%!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: pseudorel.dvi %%Pages: 13 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips pseudorel.dvi %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2005.04.06:1501 %%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 N/hsize 8.5 72 mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{ landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[ matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{ statusdict/manualfeed true put}B/@copies{/#copies X}B/FMat[1 0 0 -1 0 0] N/FBB[0 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Fr(137)e(the)j(\014ne)g(structure)f(constan)n (t.)54 b Fq(x)1714 4458 y Fl(k)1788 4446 y Fr(=)33 b Fo(j)p Fh(x)1959 4458 y Fl(k)2000 4446 y Fo(j)h Fr(giv)n(es)e(the)i(lo) r(cation)f(of)g(the)h(electron)f(with)160 4546 y(resp)r(ect)27 b(to)f(the)h(n)n(ucleus.)36 b(The)27 b(n)n(ucleus)f(is)g(assumed)g(to)g (ha)n(v)n(e)g(in\014nite)h(mass,)f(c)n(harge)e(n)n(um)n(b)r(er)j Fq(Z)32 b Fr(and)160 4645 y(to)c(sit)g(in)g(the)f(origin.)285 4816 y(It)34 b(is)f(an)g(imp)r(ortan)n(t)g(fact)h(that)f(in)n (teracting)g(relativistic)f(electrons)h(cannot)g(adequately)f(b)r(e)i (de-)160 4916 y(scrib)r(ed)28 b(b)n(y)f(an)g(op)r(erator)f(where)h(the) h(electron-electron)e(in)n(teraction)1470 5128 y Fq(V)1536 5094 y Fn(\(12\))1705 5128 y Fr(=)1949 5072 y Fq(e)1988 5042 y Fn(2)p 1825 5109 323 4 v 1825 5185 a Fo(j)p Fh(x)1898 5197 y Fn(1)1955 5185 y Fo(\000)18 b Fh(x)2088 5197 y Fn(2)2125 5185 y Fo(j)3297 5128 y Fr(\(2.3\))1793 5413 y(2)p eop %%Page: 3 3 3 2 bop 160 51 a Fr(is)28 b(simply)h(added)e(to)h(the)h(sum)f(of)g (single-particle)f(op)r(erators.)36 b(This)28 b(is)g(so)g(b)r(ecause,)g (due)g(to)g(the)g(cou-)160 150 y(pling)36 b(of)f(negativ)n(e-)f(and)h (p)r(ositiv)n(e-energy)e(con)n(tin)n(uum)i(states)g(b)n(y)g Fq(V)2442 120 y Fn(\(12\))2564 150 y Fr(,)i(no)e(stable)g(b)r(ound)h (states)160 250 y(w)n(ould)27 b(exist)h([11)o(].)37 b(Instead,)27 b(an)g(appropriate)e(op)r(erator)h(has)h(to)g(b)r(e)h(deriv)n(ed)e (from)h(quan)n(tum)h(electro-)160 349 y(dynamics)20 b(\(QED\).)h(The)g (op)r(erator)d(\(2.1\))i(is)h(due)f(to)h(Suc)n(her)f(who)g(deriv)n(ed)f (it)i(from)f(the)h(Bethe-Salp)r(eter)160 449 y(equation)28 b(of)g(QED)g(b)n(y)g(neglecting)g(pair)g(creation)f(and)h(the)h (radiation)e(\014eld)h([12])g(\(see)g(also)f([4]\).)40 b(The)160 565 y(t)n(w)n(o-particle)19 b(op)r(erator)f Fq(P)990 577 y Fn(++)1117 565 y Fr(pro)5 b(jects)19 b(on)n(to)h(the)g (p)r(ositiv)n(e)g(sp)r(ectral)g(subspace)f(of)2772 503 y Fp(P)2860 523 y Fn(2)2860 590 y Fl(k)q Fn(=1)2985 565 y Fr(\()p Fq(D)3088 522 y Fn(\()p Fl(k)q Fn(\))3086 587 y(0)3184 565 y Fr(+)t Fq(V)3319 535 y Fn(\()p Fl(k)q Fn(\))3412 565 y Fr(\))160 664 y(and)28 b(is)f(de\014ned)h(b)n(y)g(the) g(pro)r(duct)f(of)h(the)g(single-electron)e(pro)5 b(jectors)25 b([9])233 869 y Fq(P)286 881 y Fn(++)438 869 y Fr(=)46 b Fq(P)614 826 y Fn(\(1\))602 889 y(+)703 869 y Fq(P)768 826 y Fn(\(2\))756 889 y(+)857 869 y Fq(;)180 b(P)1125 826 y Fn(\()p Fl(k)q Fn(\))1113 889 y Fg(\006)1264 869 y Fr(=)1384 812 y(1)p 1384 850 42 4 v 1384 926 a(2)1459 869 y(\(1)41 b Fo(\006)1694 812 y Fr(1)p 1690 850 51 4 v 1690 926 a Fq(\031)1764 756 y Fp(Z)1847 776 y Fg(1)1810 944 y(\0001)1946 869 y Fq(d\021)2346 812 y Fr(1)p 2067 850 600 4 v 2067 946 a Fq(D)2138 903 y Fn(\()p Fl(k)q Fn(\))2136 968 y(0)2249 946 y Fr(+)18 b Fq(V)2399 922 y Fn(\()p Fl(k)q Fn(\))2510 946 y Fr(+)g Fq(i\021)2690 869 y Fr(\))p Fq(;)180 b(k)26 b Fr(=)c(1)p Fq(;)14 b Fr(2)p Fq(:)72 b Fr(\(2.4\))160 1121 y(The)21 b(pro)5 b(jectors)19 b Fq(P)773 1078 y Fn(\()p Fl(k)q Fn(\))761 1142 y(+)886 1121 y Fr(and)i Fq(P)1106 1078 y Fn(\()p Fl(k)q Fn(\))1094 1142 y Fg(\000)1221 1121 y Fr(=)i(1)5 b Fo(\000)g Fq(P)1491 1078 y Fn(\()p Fl(k)q Fn(\))1479 1142 y(+)1602 1121 y Fr(\(whic)n(h)21 b(pro)5 b(jects)20 b(on)n(to)f(the)i(negativ)n(e)f(sp)r(ectral)g(subspace\))160 1221 y(are)33 b(w)n(ell-de\014ned)h(b)r(ecause)f(of)h(the)g(existence)g (of)g(a)f(gap)g(in)h(the)g(sp)r(ectrum)g(of)g(the)g (\(single-particle\))160 1336 y(Dirac)28 b(op)r(erator.)38 b(F)-7 b(or)28 b(later)g(purp)r(ose)f(w)n(e)i(de\014ne)f Fq(P)1830 1348 y Fn(+)p Fg(\000)1962 1336 y Fr(=)c Fq(P)2116 1293 y Fn(\(1\))2104 1357 y(+)2205 1336 y Fq(P)2270 1293 y Fn(\(2\))2258 1357 y Fg(\000)2359 1336 y Fq(;)63 b(P)2498 1348 y Fg(\000\000)2630 1336 y Fr(=)24 b Fq(P)2784 1293 y Fn(\(1\))2772 1357 y Fg(\000)2873 1336 y Fq(P)2938 1293 y Fn(\(2\))2926 1357 y Fg(\000)3056 1336 y Fr(and)k Fq(P)3271 1348 y Fg(\000)p Fn(+)3403 1336 y Fr(=)160 1459 y Fq(P)225 1416 y Fn(\(1\))213 1480 y Fg(\000)314 1459 y Fq(P)379 1416 y Fn(\(2\))367 1480 y(+)468 1459 y Fq(:)285 1630 y(H)d Fr(acts)18 b(on)g(the)h(Hilb)r(ert)g(space)e(of)i (an)n(tisymmetrised)e(t)n(w)n(o-electron)g(4-spinors,)h Fo(H)2869 1642 y Fn(2)2930 1630 y Fr(:=)k Fo(A)p Fr(\()p Fq(H)3207 1642 y Fn(1)3245 1630 y Fr(\()p Fm(R)3331 1600 y Fn(3)3375 1630 y Fr(\))p Fo(\012)160 1730 y Fm(C)214 1700 y Fn(4)258 1730 y Fr(\))290 1700 y Fn(2)355 1730 y Fr(where)27 b Fq(H)664 1742 y Fn(1)701 1730 y Fr(\()p Fm(R)787 1700 y Fn(3)831 1730 y Fr(\))h(is)f(the)h(Sob)r(olev)f(space)g (of)g(\014rst)h(order,)788 1929 y Fq(H)857 1941 y Fn(1)895 1929 y Fr(\()p Fm(R)981 1895 y Fn(3)1024 1929 y Fr(\))47 b(=)e Fo(f)p Fq(')23 b Fo(2)h Fq(L)1468 1941 y Fn(2)1504 1929 y Fr(\()p Fm(R)1591 1895 y Fn(3)1634 1929 y Fr(\))f(:)1758 1816 y Fp(Z)1805 2005 y Ff(R)1852 1988 y Fe(3)1883 1929 y Fr(\(1)18 b(+)g Fq(p)2100 1895 y Fn(2)2137 1929 y Fr(\))24 b Fo(j)13 b Fr(^)-55 b Fq(')p Fr(\()p Fh(p)p Fr(\))p Fo(j)2410 1895 y Fn(2)2471 1929 y Fq(d)p Fh(p)37 b Fq(<)23 b Fo(1g)p Fq(;)457 b Fr(\(2.5\))160 2139 y Fq(L)217 2151 y Fn(2)254 2139 y Fr(\()p Fm(R)340 2108 y Fn(3)384 