Content-Type: multipart/mixed; boundary="-------------0004260402809" This is a multi-part message in MIME format. ---------------0004260402809 Content-Type: text/plain; name="00-200.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="00-200.keywords" large atoms, strong magnetic fields ---------------0004260402809 Content-Type: application/postscript; name="large_atoms_main.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="large_atoms_main.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: large_atoms_main.dvi %%Pages: 23 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips large_atoms_main %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2000.04.26:1058 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul 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4053 y Fn([)p Fj(\034)2291 4068 y Fq(B)2352 4053 y Fn(\()p Fj(\032)p Fn(\))c Fi(\000)2610 3985 y Fn(3)p 2610 4030 49 4 v 2610 4121 a(5)2669 4053 y Fj(\032\034)2772 4012 y Fg(0)2761 4077 y Fq(B)2822 4053 y Fn(\()p Fj(\032)p Fn(\)])17 b Fj(dx:)324 4309 y Fn(If)32 b(w)m(e)i(apply)e(the)h(Thomas-F)-8 b(ermi)30 b(equation)j(w)m(e)g(get:)947 4457 y Fh(Z)1063 4592 y Fj(j)1103 4607 y Fq(M)7 b(T)j(F)1310 4592 y Fi(\001)16 b Fj(~)-44 b(a)28 b Fn(=)1552 4525 y(5)p 1552 4570 V 1552 4661 a(2)1627 4457 y Fh(Z)1769 4500 y Fj(~)1753 4525 y(B)g Fi(\001)21 b Fn(rot)-6 b Fj(~)-43 b(a)p 1753 4570 327 4 v 1858 4661 a(B)1937 4632 y Fr(2)2090 4592 y Fn([)p Fj(\034)2159 4607 y Fq(B)2220 4592 y Fn(\()p Fj(\032)p Fn(\))23 b(+)2477 4525 y(3)p 2477 4570 49 4 v 2477 4661 a(5)2535 4592 y Fj(\032V)2642 4607 y Fq(ef)7 b(f)2762 4592 y Fn(])17 b Fj(dx:)324 4854 y Fn(No)m(w)27 b(it)e(only)h(remains)f(to)h(notice,)i(from)d(the)i(Thomas-F)-8 b(ermi)24 b(equation,)j(that)f Fj(\032)p Fn(\()p Fj(x)p Fn(\))j(=)324 4974 y(0)j(if)f Fj(V)551 4989 y Fq(ef)7 b(f)671 4974 y Fn(\()p Fj(x)p Fn(\))28 b Fj(>)f Fn(0.)p 3491 4974 4 66 v 3495 4912 59 4 v 3495 4974 V 3553 4974 4 66 v 1894 5251 a(13)p eop %%Page: 14 14 14 13 bop 324 548 a Fe(3.1)135 b(A)45 b(useful)g(relation)324 733 y Fn(When)28 b(w)m(e)h(calculate)e(the)h(limit)c(of)j(the)i(curren) m(t)f(in)f(quan)m(tum)h(mec)m(hanics)g(w)m(e)h(will)d(get)324 853 y(a)32 b(term)g(whic)m(h)h(lo)s(oks)f(lik)m(e)1069 916 y Fh(Z)11 b(Z)1252 1052 y Fj(\032)p Fn(\()p Fj(x)p Fn(\))1443 984 y(\()p Fj(x)23 b Fi(\000)f Fj(y)t Fn(\))f Fi(\001)h Fn(\()q(~)-50 b Fj(a)p Fn(\()p Fj(x)p Fn(\))23 b Fi(\000)h Fn(~)-51 b Fj(a)q Fn(\()p Fj(y)t Fn(\)\))p 1443 1029 935 4 v 1749 1120 a Fi(j)p Fj(x)22 b Fi(\000)h Fj(y)t Fi(j)2034 1091 y Fr(3)2388 1052 y Fj(\032)p Fn(\()p Fj(y)t Fn(\))17 b Fj(dxdy)t(:)324 1271 y Fn(W)-8 b(e)30 b(will)d(no)m(w)k(use)g(the)f(minimizing)25 b(prop)s(ert)m(y)30 b(of)g(the)g(MTF-densit)m(y)h Fj(\032)f Fn(to)f(obtain)g(an)324 1391 y(equalit)m(y)j(for)g(this)g(term:)324 1550 y Fo(Lemma)44 b(3.5.)i Fk(L)-5 b(et)41 b Fj(\032)g Fk(b)-5 b(e)41 b(the)f(minimizer)g (in)h(MTF-the)-5 b(ory)40 b(and)h(let)h Fn(~)-50 b Fj(a)39 b Fi(2)g Fj(C)3270 1514 y Fg(1)3263 1575 y Fr(0)3344 1550 y Fn(\()p Ff(R)3448 1514 y Fr(3)3494 1550 y Fn(\))p Fk(.)324 1671 y(Then)324 1734 y Fh(Z)440 1870 y Fj(\032)q Fn(~)-50 b Fj(a)q Fi(\001r)p Fj(V)21 b Fn(+)817 1802 y(1)p 817 1847 49 4 v 817 1938 a(2)892 1734 y Fh(Z)11 b(Z)1075 1870 y Fj(\032)p Fn(\()p Fj(x)p Fn(\))1266 1802 y(\()p Fj(x)23 b Fi(\000)f Fj(y)t Fn(\))g Fi(\001)f Fn(\()q(~)-50 b Fj(a)p Fn(\()p Fj(x)p Fn(\))23 b Fi(\000)h Fn(~)-50 b Fj(a)p Fn(\()p Fj(y)t Fn(\)\))p 1266 1847 935 4 v 1572 1938 a Fi(j)p Fj(x)22 b Fi(\000)h Fj(y)t Fi(j)1857 1909 y Fr(3)2211 1870 y Fj(\032)p Fn(\()p Fj(y)t Fn(\))17 b Fj(dxdy)29 b Fn(=)2744 1734 y Fh(Z)2860 1870 y Fn(tr\()p Fj(D)t Fn(~)-50 b Fj(a)p Fn(\))p Fj(P)3210 1885 y Fq(B)3270 1870 y Fn(\()p Fi(j)p Fj(V)3393 1885 y Fq(ef)7 b(f)3512 1870 y Fi(j)3540 1885 y Fg(\000)3599 1870 y Fn(\))p Fj(:)324 2094 y Fk(Pr)-5 b(o)g(of.)48 b Fn(Let)41 b(us)g(de\014ne)g Fj(\032)1284 2109 y Fq(t)1314 2094 y Fn(\()p Fj(x)p Fn(\))g(=)g(\003) 1671 2109 y Fq(t)1701 2094 y Fn(\()p Fj(x)p Fn(\))p Fj(\032)p Fn(\()p Fj(x)28 b Fn(+)f Fj(t)q Fn(~)-50 b Fj(a)q Fn(\()p Fj(x)p Fn(\)\))40 b(for)g(small)e Fj(t)p Fn(,)43 b(where)e(\003)3284 2109 y Fq(t)3314 2094 y Fn(\()p Fj(x)p