(\()p Fj(\016)q Fp(\))638 2459 y(1)789 2422 y Fr(\()q Fq(t)d Fo(\000)g Fr(1\))908 2401 y Fn(\000)934 2388 y Fm(1)943 2397 y Fl(=)953 2405 y Fm(2)978 2422 y Fq(t)998 2394 y Fl(n)p 998 2400 19 2 v 1000 2416 a Fm(2)1030 2422 y Fq(dt)f Fr(+)1115 2411 y(~)1115 2422 y Fq(h)p Fr(\(\012\))g Fo(\000)1249 2388 y Fk(\000)1268 2422 y Fq(u)p Fr(\()p Fq(\016)r Fr(\))i Fo(\000)f Fr(\012)1425 2405 y Fp(2)1444 2388 y Fk(\001)1463 2396 y Fn(\000)1489 2384 y Fm(1)1498 2392 y Fl(=)1507 2400 y Fm(2)1546 2394 y Fr(2)p Fq(u)p Fr(\()p Fq(\016)r Fr(\))p 1538 2412 115 2 v 1538 2450 a Fq(\016)g(u)1589 2438 y Fn(0)1600 2450 y Fr(\()p Fq(\016)r Fr(\))1657 2422 y Fq(;)1561 2510 y Fr(\(4)p Fq(:)p Fr(3)p Fq(a)p Fr(\))p eop 19 18 bop 249 275 a Fc(Interval)19 b(A)o(rithmetic)e(in)h(Quantum)h(Me) n(chanics)448 b Fi(19)249 516 y Fr(with)384 586 y Fk(\014)384 611 y(\014)384 636 y(\014)398 623 y Fr(~)397 634 y Fq(h)p Fr(\(\012\))483 586 y Fk(\014)483 611 y(\014)483 636 y(\014)509 634 y Fo(\024)12 b Fr(2\012)604 617 y Fj(N)s Fp(+2)677 634 y Fq(\017)694 640 y Fj(h)715 634 y Fq(x)739 640 y Fp(0)758 615 y Fn(\000)789 595 y Fp(\()p Fj(N)s Fp(+1\))p 789 606 98 2 v 829 629 a(2)900 577 y Fk(Z)942 588 y Fp(\012)966 575 y Fa(\000)p Fm(2)1004 588 y Fj(u)p Fp(\()p Fj(\016)q Fp(\))923 672 y(1)1075 634 y Fr(\()p Fq(t)d Fo(\000)g Fr(1\))1193 613 y Fn(\000)1219 600 y Fm(1)1228 609 y Fl(=)1238 617 y Fm(2)1263 634 y Fq(t)1283 595 y Fp(\()p Fj(N)s Fp(+1\))p 1283 606 V 1324 629 a(2)1394 634 y Fq(dt)135 b Fr(\(4)p Fq(:)p Fr(3)p Fq(b)p Fr(\))249 796 y(This)15 b(form)o(ula,)e(and)i(a)f(similar)f(form)o(ula)g(for)i Fq(I)1003 802 y Fp(3)1021 796 y Fr(,)g(allo)o(w)f(us)h(to)g(compute)g Fq(I)1451 802 y Fp(2)1485 796 y Fr(and)g Fq(I)1585 802 y Fp(3)1619 796 y Fr(for)249 846 y(man)o(y)e(small)g(\012)503 831 y Fn(\003)522 846 y Fr(.)22 b(There)16 b(is)f(a)g(limit)d(again,)i (ho)o(w)o(ev)o(er,)i(since)g(as)f(\012)e Fo(!)g Fr(0)i(the)h(in)o (tegrals)249 896 y(ab)q(o)o(v)o(e)d(are)g(extended)h(o)o(v)o(er)f(an)f (in)o(terv)n(als)h(spreading)g(to)f(in\014nit)o(y)m(.)k(This)d(calls)f (for)h(a)f(\014nal)249 945 y({)k(but)f(fortunately)h(easy)g({)g (analysis)f(to)g(con)o(trol)h(these)h(in)o(tegrals)e(for)h(v)o(ery)g (small)d(\012)1638 930 y Fn(\003)1657 945 y Fr(.)249 995 y(W)m(e)j(simply)e(state)i(that)h(this)f(can)g(b)q(e)h(done)f(and)g (refer)h(the)f(in)o(terested)i(reader)f(to)f([21)o(])249 1045 y(for)e(all)e(the)j(details.)249 1215 y Ft(Case)23 b(3)249 1315 y Fr(W)m(e)15 b(are)i(\014nally)d(left)i(with)f(the)i(pro) q(of)e(that)h Fq(F)1001 1300 y Fn(00)1022 1315 y Fr(\(\012\))f Fo(\024)g Fq(c)g(<)g Fr(0)g(for)h(\012)f(su\016cien)o(tly)h(close)249 1365 y(to)g(\012)332 1371 y Fj(c)349 1365 y Fr(.)26 b(The)16 