2139 y Fr(\))21 b(b)r(eing)f(the)h(space)f(of)g(of)h(square-in)n(tegrable)c (one-electron)i(functions,)j Fq(p)h Fr(=)g Fo(j)p Fh(p)p Fo(j)e Fr(and)33 b(^)-55 b Fq(')21 b Fr(denoting)160 2238 y(the)28 b(F)-7 b(ourier)27 b(transform)f(of)i Fq(':)285 2409 y Fr(The)h(w)n(a)n(y)e(to)i(construct)f(the)h(desired)f(op)r (erator)f(is)i(most)f(readily)g(displa)n(y)n(ed)g(in)h(the)g (one-electron)160 2520 y(case.)36 b(Let)28 b Fq(D)588 2477 y Fn(\()p Fl(k)q Fn(\))586 2545 y Fl(V)704 2520 y Fr(:=)23 b Fq(D)886 2477 y Fn(\()p Fl(k)q Fn(\))884 2543 y(0)997 2520 y Fr(+)18 b Fq(V)1147 2490 y Fn(\()p Fl(k)q Fn(\))1240 2520 y Fq(:)50 b Fr(Then)28 b(one)f(has)g(the)h (decomp)r(osition)1073 2697 y Fq(D)1144 2654 y Fn(\()p Fl(k)q Fn(\))1142 2722 y Fl(V)1283 2697 y Fr(=)46 b Fq(P)1459 2654 y Fn(\()p Fl(k)q Fn(\))1447 2718 y(+)1565 2697 y Fq(D)1636 2654 y Fn(\()p Fl(k)q Fn(\))1634 2722 y Fl(V)1743 2697 y 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Fq(p)1093 4500 y Fl(k)1134 4488 y Fr(\))23 b(\(1)32 b(+)g Fq(\014)1443 4454 y Fn(\()p Fl(k)q Fn(\))1590 4432 y Fk(\013)1653 4402 y Fn(\()p Fl(k)q Fn(\))1746 4432 y Fh(p)1799 4444 y Fl(k)p 1560 4469 311 4 v 1560 4545 a Fq(E)1621 4557 y Fl(p)1655 4566 y Fi(k)1714 4545 y Fr(+)18 b Fq(m)1880 4488 y Fr(\))p Fq(;)180 b(A)p Fr(\()p Fq(p)2251 4500 y Fl(k)2293 4488 y Fr(\))46 b(=)2482 4371 y Fp(\022)2553 4432 y Fq(E)2614 4444 y Fl(p)2648 4453 y Fi(k)2707 4432 y Fr(+)18 b Fq(m)p 2553 4469 V 2619 4545 a Fr(2)p Fq(E)2722 4557 y Fl(p)2756 4566 y Fi(k)2873 4371 y Fp(\023)2944 4366 y Fe(1)p 2944 4375 29 3 v 2944 4408 a(2)3297 4488 y Fr(\(2.9\))160 4728 y(with)33 b(its)f(in)n(v)n(erse)e Fq(U)819 4685 y Fn(\()p Fl(k)q Fn(\))p Fg(\000)p Fn(1)810 4750 y(0)1041 4728 y Fr(=)g(\(1)35 b Fo(\000)g Fq(\014)1396 4698 y Fn(\()p Fl(k)q Fn(\))1523 4691 y Fc(\013)1573 4666 y Fe(\()p Fi(k)q Fe(\))1654 4691 y Fj(p)1696 4700 y Fi(k)p 1512 4709 230 4 v 1512 4756 a Fl(E)1561 4764 y Fi(p)1592 4779 y(k)1632 4756 y Fn(+)p 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V 2556 5094 a(2)2621 5037 y(\(1)p Fo(\000)p Fq(\014)2811 5003 y Fn(\()p Fl(k)q Fn(\))2904 5037 y Fr(\))14 b Fq(M)3064 4981 y Fr(1)p 3064 5018 V 3064 5094 a(2)3129 5037 y(\(1)p Fo(\000)p Fq(\014)3319 5003 y Fn(\()p Fl(k)q Fn(\))3412 5037 y Fr(\))p Fq(;)3255 5164 y Fr(\(2.10\))1793 5413 y(3)p eop %%Page: 4 4 4 3 bop 160 88 a Fr(since)36 b Fq(U)438 45 y Fn(\()p Fl(k)q Fn(\))429 110 y(0)531 88 y Fr(\003)589 45 y Fn(\()p Fl(k)q Fn(\))589 109 y Fg(\006)681 88 y Fq(U)747 45 y Fn(\()p Fl(k)q Fn(\))p Fg(\000)p Fn(1)738 110 y(0)976 88 y Fr(=)1088 55 y Fn(1)p 1088 69 34 4 v 1088 117 a(2)1131 88 y Fr(\(1)24 b Fo(\006)g Fq(\014)1369 58 y Fn(\()p Fl(k)q Fn(\))1462 88 y Fr(\))52 b(=)1648 -29 y Fp(\022)1750 37 y Fq(I)90 b Fr(0)1751 137 y(0)83 b(0)1960 -29 y Fp(\023)2094 88 y Fr(resp.)2329 -29 y Fp(\022)2431 37 y Fr(0)h(0)2431 137 y(0)f Fq(I)2640 -29 y Fp(\023)2753 88 y Fo(2)38 b Fm(C)2900 58 y Fn(4)p Fl(;)p Fn(4)3032 88 y Fr(for)e(the)g(plus)160 243 y(resp.)52 b(min)n(us)33 b(sign,)g(with)h Fq(I)39 b Fr(the)33 b(unit)h(matrix)e(in)h Fm(C)1862 213 y Fn(2)p Fl(;)p Fn(2)1958 243 y Fr(.)52 b(Note)33 b(that)g(eac)n(h)f(of)h(the)g (t)n(w)n(o)f(op)r(erators)e(in)160 342 y(\(2.10\))k(e\013ectiv)n(ely)g (acts)g(on)h(a)f(2-spinor)e(space,)k(b)r(ecause)e(for)2205 275 y Fp(\000)2244 305 y Fl(u)2283 313 y Fe(+)2243 371 y Fl(u)2282 379 y Fb(\000)2331 275 y Fp(\001)2404 342 y Fo(2)h Fq(H)2563 354 y Fn(1)2600 342 y Fr(\()p Fm(R)2686 312 y Fn(3)2730 342 y Fr(\))23 b Fo(\012)f Fm(C)2926 312 y Fn(4)2970 342 y Fr(,)36 b(one)e(has)g(e.g.)170 436 y Fn(1)p 170 450 V 170 497 a(2)213 468 y Fr(\(1)19 b(+)g Fq(\014)441 438 y Fn(\()p Fl(k)q Fn(\))534 468 y Fr(\))566 401 y Fp(\000)605 431 y Fl(u)644 439 y Fe(+)604 497 y Fl(u)643 505 y Fb(\000)692 401 y Fp(\001)769 468 y Fr(=)857 401 y Fp(\000)896 431 y Fl(u)935 439 y Fe(+)922 497 y Fn(0)982 401 y Fp(\001)1020 468 y Fq(:)28 b Fr(Unitary)g (transformations)e(preserv)n(e)h(the)i(sp)r(ectrum)f(suc)n(h)g(that)h (w)n(e)f(ha)n(v)n(e)170 574 y Fn(1)p 170 588 V 170 636 a(2)213 607 y Fr(\(1)20 b(+)f Fq(\014)442 577 y Fn(\()p Fl(k)q Fn(\))535 607 y Fr(\))14 b Fq(M)695 574 y Fn(1)p 695 588 V 695 636 a(2)738 607 y Fr(\(1)20 b(+)f Fq(\014)967 577 y Fn(\()p Fl(k)q Fn(\))1060 607 y Fr(\))40 b Fo(\025)26 b Fr(0)k(from)f Fq(P)1558 564 y Fn(\()p Fl(k)q Fn(\))1546 628 y(+)1650 607 y Fq(D)1721 564 y Fn(\()p Fl(k)q Fn(\))1719 631 y Fl(V)1814 607 y Fq(P)1879 564 y Fn(\()p Fl(k)q Fn(\))1867 628 y(+)2012 607 y Fo(\025)d Fr(0)p Fq(:)55 b Fr(This)30 b(p)r(ositiv)n(e)f(op)r(erator)f(is)h(therefore)160 707 y(the)f(desired)f(one-particle)g(op)r(erator)f(for)h(describing)f (solely)h(the)h(electronic)f(degrees)f(of)i(freedom.)285 878 y(Let)37 b(us)g(turn)h(to)f(the)h(t)n(w)n(o-electron)d(case.)65 b(W)-7 b(e)37 b(follo)n(w)g(Douglas)f(and)h(Kroll)f([4])h(to)g(w)n(ork) f(with)160 977 y(pro)r(ducts)c(of)g(single-particle)e(pro)5 b(jectors.)48 b(W)-7 b(e)33 b(ha)n(v)n(e)d Fq(P)1975 989 y Fn(++)2103 977 y Fr(+)21 b Fq(P)2242 989 y Fn(+)p Fg(\000)2371 977 y Fr(+)g Fq(P)2510 989 y Fg(\000)p Fn(+)2638 977 y Fr(+)g Fq(P)2777 989 y Fg(\000\000)2916 977 y Fr(=)29 b(1)62 b(as)32 b(w)n(ell)f(as)160 1077 y Fq(P)213 1089 y Fn(+)p Fg(\000)321 1077 y Fq(P)374 1089 y Fn(++)503 1077 y Fr(=)23 b Fq(P)644 1089 y Fg(\000)p Fn(+)751 1077 y Fq(P)804 1089 y Fn(++)934 1077 y Fr(=)f Fq(P)1074 1089 y Fg(\000\000)1182 1077 y Fq(P)1235 1089 y Fn(++)1365 1077 y Fr(=)g(0)51 b(and)27 b(the)h(decomp)r(osition)628 1252 y Fq(H)53 b Fr(=)46 b Fq(P)914 1264 y Fn(++)1034 1252 y Fq(H)21 b(P)1177 1264 y Fn(++)1324 1252 y Fr(+)42 b Fq(P)1484 1264 y Fn(+)p Fg(\000)1605 1252 y Fq(H)20 b(P)1747 1264 y Fn(+)p Fg(\000)1896 1252 y Fr(+)41 b Fq(P)2055 1264 y Fg(\000)p