Fn(\))g(=)324 2214 y Fi(j)17 b Fn(det\()p Fj(I)26 b Fn(+)19 b Fj(tD)t Fn(~)-50 b Fj(a)p Fn(\()p Fj(x)p Fn(\)\))p Fi(j)28 b Fn(=)f(1)19 b(+)g Fj(t)p Fn(tr\()p Fj(D)t Fn(~)-50 b Fj(a)p Fn(\()p Fj(x)p Fn(\)\))19 b(+)g Fj(O)s Fn(\()p Fj(t)2085 2178 y Fr(2)2124 2214 y Fn(\).)43 b(Notice,)31 b(that)f(for)h(small)e Fj(t)i Fn(w)m(e)h(ma)m(y)324 2335 y(write)g Fj(x)c Fn(=)g Fj(\036)818 2350 y Fq(t)847 2335 y Fn(\()p Fj(x)23 b Fn(+)f Fj(t)q Fn(~)-50 b Fj(a)p Fn(\()p Fj(x)p Fn(\)\))p Fj(;)33 b Fn(where)h Fj(\036)1716 2350 y Fq(t)1745 2335 y Fn(\()p Fj(y)t Fn(\))27 b(=)g Fj(y)f Fi(\000)c Fj(t)q Fn(~)-50 b Fj(a)q Fn(\()p Fj(y)t Fn(\))21 b(+)h Fj(O)s Fn(\()p Fj(t)2661 2299 y Fr(2)2700 2335 y Fn(\).)470 2455 y(No)m(w)552 2648 y Fi(E)605 2663 y Fq(M)7 b(T)j(F)789 2648 y Fn([)p Fj(\032)866 2663 y Fq(t)896 2648 y Fn(])83 b(=)1165 2513 y Fh(Z)1281 2648 y Fj(\034)1323 2663 y Fq(B)1384 2648 y Fn(\(\003)1490 2663 y Fq(t)1519 2648 y Fn(\()p Fj(\036)1615 2663 y Fq(t)1645 2648 y Fn(\()p Fj(y)t Fn(\)\))p Fj(\032)p Fn(\()p Fj(y)t Fn(\)\))2178 2581 y Fj(dy)p 2037 2625 388 4 v 2037 2717 a Fn(\003)2105 2732 y Fq(t)2132 2717 y Fn(\()p Fj(\036)2228 2732 y Fq(t)2258 2717 y Fn(\()p Fj(y)t Fn(\)\))2455 2648 y(+)2553 2513 y Fh(Z)2669 2648 y Fj(V)21 b Fn(\()p Fj(\036)2843 2663 y Fq(t)2873 2648 y Fn(\()p Fj(y)t Fn(\)\))p Fj(\032)p Fn(\()p Fj(y)t Fn(\))c Fj(dy)1165 2910 y Fn(+)1251 2843 y(1)p 1251 2887 49 4 v 1251 2979 a(2)1326 2775 y Fh(Z)12 b(Z)1509 2910 y Fj(\032)p Fn(\()p Fj(x)p Fn(\))1981 2843 y(1)p 1700 2887 611 4 v 1700 2979 a Fi(j)p Fj(\036)1786 2994 y Fq(t)1815 2979 y Fn(\()p Fj(x)p Fn(\))23 b Fi(\000)f Fj(\036)2126 2994 y Fq(t)2156 2979 y Fn(\()p Fj(y)t Fn(\))p Fi(j)2320 2910 y Fj(\032)p Fn(\()p Fj(y)t Fn(\))17 b Fj(dxdy)t(:)324 3129 y Fn(Using)32 b(the)h(minimizing)28 b(prop)s(ert)m(y)33 b(of)f Fj(\032)h Fn(w)m(e)h(get:)813 3342 y(0)83 b(=)1131 3275 y Fj(d)p 1114 3319 86 4 v 1114 3410 a(dt)1209 3342 y Fi(j)1237 3357 y Fq(t)p Fr(=0)1357 3342 y Fi(E)1410 3357 y Fq(M)7 b(T)j(F)1594 3342 y Fn([)p Fj(\032)1671 3357 y Fq(t)1701 3342 y Fn(])945 3579 y(=)1104 3443 y Fh(Z)1220 3579 y Fn(\()p Fj(\034)1311 3538 y Fg(0)1300 3604 y Fq(B)1361 3579 y Fn(\()p Fj(\032)p Fn(\))p Fj(\032)p Fn(tr\()p Fj(D)t Fn(~)-50 b Fj(a)p Fn(\))22 b Fi(\000)h Fj(\034)1988 3594 y Fq(B)2049 3579 y Fn(\()p Fj(\032)p Fn(\)tr\()p Fj(D)t Fn(~)-50 b Fj(a)p Fn(\)\))22 b Fi(\000)2621 3443 y Fh(Z)2737 3579 y Fj(\032)p Fi(r)p Fj(V)45 b Fi(\001)23 b Fn(~)-51 b Fj(a)1104 3840 y Fi(\000)1191 3773 y Fn(1)p 1191 3817 49 4 v 1191 3909 a(2)1266 3705 y Fh(Z)12 b(Z)1449 3840 y Fj(\032)p Fn(\()p Fj(x)p Fn(\))1640 3773 y(\()p Fj(x)23 b Fi(\000)g Fj(y)t Fn(\))e Fi(\001)h Fn(\()q(~)-50 b Fj(a)p Fn(\()p Fj(x)p Fn(\))22 b Fi(\000)i Fn(~)-50 b Fj(a)p Fn(\()p Fj(y)t Fn(\)\))p 1640 3817 935 4 v 1946 3909 a Fi(j)p Fj(x)23 b Fi(\000)f Fj(y)t Fi(j)2231 3880 y Fr(3)2585 3840 y Fj(\032)p Fn(\()p Fj(y)t Fn(\))17 b Fj(dxdy)t(:)324 4059 y Fn(Th)m(us,)875 4142 y Fh(Z)991 4277 y Fj(\032)q Fn(~)-50 b Fj(a)23 b Fi(\001)f(r)p Fj(V)43 b Fn(+)1456 4210 y(1)p 1456 4254 49 4 v 1456 4346 a(2)1532 4142 y Fh(Z)11 b(Z)1714 4277 y Fj(\032)p Fn(\()p Fj(x)p Fn(\))1905 4210 y(\()p Fj(x)23 b Fi(\000)g Fj(y)t Fn(\))e Fi(\001)h Fn(\()q(~)-50 b Fj(a)p Fn(\()p Fj(x)p Fn(\))23 b Fi(\000)h Fn(~)-51 b Fj(a)q Fn(\()p Fj(y)t Fn(\)\))p 1905 4254 935 4 v 2212 4346 a Fi(j)p Fj(x)22 b Fi(\000)h Fj(y)t Fi(j)2497 4317 y Fr(3)2850 4277 y Fj(\032)p Fn(\()p Fj(y)t Fn(\))17 b Fj(dxdy)716 4539 y Fn(=)83 b Fi(\000)969 4404 y Fh(Z)1085 4539 y Fn(tr\()p Fj(D)t Fn(~)-50 b Fj(a)p Fn(\)\()p Fj(\034)1452 4554 y Fq(B)1513 4539 y Fn(\()p Fj(\032)p Fn(\))22 b(+)g Fj(\032V)1866 4554 y Fq(ef)7 b(f)1986 4539 y Fn(\))716 4794 y(=)875 4658 y Fh(Z)991 4794 y Fn(tr\()p Fj(D)t Fn(~)-50 b Fj(a)p Fn(\))p Fj(P)1341 4809 y Fq(B)1401 4794 y Fn(\()p Fi(j)p Fj(V)1524 4809 y Fq(ef)7 b(f)1643 4794 y Fi(j)1671 4809 y Fg(\000)1730 4794 y Fn(\))p Fj(:)p 3491 5002 4 66 v 3495 4940 59 4 v 3495 5002 V 3553 5002 4 66 v 1894 5251 a Fn(14)p eop %%Page: 15 15 15 14 bop 324 548 a Fe(3.2)135 b(Scaling)46 b(in)e(MTF-theory)324 733 y Fn(When)33 b Fj(V)22 b Fn(\()p Fj(x)p Fn(\))28 b(=)959 693 y Fg(\000)p Fq(Z)p 959 710 108 4 v 974 767 a Fg(j)p Fq(x)p Fg(j)1109 733 