b(previous)h(metho)q(d)f(fails)f(for)h(these)i(\012)e(mainly)e(b)q (ecause)k(the)f(deriv)n(ativ)o(es)249 1414 y(of)e Fq(u)g Fr(at)g(the)h(crossing)f(p)q(oin)o(ts)h Fq(r)766 1420 y Fj(i)794 1414 y Fr(degenerate)h(to)e(0.)22 b(The)16 b(idea)f(here)h(is)f(to)h(c)o(hange)f(the)249 1464 y(approac)o(h)i (completely)m(,)f(and)h(consider)h(the)g(c)o(hange)f(of)g(v)n(ariables) f(giv)o(en)h(b)o(y)g Fq(r)q Fr(\()p Fq(t)p Fr(\),)g(the)249 1514 y(in)o(v)o(erse)e(of)e Fq(t)p Fr(\()p Fq(r)q Fr(\))f(=)f Fo(\006)588 1479 y Fk(p)p 630 1479 176 2 v 35 x Fr(\012)660 1502 y Fp(2)660 1524 y Fj(c)688 1514 y Fo(\000)f Fq(u)p Fr(\()p Fq(r)q Fr(\))o(,)k(whic)o(h)g(allo)o(ws)e(us)i(to)g(rewrite)282 1637 y Fo(\000)p Fq(F)347 1620 y Fn(0)358 1637 y Fr(\(\012\))e(=)g (\012)519 1581 y Fk(Z)561 1591 y Fp(1)543 1675 y Fn(\000)p Fp(1)587 1637 y Fr(\(1)d Fo(\000)h Fq(t)690 1620 y Fp(2)708 1637 y Fr(\))724 1620 y Fn(\000)750 1608 y Fm(1)759 1616 y Fl(=)769 1624 y Fm(2)794 1637 y Fq(w)q Fr(\()p Fq(D)q(t)p Fr(\))d Fq(dt;)48 b(w)q Fr(\()p Fq(t)p Fr(\))12 b(=)1150 1609 y Fq(r)q Fr(\()p Fq(t)p Fr(\))p 1150 1628 68 2 v 1150 1666 a Fq(r)q Fr(\()p Fq(t)p Fr(\))1222 1637 y Fq(;)48 b(D)1317 1620 y Fp(2)1348 1637 y Fr(=)11 b(\012)1421 1620 y Fp(2)1421 1648 y Fj(c)1449 1637 y Fo(\000)f Fr(\012)1521 1620 y Fp(2)1539 1637 y Fq(:)32 b Fr(\(4)p Fq(:)p Fr(4\))249 1805 y(The)20 b(main)d(di\016cult)o(y)h(here)i(is)f(to)g(gain)g(con)o (trol)g(o)o(v)o(er)g(the)h(function)e Fq(r)q Fr(\()p Fq(t)p Fr(\))i(\(or)f Fq(w)q Fr(\()p Fq(t)p Fr(\)\),)249 1855 y(whic)o(h)14 b(is)g(done)g(via)f(the)h(follo)o(wing)e(tec)o (hnical)i(lemma)c(in)k(complex)e(v)n(ariables.)249 1955 y Fg(Lemma)k(4.3.)22 b Fs(L)n(et)15 b Fq(u)e Fo(2)f Fq(H)702 1940 y Fp(1)721 1955 y Fr(\()p Fo(j)p Fq(z)f Fo(\000)f Fq(r)840 1961 y Fj(c)857 1955 y Fo(j)j(\024)g Fq(R)p Fr(\))p Fs(,)i(smo)n(oth)h(on)g(the)g(b)n(oundary)h(of)e Fq(B)r Fr(\()p Fq(r)1572 1961 y Fj(c)1590 1955 y Fq(;)7 b(R)p Fr(\))p Fs(,)249 2005 y(of)15 b(the)g(form)364 2111 y Fq(u)p Fr(\()p Fq(x)p Fr(\))c(=)h(\012)529 2093 y Fp(2)529 2121 y Fj(c)557 2111 y Fo(\000)d Fq(u)622 2117 y Fp(2)641 2111 y Fq(R)673 2093 y Fp(2)698 2111 y Fq(z)719 2093 y Fp(2)747 2111 y Fr(+)g Fq(z)809 2093 y Fp(3)828 2111 y Fq(f)t Fr(\()p Fq(z)r Fr(\))p Fq(;)93 b(z)13 b Fr(=)1091 2082 y Fq(x)c Fo(\000)h Fq(r)1185 2088 y Fj(c)p 1091 2101 111 2 v 1130 2139 a Fq(R)1206 2111 y(;)92 b(f)t Fr(\(0\))12 b(=)g Fq(u)1467 2117 y Fp(3)1492 2111 y Fq(R)1524 2093 y Fp(3)1543 2111 y Fq(;)249 2214 y Fs(satisfying)279 2346 y Fr(1.)20 b Fo(j)-7 b(j)o Fq(f)t Fo(j)h(j)11 b(\024)h Fq(h)p Fs(,)i Fq(u)520 2352 y Fp(2)550 2346 y Fq(>)e Fr(0)i Fs(and)i Fq(u)734 2352 y Fp(2)752 2346 y Fq(R)784 2331 y Fp(2)814 2346 y