Fn(+)2176 1252 y Fq(H)21 b(P)2319 1264 y Fg(\000)p Fn(+)2468 1252 y Fr(+)41 b Fq(P)2627 1264 y Fg(\000\000)2749 1252 y Fq(H)20 b(P)2891 1264 y Fg(\000\000)3255 1252 y Fr(\(2.11\))160 1427 y(where)39 b(it)g(is)g(used)g(that)h(single-particle)d(op)r(erators)g(acting)i(on) g(electron)f(1)g(comm)n(ute)h(with)h(those)160 1527 y(acting)32 b(on)f(electron)g(2.)48 b(One)32 b(should)f(k)n(eep)g(in)h(mind)g(that) g(\(2.11\))f(di\013ers)h(from)f(the)h(decomp)r(osition)160 1642 y Fq(H)e Fr(=)23 b Fq(P)412 1599 y Fn(\(12\))400 1663 y(+)534 1642 y Fq(H)7 b(P)675 1599 y Fn(\(12\))663 1663 y(+)828 1642 y Fr(+)17 b Fq(P)975 1599 y Fn(\(12\))963 1663 y Fg(\000)1097 1642 y Fq(H)7 b(P)1238 1599 y Fn(\(12\))1226 1663 y Fg(\000)1387 1642 y Fr(if)28 b Fq(P)1528 1599 y Fn(\(12\))1516 1663 y Fg(\006)1677 1642 y Fr(w)n(ere)e(c)n(hosen)g (to)h(pro)5 b(ject)27 b(on)n(to)f(the)i(p)r(ositiv)n(e/negativ)n(e)160 1742 y(sp)r(ectral)g(subspace)g(of)g Fq(H)60 b Fr(\()p Fd(including)30 b Fr(the)e(electron-electron)f(in)n(teraction\).)39 b(No)n(w)28 b(w)n(e)g(translate)f(the)160 1842 y(step)h(from)e(\(2.6\)) h(to)g(\(2.7\))g(to)g(the)g(t)n(w)n(o-particle)e(case,)i(i.e.)36 b(w)n(e)27 b(searc)n(h)f(for)g(a)h(unitary)f(transformation)160 1941 y Fq(U)40 b Fr(whic)n(h)32 b(casts)e Fq(H)38 b Fr(in)n(to)31 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(transformed)e(Coulom)n(b-Dirac)g(op)r(erator)160 2439 y(has)27 b(the)h(desired)f(blo)r(c)n(k-diagonal)f(shap)r(e)h(to)h (order)e Fq(n)h Fr(in)h Fq(e)2050 2409 y Fn(2)2087 2439 y Fr(,)1291 2614 y Fq(U)1357 2580 y Fg(\000)p Fn(1)1460 2614 y Fq(H)20 b(U)55 b Fr(=)46 b Fq(H)1848 2580 y Fn(\()p Fl(n)p Fn(\))1986 2614 y Fr(+)41 b Fq(R)2156 2580 y Fn(\()p Fl(n)p Fn(+1\))3255 2614 y Fr(\(2.12\))163 2856 y Fq(H)239 2822 y Fn(\()p Fl(n)p Fn(\))382 2856 y Fr(=)k(\003)550 2868 y Fn(++)670 2856 y Fr(\()742 2752 y Fl(n)702 2777 y Fp(X)709 2956 y Fl(l)p Fn(=0)836 2856 y Fq(H)905 2868 y Fl(l)930 2856 y Fr(\))14 b(\003)1034 2868 y Fn(++)1173 2856 y Fr(+)31 b(\003)1327 2868 y Fn(+)p Fg(\000)1448 2856 y Fr(\()1520 2752 y Fl(n)1480 2777 y Fp(X)1487 2956 y Fl(l)p Fn(=0)1614 2856 y Fq(H)1683 2868 y Fl(l)1708 2856 y Fr(\))14 b(\003)1812 2868 y Fn(+)p Fg(\000)1952 2856 y Fr(+)31 b(\003)2106 2868 y Fg(\000)p Fn(+)2227 2856 y Fr(\()2299 2752 y Fl(n)2259 2777 y Fp(X)2266 2956 y Fl(l)p Fn(=0)2393 2856 y Fq(H)2462 2868 y Fl(l)2487 2856 y Fr(\))14 b(\003)2591 2868 y Fg(\000)p Fn(+)2717 2856 y Fr(+)32 b(\003)2872 2868 y Fg(\000\000)2993 2856 y Fr(\()3065 2752 y Fl(n)3025 2777 y Fp(X)3032 2956 y Fl(l)p Fn(=0)3159 2856 y Fq(H)3228 2868 y Fl(l)3254 2856 y Fr(\))14 b(\003)3358 2868 y Fg(\000\000)160 3085 y Fr(where)33 b Fq(H)475 3097 y Fl(l)533 3085 y Fr(is)g(an)g(op)r(erator) e(dep)r(ending)i(on)g Fq(U)41 b Fr(and)33 b(con)n(taining)f(the)h(p)r (oten)n(tial)g(strength)f Fq(e)3158 3055 y Fn(2)3228 3085 y Fr(to)h Fq(l)r Fr(-th)160 3185 y(order,)e Fq(l)f Fr(=)f(0)p Fq(;)14 b Fr(1)p Fq(;)g(:::;)g(n;)29 b Fr(while)j Fq(R)1205 3155 y Fn(\()p Fl(n)p Fn(+1\))1417 3185 y Fr(is)f(a)f (remainder)g(whic)n(h)h(still)g(allo)n(ws)f(for)h(transitions)f(b)r(et) n(w)n(een)160 3284 y(the)25 b(p)r(ositiv)n(e)g(and)f(negativ)n(e)g(sp)r (ectral)g(subspaces)g(of)g(the)h(electrons)f(and)h(whic)n(h)f(is)h(of)g (\()p Fq(n)13 b Fr(+)g(1\)-st)23 b(order)160 3384 y(in)k Fq(e)295 3354 y Fn(2)358 3384 y Fr(relativ)n(e)e(to)g(the)i(free)f (Dirac)f(op)r(erator.)35 b(The)26 b(t)n(w)n(o-particle)e(pro)5 b(jectors)24 b(are)h(de\014ned)i(in)f(terms)g(of)160 3495 y(pro)r(ducts)i(of)f(the)h(single-particle)e(pro)5 b(jectors)26 b(as)h(b)r(efore)g(\(e.g.)37 b(\003)2267 3507 y Fn(+)p Fg(\000)2397 3495 y Fr(=)22 b(\003)2542 3452 y Fn(\(1\))2542 3516 y(+)2631 3495 y Fr(\003)2689 3452 y Fn(\(2\))2689 3516 y Fg(\000)2778 3495 y Fr(\))p Fq(:)285 3666 y Fr(F)-7 b(or)27 b Fq(U)9 b Fr(\()p Fq(t)p Fr(\))23 b(=)g Fq(e)744 3636 y Fl(iB)s(t)849 3666 y Fq(;)60 b(t)23 b Fo(2)h Fm(R)p Fq(;)33 b Fr(w)n(e)28 b(ha)n(v)n(e)e(the)i(op)r (erator)e(iden)n(tit)n(y)i(\(see)f(e.g.)37 b(Sob)r(olev)27 b([14)o(]\))895 3899 y Fq(U)9 b Fr(\()p Fo(\000)p Fq(t)p Fr(\))14 b Fq(H)20 b(U)9 b Fr(\()p Fq(t)p Fr(\))47 b(=)f Fq(H)i Fr(+)41 b Fq(i)1807 3786 y Fp(Z)1889 3806 y Fl(t)1852 3975 y Fn(0)1932 3899 y Fq(dt)2005 3865 y Fg(0)2052 3899 y Fq(U)9 b Fr(\()p Fo(\000)p Fq(t)2245 3865 y Fg(0)2268 3899 y Fr(\))14 b([)p Fq(H)r(;)g(B)t 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Fr(can)f(b)r(e)h(decomp)r(osed)f(in)n(to)1345 4745 y Fq(A)46 b Fr(=)g(pro)5 b(j)26 b(\()p Fq(A)p Fr(\))43 b(+)e(o\013)27 b(\()p Fq(A)p Fr(\))p Fq(;)973 b Fr(\(2.14\))512 4920 y(pro)5 b(j)27 b(\()p Fq(A)p Fr(\))47 b(=)e(\003)1031 4932 y Fn(++)1151 4920 y Fq(A)14 b Fr(\003)1285 4932 y Fn(++)1432 4920 y Fr(+)41 b(\003)1596 4932 y Fn(+)p Fg(\000)1717 4920 y Fq(A)14 b Fr(\003)1851 4932 y Fn(+)p Fg(\000)1999 4920 y Fr(+)41 b(\003)2163 4932 y Fg(\000)p Fn(+)2284 4920 y Fq(A)14 b Fr(\003)2418 4932 y Fg(\000)p Fn(+)2566 4920 y Fr(+)41 b(\003)2730 4932 y Fg(\000\000)2851 4920 y Fq(A)14 b Fr(\003)2985 4932 y Fg(\000\000)3093 4920 y Fq(;)160 5065 y Fr(where)37 b(pro)5 b(j\()p Fq(A)p Fr(\))37 b(do)r(es)f(not)h(couple)f(the)i(sp)r(ectral)e(subspaces)g (while)h(o\013\()p Fq(A)p Fr(\))h(consists)e(of)g(all)h(12)f(o\013-)160 5164 y(diagonal)26 b(terms)g(\(suc)n(h)h(as)f(\003)1100 5176 y Fn(++)1206 5164 y Fq(A)p Fr(\003)1326 5176 y Fn(+)p Fg(\000)1433 5164 y Fq(;)60 b Fr(\003)1574 5176 y Fn(++)1680 5164 y Fq(A)p Fr(\003)1800 5176 y Fg(\000)p Fn(+)1907 5164 y Fq(;)g Fr(\003)2048 5176 y Fn(++)2153 5164 y Fq(A)p Fr(\003)2273 5176 y Fg(\000\000)2381 5164 y Fq(;)14 b(:::)p Fr(\).)37 b(The)27 b(op)r(erators)e Fq(B)3179 5176 y Fn(1)3216 5164 y Fq(;)14 b(:::;)g(B)3422 5176 y Fl(n)1793 5413 y Fr(4)p eop %%Page: 