y Fn(and)1315 707 y Fj(~)1299 733 y(B)33 b Fn(=)27 b Fj(B)5 b Fn(\(1)p Fj(;)17 b Fn(0)p Fj(;)g Fn(0\),)32 b(w)m(e)h(ma)m(y)g(de\014ne)g(a)g(scaled)f (functional)955 984 y(^)937 1010 y Fi(E)990 1025 y Fq(\014)1037 1010 y Fn([)p Fj(\032)p Fn(])c(=)1273 874 y Fh(Z)1394 1010 y Fn(^)-54 b Fj(\034)1431 1025 y Fq(\014)1479 1010 y Fn(\()p Fj(\032)p Fn(\()p Fj(x)p Fn(\)\))17 b Fj(dx)22 b Fn(+)1979 874 y Fh(Z)2105 942 y 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Fr(3)1615 1730 y Fn(\(1)22 b(+)g Fj(\014)6 b Fn(\))1921 1689 y Fg(\000)p Fr(2)p Fq(=)p Fr(5)2085 1730 y Fj(E)2163 1689 y Fq(M)h(T)j(F)2348 1730 y Fn(\()p Fj(N)5 b(;)17 b(B)5 b(;)17 b(Z)7 b Fn(\))p Fj(;)324 1926 y Fn(and)32 b(the)h(minimizing)28 b(densities)33 b Fj(\032)1632 1890 y Fq(M)7 b(T)j(F)1849 1926 y Fn(and)33 b Fj(\032)2089 1941 y Fq(\025;\014)2230 1926 y Fn(are)f(related)h(b)m(y)958 2122 y Fj(\032)1008 2081 y Fq(M)7 b(T)j(F)1008 2146 y(N)s(;B)s(;Z)1220 2122 y Fn(\()p Fj(x)p Fn(\))28 b(=)g Fj(Z)1557 2081 y Fr(2)1596 2122 y Fn(\(1)22 b(+)g Fj(\014)6 b Fn(\))1902 2081 y Fr(6)p Fq(=)p Fr(5)2011 2122 y Fj(\032)2061 2137 y Fq(\025;\014)2170 2122 y Fn(\()p Fj(Z)2282 2081 y Fr(1)p Fq(=)p Fr(3)2391 2122 y Fn(\(1)22 b(+)g Fj(\014)6 b Fn(\))2697 2081 y Fr(2)p Fq(=)p Fr(5)2807 2122 y Fj(x)p Fn(\))p Fj(:)324 2318 y Fn(Let)33 b(us)g(state)g(the)g(TF-equation:)1566 2514 y(^)-54 b Fj(\034)1614 2472 y Fg(0)1603 2538 y Fq(\014)1651 2514 y Fn(\()p Fj(\032)1739 2529 y Fq(\025;\014)1847 2514 y Fn(\))28 b(=)f Fi(j)p Fj(v)2091 2529 y Fq(ef)7 b(f)2210 2514 y Fi(j)2238 2529 y Fg(\000)2297 2514 y Fj(;)324 2709 y Fn(or)1551 2830 y Fj(\032)1601 2845 y Fq(\025;\014)1737 2830 y Fn(=)1862 2805 y(^)1841 2830 y Fj(P)1918 2789 y Fg(0)1904 2854 y Fq(\014)1951 2830 y Fn(\()p Fi(j)p Fj(v)2064 2845 y Fq(ef)g(f)2183 2830 y Fi(j)2211 2845 y Fg(\000)2269 2830 y Fn(\))p Fj(:)324 3014 y Fn(Here)566 2989 y(^)544 3014 y Fj(P)607 3029 y Fq(\014)654 3014 y Fn(\()p Fj(w)s Fn(\))27 b(=)h(\(1)r(+)r Fj(\014)6 b Fn(\))1200 2978 y Fg(\000)p Fr(8)p Fq(=)p Fr(5)1365 3014 y Fj(P)1428 3029 y Fq(\014)1475 3014 y Fn(\(\(1)r(+)r Fj(\014)g Fn(\))1779 2978 y Fr(2)p Fq(=)p Fr(5)1889 3014 y Fj(w)s Fn(\),)24 b(and)f Fj(v)2278 3029 y Fq(ef)7 b(f)2397 3014 y Fn(\()p Fj(x)p Fn(\))28 b(=)2670 2974 y Fg(\000)p Fr(1)p 2670 2991 91 4 v 2676 3048 a Fg(j)p Fq(x)p Fg(j)2772 3014 y Fn(+)r Fj(\032)2900 3029 y Fq(\025;\014)3011 3014 y Fi(\003)3095 2974 y Fr(1)p 3073 2991 79 4 v 3073 3048 a Fg(j)p Fq(x)p Fg(j)3165 3014 y Fn(+)r Fj(\027)6 b Fn(\()p Fj(\025;)17 b(\014)6 b Fn(\))p Fj(:)470 3134 y Fn(Notice)32 b(also)g(that)1207 3330 y Fi(\000)1307 3305 y Fn(^)1284 3330 y Fj(P)1347 3345 y Fq(\014)1395 3330 y Fn(\()p Fi(j)p Fj(v)1508 3345 y Fq(ef)7 b(f)1627 3330 y Fi(j)1655 3345 y Fg(\000)1713 3330 y Fn(\))28 b(=)33 b(^)-54 b Fj(\034)1925 3345 y Fq(\014)1972 3330 y Fn(\()p Fj(\032)2060 3345 y Fq(\025;\014)2168 3330 y Fn(\))22 b(+)g Fj(v)2373 3345 y Fq(ef)7 b(f)2493 3330 y Fj(\032)2543 3345 y Fq(\025;\014)2651 3330 y Fj(:)470 3526 y Fn(Using)42 b(the)g(iden)m(tit)m(y)g(from)f(Lemma)83 b(3.5)42 b(w)m(e)h(can)f(pro)m(v)m(e)h(the)g(Theorem)84 b(1.9)42 b(-)324 3646 y(assuming)32 b(the)h(v)-5 b(alidit)m(y)30 b(of)i(Theorem)66 b(1.8:)324 3835 y Fk(Pr)-5 b(o)g(of.)48 b Fn(F)-8 b(rom)33 b(Theorem)70 b(1.8)35 b(and)g(Theorem)69 b(1.7)35 b(w)m(e)h(get)f(that)f(the)h(quan)m(tum)g(me-)324 3955 y(c)m(hanical)c(curren)m(t)j(in)e(the)h(limit)c(b)s(eha)m(v)m(es) 35 b(lik)m(e:)1003 4181 y Fj(Z)1094 4045 y Fh(Z)1221 4113 y Fn(~)-50 b Fj(a)p Fn(\()p Fj(x)p Fn(\))22 b Fi(\001)g Fj(x)p 1220 4158 310 4 v 1300 4249 a Fi(j)p Fj(x)p Fi(j)1411 4220 y Fr(3)1540 4181 y Fj(\032)1590 4196 y Fq(\025;\014)1698 4181 y Fn(\()p Fj(x)p Fn(\))17 b Fj(dx)843 4443 y Fi(\000)1013 4376 y Fn(3)p 1013 4420 49 4 v 1013 4511 a(2)1072 4443 y Fj(Z)1162 4307 y Fh(Z)1262 4443 y Fn(\()p Fj(@)1351 4458 y Fq(x)1391 4467 y Fc(1)1430 4443 y Fj(a)1481 4458 y Fr(2)1543 4443 y Fi(\000)22 b Fj(@)1693 4458 y Fq(x)1733 4467 y Fc(2)1772 4443 y Fj(a)1823 4458 y Fr(1)1863 4443 y Fn(\))1923 4418 y(^)1901 4443 y Fj(P)1964 4458 y Fq(\014)2011 4443 y Fn(\()p Fi(j)p Fj(v)2124 4458 y Fq(ef)7 b(f)2243 4443 y Fi(j)2271 4458 y Fg(\000)2330 4443 y Fn(\))17 b Fj(dx)1003 