Fq(>)c(h)p Fs(.)279 2429 y Fr(2.)20 b Fs(F)m(or)15 b(a)g(c)n(onstant)g Fq(M)20 b Fs(we)14 b(have)828 2394 y Fk(\014)828 2419 y(\014)842 2429 y Fq(u)866 2414 y Fp(\(4\))910 2429 y Fr(\()p Fq(x)p Fr(\))966 2394 y Fk(\014)966 2419 y(\014)991 2429 y Fo(\024)e Fq(M)20 b Fs(for)14 b Fo(j)p Fq(z)r Fo(j)d(\024)h Fr(1)p Fs(.)p eop 20 19 bop 249 275 a Fi(20)1110 b Fh(Chapter)18 b(1)249 516 y Fs(Then,)i Fq(t)p Fr(\()p Fq(x)p Fr(\))f 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Fk(\014)454 775 y(\014)454 800 y(\014)454 824 y(\014)478 782 y Fq(d)500 767 y Fj(n)p Fp(+1)564 782 y Fq(r)p 472 800 118 2 v 472 838 a(dw)525 826 y Fj(n)p Fp(+1)594 810 y Fr(\(0\))647 750 y Fk(\014)647 775 y(\014)647 800 y(\014)647 824 y(\014)673 810 y Fo(\024)12 b Fq(n)p Fr(!)c Fq(T)792 793 y Fn(\000)p Fj(n)940 782 y Fr(2)961 749 y Fo(p)p 995 749 194 2 v 995 782 a Fq(u)1019 788 y Fp(2)1047 782 y Fr(+)h Fq(hR)1144 770 y Fn(\000)p Fp(2)p 855 800 419 2 v 855 840 a Fr(2)p Fq(u)900 846 y Fp(2)927 840 y Fo(\000)h Fr(3)d Fo(j)o Fq(u)1032 846 y Fp(3)1051 840 y Fo(j)f Fq(R)j Fo(\000)1157 824 y Fp(1)p 1157 831 17 2 v 1157 855 a(6)1178 840 y Fq(M)c(R)1255 828 y Fp(2)1363 810 y Fq(n)12 b Fo(\025)f Fr(0)249 974 y(As)j(a)g(consequence,)i(w)o(e) e(obtain)f(the)i(follo)o(wing:)249 1074 y Fg(Algorithm)c(4.4.)18 b Fs(Set)13 b Fq(u)p Fr(\()p Fq(x)p Fr(\))e(=)h(\012)804 1058 y Fp(2)804 1084 y Fj(c)828 1074 y Fo(\000)5 b Fq(z)886 1058 y Fp(2)905 1074 y Fq(f)t Fr(\()p Fq(z)r Fr(\))p Fs(,)15 b(with)d Fq(z)i Fr(=)e(\()p Fq(x)5 b Fo(\000)g Fq(r)1276 1080 y Fj(c)1293 1074 y Fr(\))p Fq(=R)p Fs(.)18 b(If)13 b(we)f(know)i(that)249 1123 y Fq(f)k Fo(2)12 b(U)353 1129 y Fp(0)372 1123 y Fr(\()p Fq(I)406 1129 y Fp(0)425 1123 y Fq(;)7 b(:)g(:)g(:)e(;)i(I)536 1129 y Fp(2)p Fj(N)s Fp(+1)626 1123 y Fr(;)g Fq(C)675 1129 y Fj(h)695 1123 y Fq(;)g Fr(0;)g Fo(1)p Fr(\))14 b Fs(and)j(that)e Fq(A)1024 1108 y Fp(0)1056 1123 y Fo(\024)e(j)p Fq(y)q Fr(\()p Fq(x)p Fr(\))p Fo(j)g(\024)g Fq(A)1291 1129 y Fp(0)1326 1123 y Fs(when)j Fo(j)o Fq(x)9 b Fo(\000)h Fq(r)1538 1129 y Fj(c)1554 1123 y Fo(j)j(\024)g Fq(R)p Fs(,)249 1173 y(for)g(b)n(ounds)h Fq(A)480 1158 y Fp(0)510 1173 y Fq(<)e(A)585 1179 y Fp(0)604 1173 y Fs(,)h(we)g(c)n(an)h(then)g (c)n(onstruct)f Fq(T)19 b Fs(and)14 b Fo(U)1182 1179 y Fp(1)1201 1173 y Fs(,)f(such)h(that)f Fq(w)q Fr(\()p Fq(t)p Fr(\))f(=)g Fq(g)q Fr(\()p Fq(t=T)6 b Fr(\))p Fs(,)249 1223 y(with)14 b Fq(g)f Fo(2)e(U)437 1229 y Fp(1)456 1223 y Fs(.)19 b(We)c(use)g(this)g(to)g(c)n(ompute)g(a)g(b)n (ound)h(for)e Fq(F)1184 1208 y Fn(00)1220 1223 y Fs(su\016ciently)g (close)h(to)g Fr(\012)1611 1229 y Fj(c)1628 1223 y Fs(.)249 1323 y Fg(Description)o(.)26 b Fr(The)17 b(construction)h(of)f Fo(U)937 