5 5 5 4 bop 160 51 a Fr(de\014ning)37 b Fq(U)46 b Fr(are)36 b(determined)h(in)g(the)g(follo)n(wing)f(w)n(a)n(y)-7 b(.)64 b(W)-7 b(e)37 b(decomp)r(ose)f(the)h(p)r(oten)n(tial)g(in)g Fq(H)44 b Fr(in)n(to)160 150 y(diagonal)35 b(and)h(o\013-diagonal)e (terms)i(according)e(to)i(\(2.14\))g(and)g(consider)f Fq(B)2675 162 y Fn(1)2712 150 y Fq(;)14 b(:::;)g(B)2918 162 y Fl(n)2999 150 y Fr(as)36 b(expansion)160 250 y(co)r(e\016cien)n (ts)28 b(of)f Fq(B)32 b Fr(in)c(the)g(p)r(oten)n(tial)f(strength)h Fq(e)1728 220 y Fn(2)1764 250 y Fq(;)60 b(B)1910 262 y Fl(l)1964 250 y Fr(b)r(eing)27 b(of)h(order)e(\()p Fq(e)2570 220 y Fn(2)2607 250 y Fr(\))2639 220 y Fl(l)2665 250 y Fq(;)60 b(l)24 b Fr(=)f(1)p Fq(;)14 b(:::;)g(n:)285 437 y Fr(In)32 b(order)e(to)i(obtain)g(a)f(consisten)n(t)g(p)r (erturbativ)n(e)g(expansion)g(w)n(e)g(also)g(ha)n(v)n(e)g(to)g(expand)h Fq(P)3274 394 y Fn(\()p Fl(k)q Fn(\))3262 458 y(+)3398 437 y Fr(in)160 560 y(p)r(o)n(w)n(ers)38 b(of)h Fq(e)592 530 y Fn(2)629 560 y Fr(.)72 b(This)39 b(is)g(done)g(b)n(y)g(inserting) g(the)g(resolv)n(en)n(t)f(iden)n(tit)n(y)-7 b(,)42 b(\()p Fq(D)2666 517 y Fn(\()p Fl(k)q Fn(\))2664 582 y(0)2785 560 y Fr(+)26 b Fq(V)2943 530 y Fn(\()p Fl(k)q Fn(\))3062 560 y Fr(+)f Fq(i\021)s Fr(\))3257 530 y Fg(\000)p Fn(1)3403 560 y Fr(=)160 677 y(\()p Fq(D)263 634 y Fn(\()p Fl(k)q Fn(\))261 699 y(0)369 677 y Fr(+)13 b Fq(i\021)s Fr(\))552 647 y Fg(\000)p Fn(1)667 677 y Fo(\000)g Fr(\()p Fq(D)848 634 y Fn(\()p Fl(k)q Fn(\))846 699 y(0)953 677 y Fr(+)g Fq(i\021)s Fr(\))1136 647 y Fg(\000)p Fn(1)1225 677 y Fq(V)1292 647 y Fn(\()p Fl(k)q Fn(\))1398 677 y Fr(\()p Fq(D)1501 634 y Fn(\()p Fl(k)q Fn(\))1499 699 y(0)1607 677 y Fr(+)g Fq(V)1751 647 y Fn(\()p Fl(k)q Fn(\))1856 677 y Fr(+)g Fq(i\021)s Fr(\))2039 647 y Fg(\000)p Fn(1)2128 677 y Fq(;)48 b Fr(rep)r(eatedly)24 b(in)n(to)h(the)g(de\014nition)g (\(2.4\),)716 900 y Fq(P)781 857 y Fn(\()p Fl(k)q Fn(\))769 920 y(+)920 900 y Fr(=)46 b(\003)1089 857 y Fn(\()p Fl(k)q Fn(\))1089 920 y(+)1223 900 y Fo(\000)1364 844 y Fr(1)p 1339 881 92 4 v 1339 957 a(2)p Fq(\031)1454 787 y Fp(Z)1537 807 y Fg(1)1501 975 y(\0001)1637 900 y Fq(d\021)1905 844 y Fr(1)p 1757 881 339 4 v 1757 977 a Fq(D)1828 934 y Fn(\()p Fl(k)q Fn(\))1826 999 y(0)1939 977 y Fr(+)18 b Fq(i\021)2119 900 y(V)2186 865 y Fn(\()p Fl(k)q Fn(\))2581 844 y Fr(1)p 2302 881 600 4 v 2302 977 a Fq(D)2373 934 y Fn(\()p Fl(k)q Fn(\))2371 999 y(0)2485 977 y Fr(+)g Fq(V)2634 953 y Fn(\()p Fl(k)q Fn(\))2746 977 y Fr(+)g Fq(i\021)3255 900 y Fr(\(2.15\))279 1228 y(=)46 b(\003)448 1185 y Fn(\()p Fl(k)q Fn(\))448 1249 y(+)582 1228 y Fo(\000)723 1172 y Fr(1)p 698 1209 92 4 v 698 1285 a(2)p Fq(\031)814 1115 y Fp(Z)897 1136 y Fg(1)860 1304 y(\0001)996 1228 y Fq(d\021)1241 1172 y Fr(1)p 1093 1209 339 4 v 1093 1306 a Fq(D)1164 1263 y Fn(\()p Fl(k)q Fn(\))1162 1328 y(0)1275 1306 y Fr(+)18 b Fq(i\021)1455 1228 y(V)1522 1194 y Fn(\()p Fl(k)q Fn(\))1642 1086 y Fp( )1866 1172 y Fr(1)p 1718 1209 V 1718 1306 a Fq(D)1789 1263 y Fn(\()p Fl(k)q Fn(\))1787 1328 y(0)1900 1306 y Fr(+)g Fq(i\021)2098 1228 y Fo(\000)2353 1172 y Fr(1)p 2205 1209 V 2205 1306 a Fq(D)2276 1263 y Fn(\()p Fl(k)q Fn(\))2274 1328 y(0)2387 1306 y Fr(+)g Fq(i\021)2567 1228 y(V)2634 1194 y Fn(\()p Fl(k)q Fn(\))2899 1172 y Fr(1)p 2750 1209 V 2750 1306 a Fq(D)2821 1263 y Fn(\()p Fl(k)q Fn(\))2819 1328 y(0)2933 1306 y Fr(+)g Fq(i\021)3131 1228 y Fr(+)g Fq(:::)3283 1086 y Fp(!)1281 1472 y Fr(=:)23 b(\003)1450 1429 y Fn(\()p Fl(k)q Fn(\))1450 1493 y(+)1584 1472 y Fr(+)41 b Fq(F)1755 1429 y Fn(\()p Fl(k)q Fn(\))1743 1494 y(0)1889 1472 y Fr(+)g Fq(F)2060 1429 y Fn(\()p Fl(k)q Fn(\))2048 1494 y(1)2194 1472 y Fr(+)18 b Fq(:::)160 1616 y Fr(This)28 b(leads)f(to)g(the)h(expansion)f(of)h(the)g(electron-electron)d(in)n (teraction)i(term,)836 1791 y Fq(P)889 1803 y Fn(++)1009 1791 y Fq(V)1076 1757 y Fn(\(12\))1212 1791 y Fq(P)1265 1803 y Fn(++)1417 1791 y Fr(=)46 b(\003)1586 1803 y Fn(++)1705 1791 y Fq(V)1772 1757 y Fn(\(12\))1908 1791 y Fr(\003)1966 1803 y Fn(++)2114 1791 y Fr(+)41 b Fq(W)2298 1803 y Fn(2)2377 1791 y Fr(+)g Fq(W)2561 1803 y Fn(3)2640 1791 y Fr(+)18 b Fq(:::)463 b Fr(\(2.16\))160 1982 y Fq(W)238 1994 y Fn(2)322 1982 y Fr(=)46 b Fq(F)498 1939 y Fn(\(1\))486 2004 y(0)587 1982 y Fr(\003)645 1939 y Fn(\(2\))645 2002 y(+)734 1982 y Fq(V)801 1948 y Fn(\(12\))923 1982 y Fr(\003)981 1994 y Fn(++)1124 1982 y Fr(+)38 b(\003)1285 1939 y Fn(\(1\))1285 2002 y(+)1373 1982 y Fq(F)1438 1939 y Fn(\(2\))1426 2004 y(0)1527 1982 y Fq(V)1594 1948 y Fn(\(12\))1716 1982 y Fr(\003)1774 1994 y Fn(++)1918 1982 y Fr(+)f(\003)2078 1994 y Fn(++)2184 1982 y Fq(V)2251 1948 y Fn(\(12\))2373 1982 y Fq(F)2438 1939 y Fn(\(1\))2426 2004 y(0)2527 1982 y Fr(\003)2585 1939 y Fn(\(2\))2585 2002 y(+)2712 1982 y Fr(+)g(\003)2872 1994 y Fn(++)2978 1982 y Fq(V)3044 1948 y Fn(\(12\))3167 1982 y Fr(\003)3225 1939 y Fn(\(1\))3225 2002 y(+)3313 1982 y Fq(F)3378 1939 y Fn(\(2\))3366 2004 y(0)160 2126 y Fr(with)30 b(corresp)r(onding)c(expressions)i(for)g(the) h(terms)f Fq(W)1902 2138 y Fl(l)1957 2126 y Fr(of)h Fq(l)r Fr(-th)f(order)g(in)h Fq(e)2570 2096 y Fn(2)2606 2126 y Fq(;)64 b(l)27 b(>)d Fr(2)p Fq(:)54 b Fr(Then)29 b(\(2.13\))f(is)160 2226 y(written)g(in)g(the)g(follo)n(wing)f(w)n(a)n(y)438 2475 y Fq(U)504 2441 y Fg(\000)p Fn(1)607 2475 y Fq(H)20 b(U)55 b Fr(=)963 2371 y Fn(2)919 2396 y Fp(X)919 2575 y Fl(k)q Fn(=1)1054 2475 y Fq(D)1125 2432 y Fn(\()p Fl(k)q Fn(\))1123 2497 y(0)1259 2475 y Fr(+)41 b(pro)5 b(j)13 b(\()1605 2371 y Fn(2)1561 2396 y Fp(X)1561 2575 y Fl(k)q Fn(=1)1696 2475 y Fq(V)1763 2441 y Fn(\()p Fl(k)q Fn(\))1855 2475 y Fr(\))42 b(+)f(o\013)14 b(\()2215 2371 y Fn(2)2171 2396 y Fp(X)2171 2575 y Fl(k)q Fn(=1)2306 2475 y Fq(V)2373 