4698 y Fi(\000)p Fj(Z)1171 4562 y Fh(Z)1271 4698 y Fn(\()p Fj(@)1360 4713 y Fq(x)1400 4722 y Fc(1)1439 4698 y Fj(a)1490 4713 y Fr(2)1551 4698 y Fi(\000)23 b Fj(@)1702 4713 y Fq(x)1742 4722 y Fc(2)1781 4698 y Fj(a)1832 4713 y Fr(1)1872 4698 y Fn(\))p Fj(v)1957 4713 y Fq(ef)7 b(f)2076 4698 y Fj(\032)2126 4713 y Fq(\025;\014)2251 4698 y Fj(dx)843 4959 y Fn(+)84 b Fj(Z)1087 4891 y Fn(1)p 1087 4936 V 1087 5027 a(2)1162 4823 y Fh(Z)12 b(Z)1345 4959 y Fj(\032)1395 4974 y Fq(\025;\014)1503 4959 y Fn(\()p Fj(x)p Fn(\))1644 4891 y(\()p Fj(x)23 b Fi(\000)g Fj(y)t Fn(\))e Fi(\001)h Fn(\()q(~)-50 b Fj(a)p Fn(\()p Fj(x)p Fn(\))22 b Fi(\000)i Fn(~)-50 b Fj(a)p Fn(\()p Fj(y)t Fn(\)\))p 1644 4936 935 4 v 1950 5027 a Fi(j)p Fj(x)22 b Fi(\000)h Fj(y)t Fi(j)2235 4998 y Fr(3)2589 4959 y Fj(\032)2639 4974 y Fq(\025;\014)2747 4959 y Fn(\()p Fj(y)t Fn(\))17 b Fj(dxdy)t(:)1894 5251 y Fn(15)p eop %%Page: 16 16 16 15 bop 324 548 a Fn(If)32 b(w)m(e)i(use)f(the)g(iden)m(tit)m(y)g(ab) s(o)m(v)m(e)g(on)g(the)g(\014rst)g(and)f(the)h(last)f(term)g(w)m(e)i (get:)1122 811 y(=)83 b Fi(\000)1368 743 y Fn(5)p 1368 788 49 4 v 1368 879 a(2)1428 811 y Fj(Z)1518 675 y Fh(Z)1618 811 y Fn(\()p Fj(@)1707 826 y Fq(x)1747 835 y Fc(1)1786 811 y Fj(a)1837 826 y Fr(2)1898 811 y Fi(\000)23 b Fj(@)2049 826 y Fq(x)2089 835 y Fc(2)2128 811 y Fj(a)2179 826 y Fr(1)2219 811 y Fn(\))2278 786 y(^)2257 811 y Fj(P)2320 826 y Fq(\014)2367 811 y Fn(\()p Fi(j)p Fj(v)2480 826 y Fq(ef)7 b(f)2599 811 y Fi(j)2627 826 y Fg(\000)2685 811 y Fn(\))17 b Fj(dx)1281 1065 y Fi(\000)p Fj(Z)1449 930 y Fh(Z)1549 1065 y Fn(\()p Fj(@)1638 1080 y Fq(x)1678 1089 y Fc(1)1717 1065 y Fj(a)1768 1080 y Fr(2)1830 1065 y Fi(\000)22 b Fj(@)1980 1080 y Fq(x)2020 1089 y Fc(2)2059 1065 y Fj(a)2110 1080 y Fr(1)2150 1065 y Fn(\))p Fj(v)2235 1080 y Fq(ef)7 b(f)2354 1065 y Fj(\032)2404 1080 y Fq(\025;\014)2529 1065 y Fj(dx)1122 1320 y Fn(=)1291 1253 y(5)p 1291 1297 V 1291 1388 a(2)1350 1320 y Fj(Z)1441 1184 y Fh(Z)1540 1320 y Fn(\()p Fj(@)1629 1335 y Fq(x)1669 1344 y Fc(1)1708 1320 y Fj(a)1759 1335 y Fr(2)1821 1320 y Fi(\000)22 b Fj(@)1971 1335 y Fq(x)2011 1344 y Fc(2)2050 1320 y Fj(a)2101 1335 y Fr(1)2141 1320 y Fn(\))e(^)-69 b Fj(\034)2221 1335 y Fq(\014)2269 1320 y Fn(\()p Fj(\032)2357 1335 y Fq(\025;\014)2465 1320 y Fn(\))17 b Fj(dx)1281 1575 y Fn(+)1367 1507 y(3)p 1367 1552 V 1367 1643 a(2)1426 1575 y Fj(Z)1516 1439 y Fh(Z)1616 1575 y Fn(\()p Fj(@)1705 1590 y Fq(x)1745 1599 y Fc(1)1784 1575 y Fj(a)1835 1590 y Fr(2)1897 1575 y Fi(\000)22 b Fj(@)2047 1590 y Fq(x)2087 1599 y Fc(2)2126 1575 y Fj(a)2177 1590 y Fr(1)2217 1575 y Fn(\))p Fj(v)2302 1590 y Fq(ef)7 b(f)2421 1575 y Fj(\032)2471 1590 y Fq(\025;\014)2596 1575 y Fj(dx;)324 1838 y Fn(whic)m(h)33 b(is)f(exactly)h(the)g(result)f(from)g(MTF-theory)-8 b(.)p 3491 1838 4 66 v 3495 1775 59 4 v 3495 1838 V 3553 1838 4 66 v 324 2170 a Fp(4)161 b(MTF-theory)53 b(with)h(a)g(curren)l (t)e(term)324 2389 y Fn(In)42 b(order)g(to)g(calculate)f(the)h(curren)m (t)h(w)m(e)g(need)g(to)e(in)m(tro)s(duce)h(a)g(p)s(erturb)s(ed)h(MTF-) 324 2510 y(functional)29 b(with)h(a)g(curren)m(t)i(term.)42 b(W)-8 b(e)31 b(w)m(an)m(t)h(to)e(stress)i(that)e(this)h(functional)e (is)h(not)324 2630 y(mean)m(t)37 b(to)g(ha)m(v)m(e)i(an)m(ything)e(to)g (do)g(with)h(the)f(so-called)f Fk(curr)-5 b(ent)40 b(density)f (functional)324 2751 y(the)-5 b(ories)35 b Fn(app)s(earing)f(in)h(the)h (ph)m(ysics)h(litterature.)51 b(This)35 b(functional)f(is)h(still)e(a)j (func-)324 2871 y(tional)h(of)j(the)g(densit)m(y)h(alone,)f(but)g(it)f (will)f(enable)h(us)i(to)e(calculate)g(a)g(part)h(of)f(the)324 2991 y(curren)m(t.)470 3112 y(The)34 b(functional)d(is:)467 3355 y Fi(E)520 3370 y Fq(C)5 b Fg(\000)p Fq(M)i(T)j(F)r(;t)851 3355 y Fn([)p Fj(\032)p Fn(;)988 3330 y Fj(~)972 3355 y(B)5 b(;)17 b(V)5 b(;)18 b Fn(~)-50 b Fj(a)p Fn(])28 b(=)1411 3220 y Fh(Z)1466 3445 y Fd(R)1514 3426 y Fc(3)1569 3355 y Fj(\034)1611 3371 y Fq(B)s Fr(\()p Fq(x)p Fr(\))1767 3355 y Fn(\()p Fj(\032)p Fn(\()p Fj(x)p Fn(\)\))17 b Fj(dx)22 b Fn(+)2267 3220 y Fh(Z)2323 3445 y Fd(R)2371 3426 y Fc(3)2426 3355 y Fj(V)g Fn(\()p Fj(x)p Fn(\))p Fj(\032)p Fn(\()p Fj(x)p Fn(\))17 b Fj(dx)22 b Fn(+)3083 3330 y(~)3060 3355 y