1329 y Fp(1)973 1323 y Fr(is)g(rather)h(easy)f(from)f(the)h (lemma)d(ab)q(o)o(v)o(e.)249 1372 y(Therefore,)g(w)o(e)g(kno)o(w)f (that)h Fq(w)q Fr(\()p Fq(t)p Fr(\))d(=)840 1341 y Fk(P)891 1372 y Fq(w)921 1378 y Fj(n)943 1372 y Fr(\()p Fq(t=T)6 b Fr(\))1041 1357 y Fj(n)1077 1372 y Fr(and)14 b(w)o(e)f(ha)o(v)o(e)h (go)q(o)q(d)f(con)o(trol)g(on)g(the)249 1422 y Fq(w)279 1428 y Fj(n)301 1422 y Fr(.)18 b(W)m(e)c(insert)h(this)e(in)o(to)h (\(4.4\))f(to)h(obtain)425 1537 y Fo(\000)464 1521 y Fp(1)p 462 1528 21 2 v 462 1551 a Fj(\031)488 1537 y Fq(F)521 1520 y Fn(0)532 1537 y Fr(\(\012\))e(=)f(\012)703 1485 y Fj(N)688 1497 y Fk(X)686 1585 y Fj(n)p Fp(=0)756 1537 y Fq(w)786 1543 y Fp(2)p Fj(n)832 1478 y Fk(\022)867 1509 y Fq(D)p 867 1527 36 2 v 870 1565 a(T)908 1478 y Fk(\023)938 1487 y Fp(2)p Fj(n)984 1537 y Fq(\013)1011 1543 y Fj(n)1043 1537 y Fr(+)e(\012)1129 1497 y Fk(X)1121 1586 y Fj(n>N)1204 1537 y Fq(w)1234 1543 y Fp(2)p Fj(n)1280 1478 y Fk(\022)1315 1509 y Fq(D)p 1315 1527 V 1318 1565 a(T)1356 1478 y Fk(\023)1386 1487 y Fp(2)p Fj(n)1432 1537 y Fq(\013)1459 1543 y Fj(n)1481 1537 y Fq(;)249 1650 y Fr(with)573 1719 y Fq(\013)600 1725 y Fj(n)634 1719 y Fr(=)685 1691 y(1)p 683 1710 26 2 v 683 1748 a Fq(\031)720 1663 y Fk(Z)761 1673 y Fp(1)743 1757 y Fn(\000)p Fp(1)787 1719 y Fr(\(1)g Fo(\000)h Fq(t)890 1702 y Fp(2)909 1719 y Fr(\))925 1702 y Fn(\000)951 1689 y Fm(1)960 1698 y Fl(=)969 1706 y Fm(2)988 1719 y Fq(t)1003 1702 y Fp(2)p Fj(n)1049 1719 y Fq(dt)h Fr(=)1141 1661 y Fk(\022)1171 1691 y Fr(2)p Fq(n)1181 1748 y(n)1217 1661 y Fk(\023)1247 1719 y Fr(2)1268 1702 y Fn(\000)p Fp(2)p Fj(n)1333 1719 y Fq(:)238 b Fr(\(5)p Fq(:)p Fr(5\))249 1808 y(The)15 b(\014rst)h(term)e(ab)q(o)o(v)o(e)h(is)f(computed)h(and)f(the)i(second) g(is)e(just)h(an)g(error)h(term,)e(whic)o(h)249 1858 y(is)g(easily)f(estimated)h(in)f(absolute)h(v)n(alue)f(from)f(ab)q(o)o (v)o(e.)18 b(Q.E.D.)249 2058 y Ft(Ac)n(kno)n(wledgemen)n(ts)249 2158 y Fr(Charles)g(L.)e(F)m(e\013erman)h(w)o(as)g(partially)e(supp)q (orted)k(b)o(y)e(an)g(NSF)g(gran)o(t)g(at)g(Princeton)249 2208 y(Univ)o(ersit)o(y)m(.)249 2408 y Ft(REFERENCES)270 2508 y Fr([1])i(V.)12 b(Arnold,)g Fs(Mathematic)n(al)h(Metho)n(ds)h(of) f(Classic)n(al)g(Me)n(chanics)p Fr(,)f(Graduate)g(T)m(exts)334 2558 y(in)i(Math.)f(No.)g(60,)g(Springer-V)m(erlag,)g(N.Y.,)g(1978.)p eop 21 20 bop 249 275 a Fc(Interval)19 b(A)o(rithmetic)e(in)h(Quantum)h(Me) n(chanics)448 b Fi(21)270 516 y Fr([2])19 b(V.)28 b(Bac)o(h,)g (\\Accuracy)h(of)e(Mean)h(Field)g(Appro)o(ximations)e(for)h(A)o(toms)g (and)334 565 y(Molecules",)14 b Fs(Comm.)h(Math.)g(Phys.)p Fr(,)e(1993,)g(V)m(ol.)f(155,)h(No.)g(2,)g(pp.)g(295{310.)270 648 y([3])19 b(A.)12 b(C\023)-21 