2441 y Fn(\()p Fl(k)q Fn(\))2466 2475 y Fr(\))41 b(+)g(\003)2703 2487 y Fn(++)2823 2475 y Fq(V)2890 2441 y Fn(\(12\))3026 2475 y Fr(\003)3084 2487 y Fn(++)462 2816 y Fr(+)22 b Fq(W)627 2828 y Fn(2)707 2816 y Fr(+)c Fq(:::)g Fr(+)41 b Fq(i)23 b Fr([)1102 2712 y Fn(2)1058 2737 y Fp(X)1058 2916 y Fl(k)q Fn(=1)1193 2816 y Fq(D)1264 2773 y Fn(\()p Fl(k)q Fn(\))1262 2838 y(0)1356 2816 y Fq(;)14 b(B)1456 2828 y Fn(1)1512 2816 y Fr(+)k Fq(:::)h Fr(+)f Fq(B)1829 2828 y Fl(n)1874 2816 y Fr(])41 b(+)h Fq(i)22 b Fr([)2163 2712 y Fn(2)2119 2737 y Fp(X)2119 2916 y Fl(k)q Fn(=1)2254 2816 y Fq(V)2321 2782 y Fn(\()p Fl(k)q Fn(\))2413 2816 y Fq(;)14 b(B)2513 2828 y Fn(1)2569 2816 y Fr(+)k Fq(:::)h Fr(+)f Fq(B)2886 2828 y Fl(n)2931 2816 y Fr(])301 b(\(2.17\))174 3127 y(+)23 b Fq(i)g Fr([\003)395 3139 y Fn(++)500 3127 y Fq(V)567 3093 y Fn(\(12\))689 3127 y Fr(\003)747 3139 y Fn(++)853 3127 y Fq(;)14 b(B)953 3139 y Fn(1)1009 3127 y Fr(+)k Fq(:::)g Fr(+)g Fq(B)1325 3139 y Fl(n)1371 3127 y Fr(])41 b(+)18 b Fq(:::)h Fr(+)1722 3071 y Fq(i)1751 3041 y Fn(2)p 1722 3108 66 4 v 1734 3184 a Fr(2)1821 3127 y([)14 b([)1924 3023 y Fn(2)1881 3048 y Fp(X)1881 3227 y Fl(k)q Fn(=1)2015 3127 y Fq(D)2086 3084 y Fn(\()p Fl(k)q Fn(\))2084 3149 y(0)2179 3127 y Fq(;)g(B)2279 3139 y Fn(1)2335 3127 y Fr(+)k Fq(:::)g Fr(+)g Fq(B)2651 3139 y Fl(n)2696 3127 y Fr(])p Fq(;)c(B)2819 3139 y Fn(1)2875 3127 y Fr(+)k Fq(:::)h Fr(+)f Fq(B)3192 3139 y Fl(n)3237 3127 y Fr(])41 b(+)19 b Fq(:::)160 3356 y Fr(where)28 b Fq(:::)g Fr(sym)n(b)r(olise)f(terms)g(of)h(at)g(least)f(third)h (order)f(in)h Fq(e)2059 3326 y Fn(2)2096 3356 y Fq(:)g Fr(The)g(op)r(erator)e Fq(B)2716 3368 y Fn(1)2781 3356 y Fr(is)i(determined)g(from)160 3456 y(the)g(requiremen)n(t)f(that)h (the)g(linear)f(o\013-diagonal)e(term)j(in)g(\(2.17\))f(is)g (cancelled,)1143 3709 y(o\013)14 b(\()1323 3605 y Fn(2)1279 3630 y Fp(X)1279 3809 y Fl(k)q Fn(=1)1414 3709 y Fq(V)1481 3675 y Fn(\()p Fl(k)q Fn(\))1573 3709 y Fr(\))42 b(+)f Fq(i)23 b Fr([)1872 3605 y Fn(2)1828 3630 y Fp(X)1828 3809 y Fl(k)q Fn(=1)1963 3709 y Fq(D)2034 3666 y Fn(\()p Fl(k)q Fn(\))2032 3731 y(0)2126 3709 y Fq(;)14 b(B)2226 3721 y Fn(1)2263 3709 y Fr(])47 b(=)e(0)770 b(\(2.18\))160 3968 y(and)28 b(it)g(follo)n(ws)e(that)i Fq(B)920 3980 y Fn(1)985 3968 y Fr(is)g(linear)f(in)g(the)h(p)r(oten)n(tial)g(\(i.e.) 37 b(in)28 b Fq(e)2201 3938 y Fn(2)2238 3968 y Fr(\).)285 4139 y(W)-7 b(e)38 b(con)n(tin)n(ue)f(b)n(y)g(collecting)g(all)h(terms) f(on)g(the)h(r.h.s.)67 b(of)37 b(\(2.17\))g(whic)n(h)h(con)n(tain)f (the)h(p)r(oten-)160 4248 y(tial)30 b(to)f(second)g(order)g(\(and)g (call)h(them)g Fq(V)1515 4260 y Fn(2)1579 4248 y Fr(:=)c Fq(W)1771 4260 y Fn(2)1828 4248 y Fr(+)20 b Fq(i)p Fr([)1965 4185 y Fp(P)2052 4206 y Fn(2)2052 4272 y Fl(k)q Fn(=1)2191 4248 y Fq(V)2257 4217 y Fn(\()p Fl(k)q Fn(\))2350 4248 y Fq(;)14 b(B)2450 4260 y Fn(1)2487 4248 y Fr(])34 b(+)19 b Fq(i)p Fr([\003)2738 4260 y Fn(++)2844 4248 y Fq(V)2911 4217 y Fn(\(12\))3033 4248 y Fr(\003)3091 4260 y Fn(++)3197 4248 y Fq(;)14 b(B)3297 4260 y Fn(1)3334 4248 y Fr(])46 b(+)170 4335 y Fl(i)193 4310 y Fe(2)p 170 4349 56 4 v 182 4396 a Fn(2)250 4367 y Fr([)14 b([)310 4305 y Fp(P)398 4326 y Fn(2)398 4392 y Fl(k)q Fn(=1)536 4367 y Fq(D)607 4324 y Fn(\()p Fl(k)q Fn(\))605 4390 y(0)700 4367 y Fq(;)g(B)800 4379 y Fn(1)837 4367 y Fr(])p Fq(;)g(B)960 4379 y Fn(1)997 4367 y Fr(])g(\))71 b(and)35 b(split)g(them)h(in)n(to)e Fq(V)1941 4379 y Fn(2)2014 4367 y Fr(=)14 b(pro)5 b(j\()p Fq(V)2324 4379 y Fn(2)2361 4367 y Fr(\)+)14 b(o\013\()p Fq(V)2642 4379 y Fn(2)2679 4367 y Fr(\))p Fq(:)71 b Fr(Then)35 b Fq(B)3092 4379 y Fn(2)3164 4367 y Fr(is)g(deter-)160 4467 y(mined)28 b(from)g(the)g(condition)f(that)h(the)g(second-order)d (o\013-diagonal)h(term)h(disapp)r(ears,)1236 4716 y(o\013)14 b(\()p Fq(V)1420 4728 y Fn(2)1458 4716 y Fr(\))41 b(+)g Fq(i)23 b Fr([)1756 4612 y Fn(2)1712 4637 y Fp(X)1712 4816 y Fl(k)q Fn(=1)1847 4716 y Fq(D)1918 4673 y Fn(\()p Fl(k)q Fn(\))1916 4738 y(0)2011 4716 y Fq(;)14 b(B)2111 4728 y Fn(2)2148 4716 y Fr(])46 b(=)g(0)p Fq(:)862 b Fr(\(2.19\))160 4965 y(This)25 b(pro)r(cedure,)f(con)n(tin)n(ued)f(to)i (order)e Fq(n)p Fr(,)h(determines)g(successiv)n(ely)f(all)h(op)r (erators)e Fq(B)2941 4977 y Fl(l)2967 4965 y Fq(;)60 b(l)24 b Fr(=)f(1)p Fq(;)14 b(:::;)g(n:)160 5065 y Fr(It)25 b(can)e(b)r(e)h(sho)n(wn)f(that)h(all)g Fq(B)1103 5077 y Fl(l)1152 5065 y Fr(are)f(b)r(ounded)h(self-adjoin)n(t)g(op)r (erators)d(on)j(\()p Fq(L)2610 5077 y Fn(2)2647 5065 y Fr(\()p Fm(R)2733 5035 y Fn(3)2776 5065 y Fr(\))11 b Fo(\012)g Fm(C)2949 5035 y Fn(4)2992 5065 y Fr(\))3024 5035 y Fn(2)3062 5065 y Fq(;)47 b Fr(suc)n(h)23 b(that)160 5164 y(the)28 b(transformation)e Fq(U)37 b Fr(is)27 b(w)n (ell-de\014ned.)1793 5413 y(5)p eop %%Page: 6 6 6 5 bop 160 51 a Fw(3)135 b(The)39 b(transformed)i(pseudo-relativistic) g(second-order)f(op-)362 200 y(erator)160 453 y Fr(F)-7 b(or)27 b Fq(n)c Fr(=)f(2,)27 b(w)n(e)g(obtain)f(the)i(transformed)d (op)r(erator)h(\(to)h(order)e(\()p Fq(e)2301 423 y Fn(2)2339 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4764 y(inating)30 b Fq(B)507 4776 y Fn(1)574 4764 y Fr(with)g(the)g (help)g(of)g(\(3.5\))f(from)g(the)h(t)n(w)n(o-particle)e(second-order)g (term,)i(the)g(exp)r(ectation)160 4864 y(v)-5 b(alue)28 b(of)f Fq(H)545 4833 y Fn(\(2\))662 4864 y Fr(can)g(b)r(e)h(reduced)f (to)h(the)g(follo)n(wing)e(expression)262 5098 y(\()p Fq( )s(;)14 b(H)464 5063 y Fn(\(2\))567 5098 y Fq( )s Fr(\))47 b(=)f(\()p Fq( )s(;)940 4956 y Fp( )1049 4994 y Fn(2)1006 5019 y Fp(X)1006 5198 y Fl(k)q Fn(=1)1126 5098 y Fr(\()p Fq(D)1229 5055 y Fn(\()p Fl(k)q Fn(\))1227 5120 y(0)1341 5098 y Fr(+)18 b Fq(V)1491 5063 y Fn(\()p Fl(k)q Fn(\))1616 5098 y