Fj(D)3141 3370 y Fq(t)3171 3355 y Fn(\()p Fj(\032;)17 b(\032)p Fn(\))p Fj(;)324 3650 y Fn(where)618 3625 y(~)595 3650 y Fj(D)676 3665 y Fq(t)705 3650 y Fn(\()p Fj(f)5 b(;)17 b(g)t Fn(\))27 b(=)1070 3611 y Fr(1)p 1070 3627 36 4 v 1070 3685 a(2)1132 3570 y Fh(R)r(R)1264 3650 y Fj(f)11 b Fn(\()p Fj(x)p Fn(\))1464 3603 y Fr(\()p Fq(x)p Fg(\000)p Fq(y)r Fr(\))p Fg(\001)p Fr(\()p Fq(x)p Fg(\000)p Fq(y)r Fr(+)p Fq(t)p Fr(\()q(~)-36 b Fq(a)r Fr(\()p Fq(x)p Fr(\))p Fg(\000)q Fr(~)g Fq(a)q Fr(\()p Fq(y)r Fr(\)\)\))p 1464 3627 846 4 v 1784 3685 a Fg(j)p Fq(x)p Fg(\000)p Fq(y)r Fg(j)1956 3666 y Fc(3)2319 3650 y Fj(g)t Fn(\()p Fj(y)t Fn(\))17 b Fj(dxdy)t(;)i Fn(and)j(where)h(the)f(other)324 3771 y(terms)32 b(are)h(as)g(in)e (standard)i(MTF-theory)-8 b(.)470 3891 y(It)33 b(is)f(easy)h(to)g(see)g (that)1004 4111 y(\(1)22 b Fi(\000)h Fj(tc)p Fn(\))p Fj(D)s Fn(\()p Fj(\032;)17 b(\032)p Fn(\))27 b Fi(\024)1787 4086 y Fn(~)1765 4111 y Fj(D)1846 4126 y Fq(t)1875 4111 y Fn(\()p Fj(\032;)17 b(\032)p Fn(\))28 b Fi(\024)g Fn(\(1)22 b(+)g Fj(tc)p Fn(\))p Fj(D)s Fn(\()p Fj(\032;)17 b(\032)p Fn(\))p Fj(;)324 4331 y Fn(where)23 b Fj(c)g Fn(only)f(dep)s(ends)i(on) e Fi(k)p Fj(D)t Fn(~)-50 b Fj(a)p Fi(k)1591 4346 y Fg(1)1665 4331 y Fn(.)40 b(W)-8 b(e)23 b(will)d(assume)j Fj(t)g Fn(so)f(small)e(that)i(the)h(constan)m(ts)324 4452 y(1)g Fi(\000)g Fj(tc)34 b Fn(and)g(1)23 b(+)g Fj(tc)p Fn(,)34 b(app)s(earing)f(in)g(the)h(inequalit)m(y)f(ab)s(o)m(v)m(e,)i(can)f(b)s (e)g(b)s(ounded)h(b)s(elo)m(w)324 4572 y(resp.)47 b(ab)s(o)m(v)m(e)34 b(b)m(y)g(1)p Fj(=)p Fn(2)f(resp.)47 b(3)p Fj(=)p Fn(2.)f(Therefore)34 b(the)g(pro)s(ofs)f(of)g(the)h(main)e(theorems)h(in)324 4692 y(MTF-theory)j(apply)f(to)g(C-MTF-theory)h(essen)m(tially)f (without)g(c)m(hange)h(so)g(w)m(e)g(only)324 4813 y(state)d(the)g (follo)m(wing)d(conclusion:)1894 5251 y(16)p eop %%Page: 17 17 17 16 bop 324 548 a Fo(Theorem)31 b(4.1.)39 b Fk(F)-7 b(or)29 b(su\016ciently)i(smal)5 b(l)30 b(\(dep)-5 b(ending)28 b(only)j(on)f Fi(k)p Fj(D)t Fn(~)-50 b Fj(a)p Fi(k)2992 563 y Fg(1)3066 548 y Fk(\))31 b Fj(t)f Fk(we)g(have:)324 668 y(Ther)-5 b(e)45 b(is)h(a)g(unique)g(minimizer)f Fj(\032)1678 683 y Fq(t)1757 668 y Fi(2)k(C)1924 683 y Fq(B)s(;N)2114 668 y Fk(of)d Fi(E)2293 683 y Fq(C)5 b Fg(\000)p Fq(M)i(T)j(F)2587 668 y Fk(.)78 b(This)46 b Fj(\032)2981 683 y Fq(t)3057 668 y Fk(satis\014es)g(the)324 789 y(Thomas-F)-7 b(ermi)32 b(e)-5 b(quation:)1503 1009 y Fj(\034)1556 968 y Fg(0)1545 1033 y Fq(B)1606 1009 y Fn(\()p Fj(\032)1694 1024 y Fq(t)1724 1009 y Fn(\()p Fj(x)p Fn(\)\))27 b(=)h Fi(j)p Fj(V)2109 1024 y Fq(ef)7 b(f)s(;t)2269 1009 y Fi(j)2297 1024 y Fg(\000)2356 1009 y Fj(;)324 1242 y Fk(wher)-5 b(e)28 b Fj(V)650 1257 y Fq(ef)7 b(f)s(;t)838 1242 y Fn(=)28 b Fj(V)22 b Fn(\()p Fj(x)p Fn(\))10 b(+)1248 1162 y Fh(R)1340 1194 y Fr(\()p Fq(x)p Fg(\000)p Fq(y)r Fr(\))p Fg(\001)p Fr(\()p Fq(x)p Fg(\000)p Fq(y)r Fr(+)p Fq(t)p Fr(\()q(~)-36 b Fq(a)r Fr(\()p Fq(x)p Fr(\))p Fg(\000)q Fr(~)g Fq(a)q Fr(\()p Fq(y)r Fr(\)\)\))p 1340 1219 846 4 v 1660 1276 a Fg(j)p Fq(x)p Fg(\000)p Fq(y)r Fg(j)1832 1257 y Fc(3)2195 1242 y Fj(\032)2245 1257 y Fq(t)2275 1242 y Fn(\()p Fj(y)t Fn(\))17 b Fj(dy)c Fn(+)d Fj(\027)c Fk(.)41 b(The)28 b(Thomas-F)-7 b(ermi)324 1362 y(e)i(quation)34 b(c)-5 b(an)35 b(also)f(b)-5 b(e)34 b(formulate)-5 b(d:)1024 1582 y Fi(\000)p Fj(P)1164 1597 y Fq(B)1225 1582 y Fn(\()p Fi(j)p Fj(V)1348 1597 y Fq(ef)7 b(f)s(;t)1508 1582 y Fn(\()p Fj(x)p Fn(\))p Fi(j)1667 1597 y Fg(\000)1726 1582 y Fn(\))28 b(=)f Fj(\034)1937 1597 y Fq(B)1998 1582 y Fn(\()p Fj(\032)2086 1597 y Fq(t)2116 1582 y Fn(\()p Fj(x)p Fn(\)\))c(+)f Fj(V)2463 1597 y Fq(ef)7 b(f)s(;t)2623 1582 y Fj(\032)2673 1597 y Fq(t)2703 1582 y Fn(\()p Fj(x)p Fn(\))p Fj(:)324 1814 y Fk(The)34 b(ener)-5 b(gy)35 b Fj(E)903 1829 y Fq(C)5 b Fg(\000)p Fq(M)i(T)j(F)r(;t)1234 1814 y Fn(\()p Fj(N)5 b(;)1415 1789 y(~)1399 1814 y(B)g(;)17 b(Z)7 b Fn(\))35 b Fk(is)f(di\013er)-5 b(entiable)34 b(in)g Fj(t)h Fk(at)h Fj(t)27 b Fn(=)h(0)35 b Fk(and)f(satis\014es)527 2025 y Fj(d)p 510 2069 86 4 v 510 2160 a(dt)606 2092 y(E)678 2107 y Fq(C)5 b Fg(\000)p Fq(M)i(T)j(F)r(;t)1009 2092 y Fn(\()p Fj(N)5 b(;)1190 2067 y(~)1174 2092 y(B)g(;)17 b(Z)7 b