b(ordoba,)11 b(C.)g(F)m(e\013erman,)g (and)g(L.)g(Seco,)h(\\A)g(T)m(rigonometric)e(Sum)g(relev)n(an)o(t)i(to) 334 698 y(the)k(Non{relativistic)e(Theory)h(of)g(A)o(toms",)e Fs(Pr)n(o)n(c.)i(Nat.)g(A)n(c)n(ad.)h(Sci.)f(USA)p Fr(,)g(June)334 748 y(1994,)e(V)m(ol.)f(91,)h(pp.)g(5776{5778.)270 831 y([4])19 b(A.)c(C\023)-21 b(ordoba,)13 b(C.)i(F)m(e\013erman,)e(and)i (L.)f(Seco,)g(\\W)m(eyl)g(Sums)g(and)g(A)o(tomic)f(Energy)334 881 y(Oscillations",)18 b Fs(R)n(evista)h(Matem\023)-21 b(atic)n(a)19 b(Ib)n(er)n(o)n(americ)n(ana)p Fr(,)f(1995,)f(V)m(ol.)g (11,)h(No.)f(1,)334 931 y(pp.)d(167{228.)270 1014 y([5])19 b(A.)d(C\023)-21 b(ordoba,)15 b(C.)g(F)m(e\013erman,)g(and)g(L.)h (Seco,)g(\\A)f(Num)o(b)q(er{Theoretic)i(Estimate)334 1064 y(for)d(the)g(Thomas{F)m(ermi)d(Densit)o(y",)i(1995)g(\(to)h(app)q (ear\).)270 1147 y([6])19 b(P)m(.)e(Dirac,)g(\\Note)g(on)g(Exc)o(hange) h(Phenomena)f(in)g(the)h(Thomas{F)m(ermi)c(A)o(tom",)334 1196 y Fs(Pr)n(o)n(c.)h(Cambridge)f(Philos.)h(So)n(c.)p Fr(,)e(1930,)g(V)m(ol.)f(26,)h(pp.)g(376|385.)270 1279 y([7])19 b(J.)e(P)m(.)e(Ec)o(kmann,)g(H.)h(Ko)q(c)o(h,)g(and)g(P)m(.)g (Witt)o(w)o(er,)g Fs(A)g(c)n(omputer)i(Assiste)n(d)e(Pr)n(o)n(of)h(of) 334 1329 y(Universality)11 b(in)g(A)o(r)n(e)n(a)g(Pr)n(eserving)g(Maps) p Fr(,)f(Memoirs)f(A.M.S.,)g(V)m(ol)g(289,)g(American)334 1379 y(Mathematical)j(So)q(ciet)o(y)m(,)i(Pro)o(vidence,)g(R.I.,)e (1984.)270 1462 y([8])19 b(J.)12 b(P)m(.)f(Ec)o(kmann)f(and)i(P)m(.)e (Witt)o(w)o(er,)h Fs(Computer)i(Metho)n(ds)g(and)h(Bor)n(el)e (Summability)334 1512 y(Applie)n(d)g(to)f(F)m(eigenb)n(aum's)h(e)n (quation)p Fr(,)e(Lecture)i(Notes)f(in)f(Mathematics,)f(V)m(ol.)g(227,) 334 1562 y(Springer-V)m(erlag,)k(N.Y.,)g(1985.)270 1645 y([9])19 b(C.)14 b(F)m(alcolini,)d(C.)i(F)m(e\013erman,)g(and)h(R.)f (Lla)o(v)o(e,)f(In)i(preparation.)249 1728 y([10])19 b(C.)12 b(F)m(e\013erman,)f(\\A)o(toms)g(and)h(Analytic)f(Num)o(b)q(er) h(Theory",)g(In:)g Fs(A.M.S.)h(Centen-)334 1778 y(nial)18 b(Public)n(ation,)f(V)m(ol.)f(II)p Fr(,)h(American)e(Math.)h(So)q(ciet) o(y)m(,)g(Pro)o(vidence,)i(RI,)d(1992,)334 1827 y(p.)f(27{36.)249 1910 y([11])19 b(C.)c(F)m(e\013erman,)g(\\The)g(A)o(tomic)f(and)i (Molecular)f(Structure)i(of)e(Matter",)h Fs(R)n(evista)334 1960 y(Matem\023)-21 b(atic)n(a)15 b(Ib)n(er)n(o)n(americ)n(ana)p Fr(,)f(1985,)e(V)m(ol.)g(1,)h(No.)h(1,)f(pp.)g(1{44.)249 2043 y([12])19 b(C.)f(F)m(e\013erman,)f(\\The)h Fq(N)5 b Fr({Bo)q(dy)18 b(Problem)f(in)h(Quan)o(tum)f(Mec)o(hanics",)h Fs(Comm.)334 2093 y(Pur)n(e)d(and)h(Appl.)e(Math.)p Fr(,)g(1986,)e(V)m (ol.)h(39,)f(No.)i(S,)f(pp.)h(S67{S110.)249 2176 y([13])19 b(C.)e(F)m(e\013erman)f(and)g(R.)g(Lla)o(v)o(e,)g(\\Relativistic)f (Stabilit)o(y)g(of)i(Matter,)g(I",)f Fs(R)n(evista)334 2226 y(Matematic)n(a)f(Ib)n(er)n(o)n(americ)n(ana)p Fr(,)f(1986,)e(V)m (ol.)g(2,)h(No.)h(1&2,)f(pp.)g(119{213.)249 2309 y([14])19 b(C.)14 b(F)m(e\013erman)f(and)h(L.)f(Seco,)h(\\The)g(Ground{State)g (Energy)g(of)f(a)h(Large)g(A)o(tom",)334 2359 y Fs(Bul)r(l.)h(A.M.S.)p Fr(,)e(1990,)f(V)m(ol.)h(23,)f(No.)i(2,)f(pp.)g(525|530.)249 2442 y([15])19 b(C.)e(F)m(e\013erman)f(and)h(L.)g(Seco,)g(\\Eigen)o(v)n (alues)f(and)h(Eigenfunctions)g(of)g(Ordinary)334 2492 y(Di\013eren)o(tial)d(Op)q(erators",)g Fs(A)n(dv.)h(Math.)p Fr(,)f(1992,)e(V)m(ol.)g(95,)h(No.)g(2,)g(pp.)h(145{305.)p eop 22 21 bop 249 275 a Fi(22)1110 b Fh(Chapter)18 b(1)249 516 y Fr([16])h(C.)d(F)m(e\013erman)g(and)g(L.)f(Seco,)i(\\The)f(Eigen) o(v)n(alue)f(Sum)g(for)h(a)g(One{Dimensional)334 565 y(P)o(oten)o(tial",)d Fs(A)n(dvanc)n(es)i(in)g(Math)p Fr(,)f(Oct.)g(1994,)f(V)m(ol.)f(108,)h(No.)g(2,)g(pp.)h(263{335.)249 646 y([17])19 b(C.)12 b(F)m(e\013erman)g(and)g(L.)g(Seco,)h(\\The)f (Densit)o(y)h(in)f(a)g(One-Dimensional)e(P)o(oten)o(tial",)334 696 y Fs(A)n(dvanc)n(es)16 b(in)f(Math.)p Fr(,)e(Sept.)h(1994,)f(V)m (ol.)f(107,)h(No.)g(2,)g(pp.)h(187{364.)249 776 y([18])19 b(C.)13 b(F)m(e\013erman)f(and)h(L.)f(Seco,)h(\\The)g(Eigen)o(v)n(alue) g(Sum)e(for)i(a)f(Three{Dimensional)334 826 y(Radial)g(P)o(oten)o (tial",)h Fs(A)n(dvanc)n(es)i(in)g(Math.)p Fr(,)f(1995)f(\(to)h(app)q (ear\).)249 906 y([19])19 b(C.)d(F)m(e\013erman)g(and)g(L.)g(Seco,)g (\\The)h(Densit)o(y)f(in)g(a)g(Three{Dimensional)f(Radial)334 956 y(P)o(oten)o(tial",)e Fs(A)n(dvanc)n(es)i(in)g(Math.)p Fr(,)f(1995)f(\(to)h(app)q(ear\).)249 1037 y([20])19 b(C.)11 b(F)m(e\013erman)f(and)h(L.)g(Seco,)g(\\On)g(the)h(Dirac)f(and) g(Sc)o(h)o(winger)g(Corrections)h(to)f(the)334 1087 y(Ground{State)17 b(Energy)g(of)f(an)h(A)o(tom",)d Fs(A)n(dvanc)n(es)k(in)g(Math.)p Fr(,)e(Aug.)g(1994,)f(V)m(ol.)334 1136 y(107,)e(No.)g(1,)g(pp.)h (1{185.)249 1217 y([21])19 b(C.)d(F)m(e\013erman)f(and)h(L.)f(Seco,)h (\\Ap)q(erio)q(dicit)o(y)g(of)f(the)i(Hamiltonian)c(Flo)o(w)i(in)g(the) 334 1267 y(Thomas{F)m(ermi)9 b(P)o(oten)o(tial",)h Fs(R)n(evista)j (Matem\023)-21 b(atic)n(a)14 b(Ib)n(er)n(o)n(americ)n(ana)p Fr(,)d(1993,)f(V)m(ol.)334 1316 y(9,)k(No.)f(3,)g(pp.)g(409{551.)249 1397 y([22])19 b(E.)11 b(F)m(ermi,)e(\\Un)i(Meto)q(do)h(Statistico)e(p) q(er)i(la)f(Determinazione)f(di)g(alcune)i(Priorieta)334 1447 y(dell'A)o(tome",)g Fs(R)n(end.)j(A)n(c)n(c)n(ad.)g(Naz.)f(Linc)n (ei)p Fr(,)g(1927,)e(V)m(ol.)g(6,)h(pp.)h(602|607.)249 1527 y([23])19 b(G.)9 b(M.)h(Graf)f(and)h(J.)g(P)m(.)f(Solo)o(v)o(ej,)f (\\A)i(Correlation)g(Estimate)f(with)h(Applications)f(to)334 1577 y(Quan)o(tum)15 b(Systems)h(with)f(Coulom)o(b)f(In)o(teractions",) i