Fr(+)1729 5042 y Fq(i)p 1723 5079 42 4 v 1723 5155 a Fr(2)1788 5098 y([o\013)c(\()p Fq(V)2014 5063 y Fn(\()p Fl(k)q Fn(\))2107 5098 y Fr(\))p Fq(;)g(B)2243 5055 y Fn(\()p Fl(k)q Fn(\))2239 5120 y(1)2336 5098 y Fr(]\))32 b(+)g Fq(V)2587 5063 y Fn(\(12\))2742 5098 y Fr(+)f Fq(C)2903 5063 y Fn(\(12\))3026 4956 y Fp(!)3105 5098 y Fq( )s Fr(\))103 b(\(3.6\))1793 5413 y(6)p eop 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Fn(\))1506 1071 y(+)1613 1050 y Fr(\()p Fq(H)1714 1062 y Fn(1)1751 1050 y Fr(\()p Fm(R)1837 1020 y Fn(3)1881 1050 y Fr(\))18 b Fo(\012)g Fm(C)2068 1020 y Fn(4)2112 1050 y Fr(\))51 b(can)27 b(b)r(e)h(expressed)e(b)n(y)925 1280 y Fq(')46 b Fr(=)g Fq(U)1202 1237 y Fn(\()p Fl(k)q Fn(\))p Fg(\000)p Fn(1)1193 1302 y(0)1393 1163 y Fp(\022)1454 1224 y Fq(u)1502 1236 y Fn(+)1485 1337 y Fr(0)1557 1163 y Fp(\023)1664 1280 y Fr(=)g Fq(U)1841 1237 y Fn(\()p Fl(k)q Fn(\))p Fg(\000)p Fn(1)1832 1302 y(0)2051 1224 y Fr(1)p 2051 1261 42 4 v 2051 1337 a(2)2103 1280 y(\(1)18 b(+)g Fq(\014)2329 1246 y Fn(\()p Fl(k)q Fn(\))2422 1280 y Fr(\))2477 1163 y Fp(\022)2539 1224 y Fq(u)2587 1236 y Fn(+)2539 1337 y Fq(u)2587 1349 y Fg(\000)2642 1163 y Fp(\023)3297 1280 y Fr(\(3.7\))160 1528 y(with)29 b Fq(u)398 1540 y Fn(+)453 1528 y Fq(;)14 b(u)538 1540 y Fg(\000)618 1528 y Fo(2)25 b Fq(H)767 1540 y Fn(1)804 1528 y Fr(\()p Fm(R)891 1498 y Fn(3)934 1528 y Fr(\))19 b Fo(\012)g Fm(C)1123 1498 y Fn(2)1195 1528 y Fr(and)28 b Fq(U)1423 1485 y Fn(\()p Fl(k)q Fn(\))1414 1550 y(0)1544 1528 y Fr(the)h(F)-7 b(oldy-W)g(outh)n(uysen)28 b(transformation)f (from)h(\(2.9\).)39 b(In)160 1627 y(the)28 b(t)n(w)n(o-particle)e(case) h(w)n(e)g(de\014ne)h(for)f Fq(u)c Fo(2)g(A)14 b Fr(\()p Fq(H)1755 1639 y Fn(1)1793 1627 y Fr(\()p Fm(R)1879 1597 y Fn(3)1922 1627 y Fr(\))19 b Fo(\012)f Fm(C)2110 1597 y Fn(2)2153 1627 y Fr(\))2185 1597 y Fn(2)2250 1627 y Fr(the)28 b(op)r(erator)e Fq(h)2776 1597 y Fn(\(2\))2893 1627 y Fr(b)n(y)h(means)g(of)1365 1801 y(\()p Fq( )s(;)14 b(H)1567 1767 y Fn(\(2\))1670 1801 y Fq( )s Fr(\))46 b(=)g(\()p Fq(u;)14 b(h)2081 1767 y Fn(\(2\))2183 1801 y Fq(u)p Fr(\))1034 b(\(3.8\))160 1909 y Fp(\022)263 1978 y Fq(h)311 1947 y Fn(\(2\))483 1978 y Fr(0)311 2077 y(0)130 b(0)566 1909 y Fp(\023)673 2026 y Fr(=)794 1970 y(1)p 794 2007 V 794 2083 a(2)845 2026 y(\(1)17 b(+)g Fq(\014)1069 1992 y Fn(\(1\))1159 2026 y Fr(\))1215 1970 y(1)p 1215 2007 V 1215 2083 a(2)1266 2026 y(\(1)g(+)g Fq(\014)1490 1992 y Fn(\(2\))1579 2026 y Fr(\))24 b Fq(U)1701 1983 y Fn(\(1\))1692 2048 y(0)1790 2026 y Fq(U)1856 1983 y Fn(\(2\))1847 2048 y(0)1958 2026 y Fq(H)2034 1992 y Fn(\(2\))2137 2026 y Fq(U)2203 1983 y Fn(\(1\))p Fg(\000)p Fn(1)2194 2048 y(0)2377 2026 y Fq(U)2443 1983 y Fn(\(2\))p Fg(\000)p Fn(1)2434 2048 y(0)2650 1970 y Fr(1)p 2650 2007 V 2650 2083 a(2)2701 2026 y(\(1)17 b(+)g Fq(\014)2925 1992 y Fn(\(1\))3014 2026 y Fr(\))3071 1970 y(1)p 3071 2007 V 3071 2083 a(2)3122 2026 y(\(1)g(+)g Fq(\014)3346 1992 y Fn(\(2\))3435 2026 y Fr(\))160 2233 y(where)22 b Fq(h)443 2203 y Fn(\(2\))554 2233 y Fr(is)g(a)g(4,4)f(matrix-v)-5 b(alued)22 b(op)r(erator.)33 b(It)22 b(can)g(b)r(e)h(sho)n(wn)e(that)i Fq(h)2523 2203 y Fn(\(2\))2634 2233 y Fr(agrees)d(with)j(the)g(second-) 160 2333 y(order)29 b(op)r(erator)g(deriv)n(ed)g(in)i([4)o(])g (\(except)f(for)g(an)g(error)f(corrected)f(b)n(y)i(Jansen)g(and)g(Hess) g([15)o(]\).)46 b(The)160 2433 y(single-particle)37 b(part)g(of)g Fq(h)1044 2403 y Fn(\(2\))1171 2433 y Fr(is)g(termed)h(Jansen-Hess)e (op)r(erator)g(and)h(its)h(semib)r(oundedness)g(and)160 2532 y(sp)r(ectral)27 b(prop)r(erties)g(are)g(deriv)n(ed)g(in)g([16)o (,)h(17)o(].)160 2877 y Fw(4)135 b(The)45 b(relativ)l(e)i(form)e(b)t (oundedness)f(of)h(the)g(p)t(oten)l(tial)160 3130 y Fr(F)-7 b(or)35 b(a)h(mathematical)f(analysis)f(the)i(pseudo-relativistic)e(op) r(erator)g(in)i(its)f(represen)n(tation)f Fq(H)3287 3100 y Fn(\(2\))3412 3130 y Fr(is)160 3230 y(more)d(appropriate)f(than)i (the)g(represen)n(tation)e Fq(h)1764 3200 y Fn(\(2\))1853 3230 y Fq(;)i Fr(in)g(particular)e(b)r(ecause)h(the)h(t)n(w)n (o-particle)e(con-)160 3329 y(tribution)f(to)g Fq(h)663 3299 y Fn(\(2\))781 3329 y Fr(is)g(v)n(ery)e(in)n(v)n(olv)n(ed)g(\(its) j(second-order)c(term)j(has)f(up)h(to)g(no)n(w)f(b)r(een)h(dropp)r(ed)g (in)g(all)160 3429 y(atomic)g(structure)g(calculations,)g(while)h(the)g (\014rst-order)e(term)h(has)g(b)r(een)h(replaced)f(b)n(y)g(the)h(un)n (trans-)160 3529 y(formed)e(op)r(erator)e Fq(V)841 3499 y Fn(\(12\))991 3529 y Fr([6]\).)285 3700 y(The)i(op)r(erator)d Fq(H)866 3670 y Fn(\(2\))983 3700 y Fr(de\014ned)j(in)g(\(3.6\))f(is)h (for)f(brevit)n(y)g(written)g(in)h(the)g(follo)n(wing)f(w)n(a)n(y)1500 3873 y Fq(H)1576 3839 y Fn(\(2\))1711 3873 y Fr(=)45 b Fq(T)53 b Fr(+)41 b Fq(W)n(;)1169 b Fr(\(4.1\))622 4126 y Fq(T)57 b Fr(=)45 b Fq(D)909 4083 y Fn(\(1\))907 4148 y(0)1031 4126 y Fr(+)32 b Fq(D)1199 4083 y Fn(\(2\))1197 4148 y(0)1288 4126 y Fq(;)180 b(W)58 b Fr(=)1782 4022 y Fn(2)1738 4047 y Fp(X)1738 4226 y Fl(k)q Fn(=1)1858 4126 y Fr(\()p Fq(V)1958 4091 y Fn(\()p Fl(k)q Fn(\))2083 4126 y Fr(+)31 b Fq(V)2246 4083 y Fn(\()p Fl(k)q Fn(\))2227 4148 y(2)2339 4126 y Fr(\))i(+)f Fq(V)2567 4091 y Fn(\(12\))2722 4126 y Fr(+)g Fq(C)2884 4091 y Fn(\(12\))160 4373 y Fr(with)d Fq(V)416 4330 y Fn(\()p Fl(k)q Fn(\))398 4395 y(2)546 4373 y Fr(=)662 4341 y Fl(i)p 657 4355 34 4 v 657 4402 a Fn(2)714 4373 y Fr([\003)795 4330 y Fn(\()p Fl(k)q Fn(\))795 4394 y(+)888 4373 y Fq(V)955 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Fr(and)k Fq(C)j Fo(\025)22 b Fr(0)27 b(a)h(real)e(n)n(um)n(b)r(er.)37 b(This)27 b(pro)n(v)n(es)f(that)i Fq(H)2388 4961 y Fn(\(2\))2505 4991 y Fr(is)f(b)r(ounded)h(from)f(b)r (elo)n(w,)696 5164 y(\()p