Fn(\))28 b(=)1550 2025 y(1)p 1550 2069 49 4 v 1550 2160 a(2)1625 1956 y Fh(Z)1742 2092 y Fj(\032)1792 2107 y Fr(0)1831 2092 y Fn(\()p Fj(x)p Fn(\))1972 2025 y(\()p Fj(x)23 b Fi(\000)g Fj(y)t Fn(\))e Fi(\001)h Fn(\()q(~)-50 b Fj(a)p Fn(\()p Fj(x)p Fn(\))22 b Fi(\000)i Fn(~)-50 b Fj(a)p Fn(\()p Fj(y)t Fn(\)\))p 1972 2069 935 4 v 2278 2160 a Fi(j)p Fj(x)23 b Fi(\000)f Fj(y)t Fi(j)2563 2132 y Fr(3)2917 2092 y Fj(\032)2967 2107 y Fr(0)3007 2092 y Fn(\()p Fj(y)t Fn(\))17 b Fj(dxdy)t(:)324 2416 y Fe(4.1)135 b(Scaled)45 b(C-MTF-theory)324 2601 y Fn(Supp)s(ose)32 b(no)m(w)h(~)-51 b Fj(a)q Fn(\()p Fj(x)p Fn(\))28 b(=)f Fj(l)s Fn(~)-50 b Fj(a)1302 2616 y Fr(0)1342 2601 y Fn(\()p Fj(x=l)r Fn(\))31 b(and)h(de\014ne)37 b(^)-54 b Fj(\034)2095 2616 y Fq(\014)2142 2601 y Fn(\()p Fj(t)p Fn(\))28 b(=)g(\(1)18 b(+)h Fj(\014)6 b Fn(\))2684 2565 y Fg(\000)p Fr(8)p Fq(=)p Fr(5)2848 2601 y Fj(\034)2890 2616 y Fq(\014)2938 2601 y Fn(\(\(1)19 b(+)f Fj(\014)6 b Fn(\))3275 2565 y Fr(6)p Fq(=)p Fr(5)3385 2601 y Fj(t)p Fn(\))28 b(=)324 2721 y(\(1)22 b(+)g Fj(\014)6 b Fn(\))630 2685 y Fr(2)p Fq(=)p Fr(5)739 2721 y Fj(\034)781 2744 y Fq(\014)s Fr(\(1+)p Fq(\014)s Fr(\))1011 2724 y Fa(\000)p Fc(4)p Fb(=)p Fc(5)1162 2721 y Fn(\()p Fj(t)p Fn(\).)43 b(De\014ne)33 b(furthermore,)894 2979 y(^)877 3004 y Fi(E)930 3019 y Fq(\014)s(;t)1022 3004 y Fn([)p Fj(\032)p Fn(])28 b(=)1258 2868 y Fh(Z)1379 3004 y Fn(^)-54 b Fj(\034)1416 3019 y Fq(\014)1463 3004 y Fn(\()p Fj(\032)p Fn(\()p Fj(x)p Fn(\)\))17 b Fj(dx)23 b Fn(+)1964 2868 y Fh(Z)2090 2937 y Fi(\000)p Fn(1)p 2090 2981 127 4 v 2098 3072 a Fi(j)p Fj(x)p Fi(j)2226 3004 y Fj(\032)p Fn(\()p Fj(x)p Fn(\))17 b Fj(dx)23 b Fn(+)f Fj(D)2732 3019 y Fq(t)2761 3004 y Fn(\()p Fj(\032;)17 b(\032)p Fn(\))p Fj(;)324 3302 y Fn(where)33 b Fj(D)686 3317 y Fq(t)716 3302 y Fn(\()p Fj(f)5 b(;)17 b(g)t Fn(\))27 b(=)1080 3263 y Fr(1)p 1080 3280 36 4 v 1080 3337 a(2)1142 3222 y Fh(R)s(R)1275 3302 y Fj(f)11 b Fn(\()p Fj(x)p Fn(\))1475 3255 y Fr(\()p Fq(x)p Fg(\000)p Fq(y)r Fr(\))p Fg(\001)p Fr(\()p Fq(x)p Fg(\000)p Fq(y)r Fr(+)p Fq(t)p Fr(\()q(~)-36 b Fq(a)1984 3264 y Fc(0)2021 3255 y Fr(\()p Fq(x)p Fr(\))p Fg(\000)q Fr(~)g Fq(a)2207 3264 y Fc(0)2242 3255 y Fr(\()p Fq(y)r Fr(\)\)\))p 1475 3280 915 4 v 1829 3337 a Fg(j)p Fq(x)p Fg(\000)p Fq(y)r Fg(j)2001 3318 y Fc(3)2399 3302 y Fj(g)t Fn(\()p Fj(y)t Fn(\))17 b Fj(dxdy)t(:)29 b Fn(Then)34 b(the)f(energy)324 3456 y(corresp)s(onding)f(to)1087 3431 y(^)1070 3456 y Fi(E)1123 3471 y Fq(\014)s(;t)1242 3456 y Fn(:)698 3702 y(^)676 3727 y Fj(E)748 3742 y Fq(C)5 b Fg(\000)p Fq(M)i(T)j(F)r(;t)1079 3727 y Fn(\()p Fj(\025;)17 b(\014)6 b Fn(\))27 b(=)h(inf)6 b Fi(f)1633 3702 y Fn(^)1616 3727 y Fi(E)1669 3742 y Fq(\014)s(;t)1760 3727 y Fn([)p Fj(\032)p Fn(])p Fi(j)p Fj(\032)28 b Fi(2)g Fj(L)2130 3686 y Fr(1)2192 3727 y Fi(\\)23 b Fj(L)2347 3686 y Fr(5)p Fq(=)p Fr(3)2457 3727 y Fj(;)17 b(\032)28 b Fi(\025)g Fn(0)p Fj(;)2777 3592 y Fh(Z)2893 3727 y Fj(\032)g Fi(\024)g Fj(\025)p Fi(g)p Fj(;)324 3990 y Fn(satis\014es)1022 4111 y Fj(E)1094 4126 y Fq(C)5 b Fg(\000)p Fq(M)i(T)j(F)r(;t)1426 4111 y Fn(\()p Fj(N)5 b(;)17 b(B)5 b(;)17 b(Z)7 b Fn(\))27 b(=)g Fj(Z)2030 4070 y Fr(2)2070 4111 y Fj(l)2101 4070 y Fg(\000)p Fr(1)2218 4086 y Fn(^)2195 4111 y Fj(E)2267 4126 y Fq(C)5 b Fg(\000)p Fq(M)i(T)j(F)r(;t)2599 4111 y Fn(\()p Fj(\025;)17 b(\014)6 b Fn(\))p Fj(;)324 4295 y Fn(with)36 b Fj(\025)f Fn(=)g Fj(N)r(=)-5 b(Z)44 b Fn(and)37 b(the)g(minimizers)d Fj(\032)i Fn(=)f Fj(\032)2099 4310 y Fq(C)5 b Fg(\000)p Fq(M)i(T)j(F)r(;t)2431 4295 y Fn(\()p Fj(N)5 b(;)17 b(B)5 b(;)17 b(Z)7 b Fn(\))36 b(and)h Fj(\032)3111 4310 y Fq(\025;\014)s(;t)3301 4295 y Fn(of)3434 4270 y(^)3417 4295 y Fi(E)3470 4310 y Fq(\014)s(;t)324 4416 y Fn(satisfy)1466 4536 y Fj(\032)p Fn(\()p Fj(x)p Fn(\))28 b(=)g Fj(Z)7 b(l)1884 4495 y Fg(\000)p Fr(3)1978 4536 y Fj(\032)2028 4551 y Fq(\025;\014)s(;t)2181 4536 y Fn(\()p Fj(x=l)r Fn(\))p Fj(:)1894 5251 y Fn(17)p eop %%Page: 18 18 18 17 bop 324 548 a Fp(5)161 b(A)54 b(lo)l(w)l(er)f(b)t(ound)g(on)h (the)f(p)t(erturb)t(ed)g(quan)l(tum)566 731 y(energy)324 950 y Fn(In)33 b(this)f(section)h(w)m(e)g(will)d(pro)m(v)m(e)k(Theorem) 66 b(1.8.)324 1139 y Fo(Lemma)36 b(5.1.)41 b