Fs(R)n(eviews)g(in)h(Math.)g(Phys.)p Fr(,)334 1627 y(1994,)c(V)m(ol.)f(6,)h(No.)g(5a,)g(pp.)h(977{997.)249 1707 y([24])19 b(W.)e(Hughes,)g(\\An)g(A)o(tomic)f(Energy)i(Lo)o(w)o (er)f(Bound)h(that)f(Agrees)h(with)f(Scott's)334 1757 y(Correction",)d Fs(A)n(dvanc)n(es)i(in)f(Mathematics)p Fr(,)e(1990,)g(V)m(ol.)f(79,)h(pp.)g(213{270.)249 1838 y([25])19 b(B.)13 b(Hel\013er,)f(A.)g(Knauf,)g(H.)g(Sieden)o(top,)g (and)g(R.)f(W)m(eik)n(ard,)g(\\On)h(the)h(Absence)h(of)e(a)334 1887 y(First)k(Order)g(Correction)f(for)g(the)g(Num)o(b)q(er)g(of)f (Bound)h(States)h(of)f(a)f(Sc)o(hr\177)-21 b(odinger)334 1937 y(Op)q(erator)17 b(with)f(Coulom)o(b)d(Singularit)o(y",)h Fs(Comm.)i(P.D.E.)p Fr(,)g(1992,)e(V)m(ol.)g(17,)h(No.)334 1987 y(3&4,)f(pp.)f(615{639.)249 2067 y([26])19 b(E.)d(Hille,)e(\\On)h (the)i(Thomas{F)m(ermi)12 b(Equation",)i Fs(Pr)n(o)n(c.)i(Nat.)g(A)n(c) n(ad.)g(Sci.)g(USA)p Fr(,)334 2117 y(1969,)d(V)m(ol.)f(62,)h(pp.)g (7|10.)249 2198 y([27])19 b(V.)f(Ivrii)f(and)h(I.)f(M.)h(Sigal,)e (\\Asymptotics)h(of)g(the)i(Ground)f(State)g(Energies)h(of)334 2248 y(Large)h(Coulom)o(b)e(Systems",)h Fs(A)o(nnals)i(of)f(Math.)p Fr(,)g(1993,)e(V)m(ol.)h(138,)g(No.)g(2,)g(pp.)334 2297 y(243{335.)249 2378 y([28])g(E.)11 b(W.)g(Kauc)o(her)h(and)f(W.)f(L.)h (Mirank)o(er,)f Fs(Self-validating)i(Numerics)g(for)g(F)m(unction)334 2428 y(Sp)n(ac)n(e)k(Pr)n(oblems)p Fr(,)d(Academic)g(Press,)i(N.Y.,)e (1984.)249 2508 y([29])19 b(O.)12 b(Lanford,)f(\\A)h(Computer{Assisted) g(Pro)q(of)g(of)f(the)i(F)m(eigen)o(baum)d(Conjecture",)334 2558 y Fs(Bul)r(l.)15 b(AMS)p Fr(,)f(1986,)e(V)m(ol.)g(6,)h(pp.)h (427{434.)p eop 23 22 bop 249 275 a Fc(Interval)19 b(A)o(rithmetic)e(in)h(Quantum)h(Me) n(chanics)448 b Fi(23)249 516 y Fr([30])19 b(E.)14 b(Lieb,)f (\\Thomas{F)m(ermi)e(and)j(Related)f(Theories)i(of)e(A)o(toms)g(and)h (Molecules",)334 565 y Fs(R)n(eviews)h(of)g(Mo)n(dern)g(Physics)p Fr(,)f(1981,)e(V)m(ol.)g(53,)h(No.)h(4,)f(pp.)g(603{641.)249 643 y([31])19 b(E.)d(Lieb)f(and)h(B.)f(Simon,)e(\\Thomas{F)m(ermi)f (Theory)k(of)f(A)o(toms,)f(Molecules)i(and)334 693 y(Solids",)d Fs(A)n(dv.)h(Math.)p Fr(,)g(1977,)e(V)m(ol.)h(23,)g(pp.)g(22|116.)249 771 y([32])19 b(R.)c(Lla)o(v)o(e,)f(\\Computer)g(Assisted)j(Bounds)f (in)e(Stabilit)o(y)g(of)h(Matter",)g Fs(Computer)334 820 y(A)o(ide)n(d)d(Pr)n(o)n(ofs)g(in)g(A)o(nalysis)p Fr(,)e(IMA)g(Series)i(in)e(Math.)h(and)f(Appl.,)g(V)m(ol)f(28,)h (Springer,)334 870 y(Cincinatti,)j(1989.)249 948 y([33])19 b(R.)d(Lohner,)i Fs(Einschlie\031ung)g(der)g(L\177)-21 b(osung)18 b(gew\177)-21 b(ohnlicher)18 b(A)o(nfangs{)g(und)h(R)n(and-) 334 998 y(wertaufgab)n(en)32 b(und)h(A)o(nwendungen)p Fr(,)g(Dissertation,)e(Univ)o(ersit\177)-21 