Fq( )s(;)14 b Fr(\()p Fq(T)30 b Fr(+)18 b Fq(W)12 b Fr(\))i Fq( )s Fr(\))47 b Fo(\025)f Fr(\(1)18 b Fo(\000)g Fq(c)p Fr(\))23 b(\()p Fq( )s(;)14 b(T)e( )s Fr(\))41 b Fo(\000)g Fq(C)30 b Fr(\()p Fq( )s(;)14 b( )s Fr(\))46 b Fo(\025)g(\000)14 b Fq(C)29 b Fr(\()p Fq( )s(;)14 b( )s Fr(\))366 b(\(4.3\))1793 5413 y(7)p eop %%Page: 8 8 8 7 bop 160 51 a Fr(since)41 b(1)27 b Fo(\000)g Fq(c)45 b(>)g Fr(0)c(and)g Fo(j)p Fq(T)12 b Fo(j)44 b(\025)h Fr(0)p Fq(:)86 b Fr(Therefore,)44 b(the)d(symmetric)g(op)r(erator)e Fq(H)2814 20 y Fn(\(2\))2944 51 y Fr(is)i(w)n(ell-de\014ned)160 150 y(in)e(the)f(form)f(sense,)j(and)e(there)g(exists)f(its)h(F)-7 b(riedric)n(hs)37 b(extension)g(to)h(a)f(self-adjoin)n(t)h(op)r(erator) e(on)160 250 y(\003)218 262 y Fn(++)338 250 y Fr(\()p Fo(A)p Fr(\()p Fq(L)525 262 y Fn(2)563 250 y Fr(\()p Fm(R)649 220 y Fn(3)692 250 y Fr(\))19 b Fo(\012)f Fm(C)880 220 y Fn(4)923 250 y Fr(\))955 220 y Fn(2)993 250 y Fr(\))p Fq(:)285 421 y Fr(W)-7 b(e)26 b(recall)f(that)i(for)e(an)n(y)h(state)g Fq( )j Fr(in)d(the)h(p)r(ositiv)n(e)e(sp)r(ectral)h(subspace)f Fo(H)2659 433 y Fn(+)p Fl(;)p Fn(2)2793 421 y Fr(one)h(has)f(\()p Fq( )s(;)14 b(T)e( )s Fr(\))37 b(=)160 530 y(\()p Fq( )s(;)14 b Fr(\()p Fq(E)379 542 y Fl(p)413 550 y Fe(1)474 530 y Fr(+)23 b Fq(E)623 542 y Fl(p)657 550 y Fe(2)694 530 y Fr(\))14 b Fq( )s Fr(\))p Fq(:)72 b Fr(Since)35 b Fq(E)1209 542 y Fl(p)1243 551 y Fi(k)1319 530 y Fr(=)1419 460 y Fp(p)p 1502 460 295 4 v 70 x Fq(p)1544 502 y Fn(2)1544 555 y Fl(k)1604 530 y Fr(+)18 b Fq(m)1760 506 y Fn(2)1846 530 y Fo(\025)35 b Fq(p)1988 542 y Fl(k)2099 530 y Fr(it)h(is)f (therefore)f(su\016cien)n(t)h(to)g(sho)n(w)f(that)160 630 y(all)d(con)n(tributions)g(to)g Fq(W)43 b Fr(are)30 b(form)h(b)r(ounded)g(relativ)n(e)f(to)h Fq(p)2145 642 y Fn(1)2203 630 y Fr(+)20 b Fq(p)2330 642 y Fn(2)2367 630 y Fq(:)61 b Fr(Moreo)n(v)n(er,)29 b(the)i(b)r(ounds)h(of)f(an)160 730 y(op)r(erator)19 b(are)h(not)h(c)n(hanged)f(b)n(y)g(a)h(unitary)f (transformation,)h(suc)n(h)f(that)h(the)h(single-particle)d(estimates) 160 829 y(can)28 b(b)r(e)g(tak)n(en)f(from)g(existing)g(w)n(ork)f(on)i (the)g(Jansen-Hess)e(op)r(erator.)285 1012 y(F)-7 b(or)25 b(the)h(one-particle)f(op)r(erator)f Fq(D)1428 969 y Fn(\()p Fl(k)q Fn(\))1426 1036 y Fl(V)1546 1012 y Fr(acting)i(on)f Fq(')f Fo(2)f Fr(\003)2121 1024 y Fn(+)2176 1012 y Fr(\()p Fq(H)2277 1024 y Fn(1)2314 1012 y Fr(\()p Fm(R)2401 982 y Fn(3)2444 1012 y Fr(\))15 b Fo(\012)f Fm(C)2624 982 y Fn(4)2668 1012 y Fr(\))26 b(\(kno)n(wn)f(b)n(y)h(the)g(name)160 1112 y(Bro)n(wn-Ra)n(v)n(enhall)f(op)r(erator\),)h(Burenk)n(o)n(v)g (and)h(Ev)-5 b(ans)27 b([18)o(])h(deriv)n(ed)f(the)h(follo)n(wing)e(b)r (ound,)996 1316 y(\()p Fq(';)14 b Fr(\()p Fq(D)1222 1272 y Fn(\()p Fl(k)q Fn(\))1220 1338 y(0)1348 1316 y Fo(\000)1475 1259 y Fq(\015)p 1455 1296 89 4 v 1455 1372 a(x)1502 1384 y Fl(k)1553 1316 y Fr(\))g Fq(')p Fr(\))47 b Fo(\025)f Fr(\(1)32 b Fo(\000)2107 1259 y Fq(\015)p 2056 1296 151 4 v 2056 1372 a(\015)2099 1384 y Fl(B)s(R)2216 1316 y Fr(\))24 b(\()p Fq(';)14 b(E)2456 1328 y Fl(p)2490 1337 y Fi(k)2545 1316 y Fq(')p Fr(\))666 b(\(4.4\))160 1567 y(with)36 b Fq(\015)400 1579 y Fl(B)s(R)544 1567 y Fr(=)772 1534 y Fn(2)p 655 1548 268 4 v 655 1595 a Fl(\031)r(=)p Fn(2+2)p Fl(=\031)968 1567 y Fr(=)g(0)p Fq(:)p Fr(906)p Fq(:)70 b Fr(When)36 b Fq(D)1674 1524 y Fn(\()p Fl(k)q Fn(\))1672 1591 y Fl(V)1802 1567 y Fr(is)f(estimated)h(b)n(y)f(using)g (a)g(t)n(w)n(o-particle)f(function)160 1680 y Fq( )40 b Fo(2)c(H)415 1692 y Fn(+)p Fl(;)p Fn(2)523 1680 y Fr(,)i(w)n(e)d(set) g Fq( )s Fr(\()p Fh(x)990 1692 y Fn(1)1028 1680 y Fq(;)14 b Fh(x)1115 1692 y Fn(2)1153 1680 y Fr(\))36 b(=:)g Fq( )1399 1692 y Fl(x)1437 1700 y Fe(2)1473 1680 y Fr(\()p Fh(x)1555 1692 y Fn(1)1593 1680 y Fr(\))g(and)f(consider)f Fh(x)2211 1692 y Fn(2)2285 1680 y Fr(as)g(parameter.)59 b(Then)36 b(\(4.4\))f(holds)160 1780 y(for)d Fq(')e Fr(:=)g Fq( )548 1792 y Fl(x)586 1800 y Fe(2)623 1780 y Fr(\()p Fh(x)705 1792 y Fn(1)743 1780 y Fr(\))i(and)g Fq(k)h Fr(=)d(1,)i(and)g (subsequen)n(t)g(in)n(tegration)e(o)n(v)n(er)g Fh(x)2495 1792 y Fn(2)2565 1780 y Fr(pro)n(v)n(es)g(the)i(general)f(result)160 1880 y(that)k(the)f(estimates)g(of)g(single-particle)f(op)r(erators)f (are)i(unc)n(hanged)f(if)i(t)n(w)n(o-particle)d(functions)j(are)160 1979 y(used.)51 b(The)32 b(estimate)g(of)h(the)f(electron-electron)f (in)n(teraction)g(follo)n(ws)g(immediately)h(from)g(\(4.4\))g(for)160 2079 y Fq( )27 b Fo(2)c(H)389 2091 y Fn(+)p Fl(;)p Fn(2)497 2079 y Fq(;)712 2235 y Fr(\()p Fq( )s(;)14 b(V)905 2200 y Fn(\(12\))1041 2235 y Fq( )s Fr(\))46 b Fo(\024)1334 2179 y Fq(e)1373 2148 y Fn(2)p 1297 2216 151 4 v 1297 2292 a Fq(\015)1340 2304 y Fl(B)s(R)1480 2235 y Fr(\()p Fq( )s(;)14 b(E)1667 2247 y Fl(p)1701 2255 y Fe(1)1753 2235 y Fq( )s Fr(\))46 b(=)2067 2179 y Fq(e)2106 2148 y Fn(2)p 2009 2216 192 4 v 2009 2292 a Fr(2)p Fq(\015)2094 2304 y Fl(B)s(R)2234 2235 y Fr(\()p Fq( )s(;)14 b Fr(\()p Fq(E)2453 2247 y Fl(p)2487 2255 y Fe(1)2543 2235 y Fr(+)k Fq(E)2687 2247 y Fl(p)2721 2255 y Fe(2)2758 2235 y Fr(\))c Fq( )s Fr(\))p Fq(;)381 b Fr(\(4.5\))160 2424 y(where)27 b(in)h(the)g(last)g(step)f(the)h(an)n(tisymmetry)f(of)h Fq( )i Fr(with)f(resp)r(ect)e(to)g(particle)g(exc)n(hange)g(is)g(used.) 285 2595 y(Next)32 b(w)n(e)e(estimate)i(the)f(p)r(oten)n(tial)g Fq(C)1519 2565 y Fn(\(12\))1642 2595 y Fq(:)g Fr(It)h(is)f(su\016cien)n (t)g(to)g(pro)n(v)n(e)f(the)i(relativ)n(e)e(b)r(oundedness)160 2706 y(for)e(the)g Fq(k)d Fr(=)e(1)k(con)n(tribution)g(to)h Fq(C)1296 2676 y Fn(\(12\))1418 2706 y Fq(;)60 b(V)1568 2676 y Fn(\(12\))1691 2706 y Fr(\003)1749 2663 y Fn(\(1\))1749 2726 y Fg(\000)1837 2706 y Fq(F)1902 2663 y Fn(\(1\))1890 2728 y(0)2024 2706 y