Fk(Supp)-5 b(ose)33 b(ther)-5 b(e)34 b(exists)g Fj(C)g(<)28 b Fi(1)33 b Fk(such)h(that)h Fj(\026h)27 b Fi(\024)h Fj(C)7 b Fk(,)34 b(then)g(as)g Fj(h)28 b Fi(!)f Fn(0)324 1259 y Fk(we)34 b(get)1282 1463 y Fi(h)p Fn(\011)1397 1478 y Fq(scal)q(ed)1592 1463 y Fi(j)1644 1438 y Fn(~)1620 1463 y Fj(J)1674 1478 y Fq(N)s(;K)5 b(I)g(N)1920 1463 y Fn(\()p Fj(h;)17 b(\026;)g Fn(~)-49 b Fj(a)o Fn(\))p Fi(j)p Fn(\011)2353 1478 y Fq(scal)q(ed)2549 1463 y Fi(i)1122 1660 y(\031)83 b(\000)1369 1593 y Fn(3)p 1369 1637 49 4 v 1369 1728 a(2)1428 1660 y Fj(Z)1519 1524 y Fh(Z)1618 1660 y Fn(\()p Fj(@)1707 1675 y Fq(x)1747 1684 y Fc(1)1786 1660 y Fj(a)1837 1675 y Fr(2)1899 1660 y Fi(\000)23 b Fj(@)2050 1675 y Fq(x)2090 1684 y Fc(2)2129 1660 y Fj(a)2180 1675 y Fr(1)2219 1660 y Fn(\))2279 1635 y(^)2257 1660 y Fj(P)2320 1675 y Fq(\014)2367 1660 y Fn(\()p Fi(j)p Fj(v)2480 1675 y Fq(ef)7 b(f)2599 1660 y Fi(j)2627 1675 y Fg(\000)2686 1660 y Fn(\))17 b Fj(dx)1282 1915 y Fi(\000)p Fj(Z)1450 1779 y Fh(Z)1550 1915 y Fn(\()p Fj(@)1639 1930 y Fq(x)1679 1939 y Fc(1)1718 1915 y Fj(a)1769 1930 y Fr(2)1830 1915 y Fi(\000)23 b Fj(@)1981 1930 y Fq(x)2021 1939 y Fc(2)2060 1915 y Fj(a)2111 1930 y Fr(1)2151 1915 y Fn(\))p Fj(v)2236 1930 y Fq(ef)7 b(f)2355 1915 y Fj(\032)2405 1930 y Fq(\025;\014)2530 1915 y Fj(dx;)324 2161 y Fk(and)1037 2365 y Fi(h)p Fn(\011)1152 2380 y Fq(scal)q(ed)1348 2365 y Fi(j)1399 2340 y Fn(~)1376 2365 y Fj(J)1430 2380 y Fq(N)s(;I)e(N)i(T)1662 2365 y Fn(\()p Fj(h;)17 b(\026;)g Fn(~)-49 b Fj(a)o Fn(\))p Fi(j)p Fn(\011)2095 2380 y Fq(scal)q(ed)2291 2365 y Fi(i)877 2568 y(\031)83 b Fj(Z)1128 2433 y Fh(Z)11 b(Z)1311 2568 y Fj(\032)1361 2583 y Fq(\025;\014)1469 2568 y Fn(\()p Fj(x)p Fn(\))1610 2501 y(\()p Fj(x)23 b Fi(\000)f Fj(y)t Fn(\))f Fi(\001)h Fn(\()q(~)-50 b Fj(a)p Fn(\()p Fj(x)p Fn(\))23 b Fi(\000)h Fn(~)-50 b Fj(a)p Fn(\()p Fj(y)t Fn(\)\))p 1610 2545 935 4 v 1916 2636 a Fi(j)p Fj(x)22 b Fi(\000)h Fj(y)t Fi(j)2201 2608 y Fr(3)2555 2568 y Fj(\032)2605 2583 y Fq(\025;\014)2713 2568 y Fn(\()p Fj(y)t Fn(\))17 b Fj(dxdy)t(:)470 2828 y Fn(In)33 b(the)g(pro)s(of)f(w) m(e)h(will)e(use)i(the)g(semiclassical)d(lo)m(w)m(er)j(b)s(ound)g(from) 64 b([F)-8 b(ou00)o(]:)324 3027 y Fo(Lemma)37 b(5.2.)42 b Fk(L)-5 b(et)36 b Fj(v)1153 3042 y Fq(ef)7 b(f)s(;t)1342 3027 y Fn(=)28 b Fi(\000j)p Fj(x)p Fi(j)1634 2991 y Fg(\000)p Fr(1)1751 3027 y Fn(+)1849 2947 y Fh(R)1942 2980 y Fr(\()p Fq(x)p Fg(\000)p Fq(y)r Fr(\))p Fg(\001)p Fr(\()p Fq(x)p Fg(\000)p Fq(y)r Fr(+)p Fq(t)p Fr(\()q(~)-36 b Fq(a)r Fr(\()p Fq(x)p Fr(\))p Fg(\000)q Fr(~)g Fq(a)q Fr(\()p Fq(y)r Fr(\)\)\))p 1942 3004 846 4 v 2262 3062 a Fg(j)p Fq(x)p Fg(\000)p Fq(y)r Fg(j)2434 3043 y Fc(3)2797 3027 y Fj(\032)2847 3042 y Fq(t)2877 3027 y Fn(\()p Fj(y)t Fn(\))17 b Fj(dy)24 b Fn(+)e Fj(\027)6 b Fn(\()p Fj(\025;)17 b(\014)6 b Fn(\))p Fj(;)324 3148 y Fk(and)394 3419 y Fj(H)483 3378 y Fr(0)475 3444 y Fq(N)542 3419 y Fn(\()p Fj(t;)17 b(h;)g(\026;)g(v)909 3434 y Fq(ef)7 b(f)s(;t)1069 3419 y Fn(\))28 b(=)1279 3295 y Fq(N)1238 3325 y Fh(X)1249 3535 y Fq(j)t Fr(=1)1382 3419 y Fn(\()p Fj(p)1482 3432 y Fq(~)1469 3449 y(A)1526 3419 y Fn(\))1564 3434 y Fq(j)1623 3419 y Fi(\001)21 b Fj(S)1732 3434 y Fq(t)1762 3419 y Fn(\()p Fj(x)1855 3434 y Fq(j)1892 3419 y Fn(\)\()p Fj(p)2030 3432 y Fq(~)2017 3449 y(A)2074 3419 y Fn(\))2112 3434 y Fq(j)2170 3419 y Fi(\000)i Fj(\026h)p Fn(\(1)f(+)g Fj(tb)2668 3378 y Fr(\(3\))2763 3419 y Fn(\()p Fj(x)2856 3434 y Fq(j)2893 3419 y Fn(\)\))g(+)g Fj(v)3136 3434 y Fq(ef)7 b(f)s(;t)3296 3419 y Fn(\()p Fj(x)3389 3434 y Fq(j)3426 3419 y Fn(\))p Fj(;)324 3734 y Fk(wher)-5 b(e)34 b Fj(S)659 3749 y Fq(t)689 3734 y Fn(\()p Fj(x)p Fn(\))28 b(=)f(1)22 b(+)g Fj(tM)10 b Fn(\()p Fj(x)p Fn(\))p Fk(,)36 b Fj(M)10 b Fn(\()p Fj(x)p Fn(\))29 b(=)e Fi(\000)1917 3653 y Fh(\000)1963 3734 y Fj(D)t Fn(~)-50 b Fj(a)p Fn(\()p Fj(x)p Fn(\))23 b(+)f(\()p Fj(D)t Fn(~)-50 b Fj(a)o Fn(\()p Fj(x)p Fn(\)\))2691 3698 y Fq(T)2747 3653 y Fh(\001)2792 3734 y Fk(.)45 b(Then)607 3938 y Fn(inf)23 b(Sp)s(ec)17 b Fj(H)1045 3896 y Fr(0)1037 3962 y Fq(N)1105 3938 y Fn(\()p Fj(t;)g(h;)g(\026;)g(v)1472 3953 y Fq(ef)7 b(f)s(;t)1631 3938 y Fn(\))28 b Fi(\025)g Fj(E)1874 3953 y Fq(scl)1963 3938 y