b(at)33 b(Karlsruhe)334 1047 y(\(TH\),)14 b(1988.)249 1125 y([34])19 b(R.)25 b(E.)h(Mo)q(ore,)f Fs(Metho)n(ds)i(and)f(Applic)n(ations)g(of)g (Interval)f(A)o(nalysis)p Fr(,)g(SIAM,)334 1175 y(Philadelphia,)12 b(P)m(A,)i(1979.)249 1253 y([35])19 b(D.)10 b(Rana,)e Fs(Pr)n(o)n(of)j(of)g(A)n(c)n(cur)n(ate)g(Upp)n(er)g(and)h(L)n(ower)e (Bounds)j(for)d(Stability)h(Domains)334 1302 y(in)k(Denominator)h(Pr)n (oblems)p Fr(,)d(Thesis,)h(Princeton)h(Univ)o(ersit)o(y)m(,)e(1987.)249 1380 y([36])19 b(J.)f(Sc)o(h)o(winger,)f(\\Thomas{F)m(ermi)d(Mo)q(del:) i(The)i(Second)g(Correction",)g Fs(Physic)n(al)334 1430 y(R)n(eview)p Fr(,)c(1981,)e(V)m(ol.)g(A24,)h(No.)h(5,)f(pp.)g (2353|2361.)249 1508 y([37])19 b(J.)c(M.)f(C.)g(Scott,)h(\\The)f (Binding)g(Energy)i(of)e(the)h(Thomas{F)m(ermi)c(A)o(tom",)h Fs(Phil.)334 1557 y(Mag.)p Fr(,)i(1952,)e(V)m(ol.)h(43,)g(pp.)g (859|867.)249 1635 y([38])19 b(L.)14 b(Seco,)h(\\Lo)o(w)o(er)f(Bounds)h (for)g(the)g(Ground)f(State)h(Energy)g(of)f(A)o(toms",)e(Thesis,)334 1685 y(Princeton)j(Univ)o(ersit)o(y)m(,)e(1989.)249 1763 y([39])19 b(L.)d(Seco,)f(\\Computer)g(Assisted)i(Lo)o(w)o(er)f(Bounds)g (for)g(A)o(tomic)e(Energies",)i Fs(Com-)334 1812 y(puter)i(A)o(ide)n(d) f(Pr)n(o)n(ofs)g(in)g(A)o(nalysis)p Fr(,)f(IMA)h(Series)g(in)f(Math.)g (and)h(Appl.,)e(V)m(ol)h(28,)334 1862 y(Springer,)e(Cincinatti,)f (1989,)f(pp.)i(241|251.)249 1940 y([40])19 b(H.)h(Sieden)o(top)g(and)g (R.)f(W)m(eik)n(ard,)f(\\On)i(the)h(Leading)e(Energy)i(Correction)f (for)334 1990 y(the)14 b(Statistical)f(Mo)q(del)h(of)e(the)i(A)o(tom:)e (In)o(teracting)i(Case",)f Fs(Communic)n(ations)i(in)334 2040 y(Mathematic)n(al)g(Physics)p Fr(,)f(1987,)e(V)m(ol.)h(112,)f(pp.) i(471{490.)249 2117 y([41])19 b(H.)11 b(Sieden)o(top)g(and)g(R.)f(W)m (eik)n(ard,)f(\\On)i(the)h(Leading)e(Correction)i(of)e(the)h(Thomas{) 334 2167 y(F)m(ermi)j(Mo)q(del:)h(Lo)o(w)o(er)h(Bound",)f(and)h(an)f (app)q(endix)h(b)o(y)f(A.)g(M.)g(K.)h(M)q(\177)-22 b(uller,)15 b Fs(Inv.)334 2217 y(Math.)p Fr(,)f(1989,)e(V)m(ol.)h(97,)g(pp.)g (159{193.)249 2295 y([42])19 b(H.)10 b(Sieden)o(top)g(and)g(R.)f(W)m (eik)n(ard,)g(\\A)h(New)g(Phase)h(Space)g(Lo)q(calization)d(T)m(ec)o (hnique)334 2344 y(with)i(Applications)f(to)g(the)i(Sum)d(of)h(Negativ) o(e)h(Eigen)o(v)n(alues)f(of)g(Sc)o(hr\177)-21 b(odinger)11 b(Op)q(er-)334 2394 y(ators",)g Fs(A)o(nn.)h(Scient.)g(Ec)n(ole)g (Normale)g(Sup)n(erieur)n(e)p Fr(,)e(1991,)f(V)m(ol.)h(24,)f(pp.)i (215{225.)249 2472 y([43])19 b(L.)12 b(H.)h(Thomas,)d(\\The)j (Calculation)e(of)h(A)o(tomic)f(Fields",)h Fs(Pr)n(o)n(c.)h(Cambridge)g (Phi-)334 2522 y(los.)i(So)n(c.)p Fr(,)e(1927,)g(V)m(ol.)f(23,)h(pp.)g (542|548.)p eop end userdict /end-hook known{end-hook}if