Fn(\()p Fj(t;)17 b(h;)g(\026;)g(v)2330 3953 y Fq(ef)7 b(f)s(;t)2490 3938 y Fn(\))22 b(+)g Fj(o)p Fn(\()p Fj(h)2789 3896 y Fg(\000)p Fr(3)2906 3938 y Fn(+)g Fj(\026h)3119 3896 y Fg(\000)p Fr(2)3213 3938 y Fn(\))p Fj(;)324 4141 y Fk(wher)-5 b(e)596 4345 y Fj(E)668 4360 y Fq(scl)758 4345 y Fn(\()p Fj(t;)17 b(h;)g(\026;)g(v)1125 4360 y Fq(ef)7 b(f)s(;t)1285 4345 y Fn(\))437 4569 y(=)625 4502 y Fj(\026)p 606 4546 96 4 v 606 4638 a(h)662 4609 y Fr(2)765 4445 y Fg(1)728 4475 y Fh(X)736 4684 y Fq(\027)t Fr(=0)889 4569 y Fj(d)940 4584 y Fq(\027)999 4434 y Fh(Z)1168 4502 y Fn(4)p Fj(b)1258 4517 y Fq(u;t)p 1126 4546 266 4 v 1126 4638 a Fn(6)p Fj(\031)t Fn(\003)1302 4653 y Fq(u;t)1418 4569 y Fn([\(2)p Fj(\027)28 b Fn(+)22 b(1\))p Fj(\026hb)1949 4584 y Fq(u;t)2061 4569 y Fi(\000)h Fj(\026h)p Fn(\(1)f(+)g Fj(tb)2559 4584 y Fr(3)2599 4569 y Fn(\()p Fj(u)p Fn(\)\))f(+)h Fj(v)2935 4584 y Fq(ef)7 b(f)s(;t)3096 4569 y Fn(\()p Fj(u)p Fn(\)])3254 4521 y Fr(3)p Fq(=)p Fr(2)3254 4601 y Fg(\000)3398 4569 y Fj(du;)324 4882 y Fk(wher)-5 b(e)42 b Fj(b)648 4897 y Fq(u;t)780 4882 y Fn(=)g Fi(j)p Fn(rot)1050 4897 y Fq(x)1094 4797 y Fh(p)p 1194 4797 441 4 v 85 x Fn(1)22 b(+)g Fj(tM)10 b Fn(\()p Fj(u)p Fn(\))1652 4857 y Fj(~)1634 4882 y(A)p Fn(\()1745 4797 y Fh(p)p 1845 4797 V 85 x Fn(1)22 b(+)g Fj(tM)10 b Fn(\()p Fj(u)p Fn(\))p Fj(x)p Fn(\))p Fi(j)p Fk(,)44 b Fn(\003)2548 4897 y Fq(u;t)2680 4882 y Fn(=)e Fi(j)17 b Fn(det)2994 4797 y Fh(p)p 3094 4797 V 85 x Fn(1)22 b(+)g Fj(tM)10 b Fn(\()p Fj(u)p Fn(\))p Fi(j)324 5002 y Fk(and)34 b Fj(d)564 5017 y Fr(0)631 5002 y Fn(=)766 4963 y Fr(1)p 744 4979 79 4 v 744 5036 a(2)p Fq(\031)833 5002 y Fk(,)g Fj(d)948 5017 y Fq(\027)1019 5002 y Fn(=)1136 4963 y Fr(1)p 1132 4979 43 4 v 1132 5036 a Fq(\031)1220 5002 y Fk(for)h Fj(\027)f Fi(\025)28 b Fn(1)p Fk(.)1894 5251 y Fn(18)p eop %%Page: 19 19 19 18 bop 324 548 a Fo(Remark)52 b(5.3.)c Fn(Notice,)h(that)c(the)h(b)s (ound)g(ab)s(o)m(v)m(e)g(only)f(w)m(as)i(pro)m(v)m(ed)g(for)e(a)g (\014xed)324 668 y(\(indep)s(enden)m(t)31 b(of)f Fj(h)p Fn(\))h Fj(v)t Fn(.)42 b(In)31 b(the)g(presen)m(t)h(case)g Fj(v)2141 683 y Fq(ef)7 b(f)s(;t)2332 668 y Fn(dep)s(ends)32 b(on)e Fj(h)p Fn(.)43 b(It)31 b(is,)f(ho)m(w)m(ev)m(er,)324 789 y(easy)g(\(see)58 b([LSY94,)29 b(p.99]\))g(to)f(see)i(that)f(the)g (pro)s(of)f(of)g(the)h(b)s(ound)g(holds)f(in)g(our)h(case)324 909 y(as)k(w)m(ell.)324 1112 y Fk(Pr)-5 b(o)g(of.)48 b Fn(De\014ne)38 b Fj(H)1017 1127 y Fq(N)1084 1112 y Fn(\()p Fj(t;)17 b(h;)g(\026)p Fn(\))34 b(=)h Fj(H)1624 1127 y Fq(N)1692 1112 y Fn(\()p Fj(h;)17 b(\026)p Fn(\))24 b(+)h Fj(t)p Fn(\()2149 1087 y(~)2125 1112 y Fj(J)2179 1127 y Fq(N)s(;K)5 b(I)g(N)2425 1112 y Fn(\()p Fj(h;)17 b(\026;)h Fn(~)-50 b Fj(a)p Fn(\))25 b(+)2904 1087 y(~)2881 1112 y Fj(J)2935 1127 y Fq(N)s(;I)5 b(N)i(T)3167 1112 y Fn(\()p Fj(h;)17 b(\026;)h Fn(~)-50 b Fj(a)p Fn(\)\))p Fj(:)324 1233 y Fn(W)-8 b(e)33 b(seek)h(a)e(lo)m(w)m(er)h(b)s(ound)g (on)1168 1453 y Fj(E)1240 1468 y Fq(N)1308 1453 y Fn(\()p Fj(t;)17 b(h;)g(\026)p Fn(\))27 b(=)73 b(inf)1752 1519 y Fg(k)p Fq( )r Fg(k)p Fr(=1)1962 1453 y Fi(h)p Fj( )t Fi(j)p Fj(H)2177 1468 y Fq(N)2243 1453 y Fn(\()p Fj(t;)17 b(h;)g(\026)p Fn(\))p Fi(j)p Fj( )t Fi(i)p Fj(:)470 1735 y Fn(Lo)m(w)m(er)34 b(b)s(ound)p 470 1751 557 4 v 24 w(Let)25 b Fj( )32 b Fi(2)c(^)1473 1699 y Fq(N)1473 1760 y(j)t Fr(=1)1600 1735 y Fj(L)1666 1699 y Fr(2)1706 1735 y Fn(\()p Ff(R)1810 1699 y Fr(3)1855 1735 y Fj(;)17 b Ff(C)1965 1699 y Fr(2)2010 1735 y Fn(\),)26 b(and)f(let)e Fj(\032)2465 1750 y Fq(t)2523 1735 y Fn(=)28 b Fj(\032)2677 1750 y Fq(t;\025;\014)2854 1735 y Fn(b)s(e)d(the)g(minimizer)324 1856 y(of)32 b(scaled)h(C-MTF)g(theory)-8 b(,)33 b(then)887 2076 y Fi(h)p Fj( )t Fi(j)p Fj(H)1102 2091 y Fq(N)1169 2076 y Fn(\()p Fj(h;)17 b(\026)p Fn(\))k(+)h Fj(t)1582 2051 y Fn(~)1558 2076 y Fj(J)1612 2091 y Fq(N)1680 2076 y Fn(\()p Fj(h;)17 b(\026)p Fn(\))p Fi(j)p Fj( )t Fi(i)727 2221 y(\025)83 b Fn(inf)23 b(Sp)s(ec)17 b Fj(H)1325 2180 y Fr(0)1317 2246 y Fq(N)1385 2221 y Fn(\()p Fj(t;)g(h;)g(\026;)g(v)1752 2236 y Fq(ef)7 b(f)s(;t)1911 2221 y Fn(\))887 2424 y(+)p Fj(Z)1037 2383 y Fg(\000)p Fr(1)1132 2424 y Fi(h)p Fj( )t Fi(j)1283 2330 y Fh(X)1291 2542 y Fq(j)t(