Fr(+)18 b Fq(F)2172 2663 y Fn(\(1\))2160 2728 y(0)2261 2706 y Fr(\003)2319 2663 y Fn(\(1\))2319 2726 y Fg(\000)2408 2706 y Fq(V)2474 2676 y Fn(\(12\))2620 2706 y Fr(=:)23 b Fq(C)2796 2663 y Fn(\(12\))2790 2728 y(1)2918 2706 y Fq(:)51 b Fr(The)28 b(estimate)160 2828 y(is)d(p)r(erformed)f(in)h(momen)n(tum)g(space)f(where)g Fq(T)36 b Fr(is)25 b(diagonal.)34 b Fq(C)2212 2785 y Fn(\(12\))2206 2851 y(1)2359 2828 y Fr(is)25 b(represen)n(ted)e(b)n(y)i (its)g(k)n(ernel)e Fq(k)3388 2840 y Fl(C)3444 2828 y Fr(,)741 3059 y(\()798 3010 y Fm(\\)773 3059 y Fq(C)838 3016 y Fn(\(12\))832 3081 y(1)961 3059 y Fq( )s Fr(\)\()p Fh(p)1135 3071 y Fn(1)1172 3059 y Fq(;)14 b Fh(p)1262 3071 y Fn(2)1300 3059 y Fr(\))46 b(=)1489 2946 y Fp(Z)1586 3059 y Fq(d)p Fh(p)1682 3024 y Fg(0)1682 3079 y Fn(1)1733 3059 y Fq(d)p Fh(p)1829 3024 y Fg(0)1829 3079 y Fn(2)1890 3059 y Fq(k)1933 3071 y Fl(C)1989 3059 y Fr(\()p Fh(p)2074 3071 y Fn(1)2111 3059 y Fq(;)14 b Fh(p)2201 3071 y Fn(2)2238 3059 y Fr(;)g Fh(p)2328 3024 y Fg(0)2328 3079 y Fn(1)2366 3059 y Fq(;)g Fh(p)2456 3024 y Fg(0)2456 3079 y Fn(2)2493 3059 y Fr(\))2565 3037 y(^)2548 3059 y Fq( )s Fr(\()p Fh(p)2690 3024 y Fg(0)2690 3079 y Fn(1)2728 3059 y Fq(;)g Fh(p)2818 3024 y Fg(0)2818 3079 y Fn(2)2855 3059 y Fr(\))410 b(\(4.6\))160 3308 y(where)30 b(the)g(in)n(tegral)e(is)i(o)n(v)n(er)e (6-dimensional)h(space)g Fm(R)1925 3278 y Fn(3)1988 3308 y Fo(\002)19 b Fm(R)2126 3278 y Fn(3)2170 3308 y Fq(:)29 b Fr(Equiv)-5 b(alen)n(tly)e(,)30 b Fq(C)2787 3265 y Fn(\(12\))2781 3330 y(1)2940 3308 y Fr(can)f(b)r(e)h(de\014ned)160 3408 y(through)d(its)h(sym)n(b)r(ol)f Fq(\036)923 3420 y Fl(C)979 3408 y Fq(;)584 3628 y Fr(\()p Fq(C)681 3585 y Fn(\(12\))675 3650 y(1)804 3628 y Fq( )s Fr(\)\()p Fh(x)975 3640 y Fn(1)1013 3628 y Fq(;)14 b Fh(x)1100 3640 y Fn(2)1138 3628 y Fr(\))46 b(=)1413 3572 y(1)p 1337 3609 194 4 v 1337 3685 a(\(2)p Fq(\031)s Fr(\))1493 3661 y Fn(6)1554 3515 y Fp(Z)1651 3628 y Fq(d)p Fh(p)1747 3594 y Fg(0)1747 3649 y Fn(1)1799 3628 y Fq(d)p Fh(p)1895 3594 y Fg(0)1895 3649 y Fn(2)1946 3628 y Fq(d)p Fh(p)2042 3640 y Fn(1)2093 3628 y Fq(d)p Fh(p)2189 3640 y Fn(2)2250 3628 y Fq(e)2289 3594 y Fl(i)p Fn(\()p Fj(p)2380 3569 y Fb(0)2380 3611 y Fe(1)2412 3594 y Fn(+)p Fj(p)2505 3602 y Fe(1)2537 3594 y Fn(\))p Fj(x)2603 3602 y Fe(1)2653 3628 y Fq(e)2692 3594 y Fl(i)p Fn(\()p Fj(p)2783 3569 y Fb(0)2783 3611 y Fe(2)2815 3594 y Fg(\000)p Fj(p)2909 3602 y Fe(2)2941 3594 y Fn(\))p Fj(x)3007 3602 y Fe(2)3297 3628 y Fr(\(4.7\))1300 3877 y Fo(\001)p Fq(\036)1372 3889 y Fl(C)1429 3877 y Fr(\()p Fh(p)1514 3889 y Fn(1)1551 3877 y Fq(;)14 b Fh(p)1641 3889 y Fn(2)1679 3877 y Fr(;)g Fh(p)1769 3843 y Fg(0)1769 3897 y Fn(1)1806 3877 y Fq(;)g Fh(p)1896 3843 y Fg(0)1896 3897 y Fn(2)1933 3877 y Fr(\))2005 3855 y(^)1988 3877 y Fq( )s Fr(\()p Fh(p)2130 3843 y Fg(0)2130 3897 y Fn(1)2168 3877 y Fq(;)g Fh(p)2258 3843 y Fg(0)2258 3897 y Fn(2)2295 3877 y Fr(\))160 4026 y(and)22 b Fq(k)359 4038 y Fl(C)436 4026 y Fr(is)g(related)f(to)g(the)h(sym)n(b) r(ol)f(via)g Fq(k)1468 4038 y Fl(C)1524 4026 y Fr(\()p Fh(p)1609 4038 y Fn(1)1647 4026 y Fq(;)14 b Fh(p)1737 4038 y Fn(2)1774 4026 y Fr(;)g Fh(p)1864 3996 y Fg(0)1864 4047 y Fn(1)1901 4026 y Fq(;)g Fh(p)1991 3996 y Fg(0)1991 4047 y Fn(2)2028 4026 y Fr(\))38 b(=)22 b(\(2)p Fq(\031)s Fr(\))2341 3996 y Fg(\000)p Fn(3)2445 4026 y Fq(\036)2494 4038 y Fl(C)2550 4026 y Fr(\()p Fh(p)2635 4038 y Fn(1)2679 4026 y Fo(\000)6 b Fh(p)2803 3996 y Fg(0)2803 4047 y Fn(1)2840 4026 y Fq(;)14 b Fh(p)2930 3996 y Fg(0)2930 4047 y Fn(2)2974 4026 y Fo(\000)6 b Fh(p)3098 4038 y Fn(2)3135 4026 y Fr(;)14 b Fh(p)3225 3996 y Fg(0)3225 4047 y Fn(1)3262 4026 y Fq(;)g Fh(p)3352 3996 y Fg(0)3352 4047 y Fn(2)3389 4026 y Fr(\))p Fq(:)160 4126 y Fr(With)29 b(the)f(help)g(of)f(\(3.5\),)h(\(3.3\))f(and)g(the)h(F)-7 b(ourier)27 b(represen)n(tation)f(of)h Fq(V)2508 4096 y Fn(\(12\))2631 4126 y Fq(;)g Fr(one)g(obtains)667 4362 y Fq(k)710 4374 y Fl(C)766 4362 y Fr(\()p Fh(p)851 4374 y Fn(1)889 4362 y Fq(;)14 b Fh(p)979 4374 y Fn(2)1016 4362 y Fr(;)g Fh(p)1106 4327 y Fg(0)1106 4382 y Fn(1)1143 4362 y Fq(;)g Fh(p)1233 4327 y Fg(0)1233 4382 y Fn(2)1270 4362 y Fr(\))47 b(=)e Fo(\000)1592 4306 y Fq(\015)5 b(e)1679 4275 y Fn(2)p 1557 4343 V 1557 4419 a Fr(\(2)p Fq(\031)s Fr(\))1713 4395 y Fn(4)1956 4306 y Fr(1)p 1794 4343 366 4 v 1794 4419 a Fo(j)p Fh(p)1870 4431 y Fn(2)1925 4419 y Fo(\000)18 b Fh(p)2061 4390 y Fg(0)2061 4441 y Fn(2)2099 4419 y Fo(j)2122 4395 y Fn(2)2556 4306 y Fr(1)p 2202 4343 749 4 v 2202 4419 a Fo(j)p Fh(p)2278 4431 y Fn(2)2334 4419 y Fo(\000)g Fh(p)2470 4390 y Fg(0)2470 4441 y Fn(2)2526 4419 y Fr(+)g Fh(p)2662 4431 y Fn(1)2717 4419 y Fo(\000)g Fh(p)2853 4390 y Fg(0)2853 4441 y Fn(1)2891 4419 y Fo(j)2914 4395 y Fn(2)650 4677 y Fo(\001)687 4536 y Fp(\()1075 4621 y Fr(1)p 764 4658 664 4 v 764 4734 a Fq(E)825 4749 y Fg(j)p Fj(p)887 4757 y Fe(2)919 4749 y Fg(\000)p Fj(p)1013 4729 y Fb(0)1013 4767 y Fe(2)1045 4749 y Fn(+)p Fj(p)1138 4757 y Fe(1)1170 4749 y Fg(j)1213 4734 y Fr(+)g Fq(E)1357 4748 y Fl(p)1391 4729 y Fb(0)1391 4767 y Fe(1)1461 4677 y Fr(\(1)32 b Fo(\000)1683 4656 y Fr(~)1664 4677 y Fq(D)1735 4634 y Fn(\(1\))1733 4700 y(0)1824 4677 y Fr(\()p Fh(p)1909 4689 y Fn(2)1965 4677 y Fo(\000)18 b Fh(p)2101 4643 y Fg(0)2101 4698 y Fn(2)2157 4677 y Fr(+)g Fh(p)2293 4689 y Fn(1)2330 4677 y Fr(\)\))23 b(\(1)33 b(+)2640 4656 y(~)2620 4677 y Fq(D)2691 4634 y Fn(\(1\))2689 4700 y(0)2781 4677 y Fr(\()p Fh(p)2866 4643 y Fg(0)2866 4698 y Fn(1)2903 4677 y Fr(\)\))330 b(\(4.8\))616 4985 y(+)1025 4928 y(1)p 714 4966 V 714 5042 a Fq(E)775 5054 y Fl(p)809 5062 y Fe(1)864 5042 y Fr(+)18 b Fq(E)1008 5057 y Fg(j)p Fj(p)1070 5037 y Fb(0)1070 5075 y Fe(2)1103 5057 y Fg(\000)p Fj(p)1197 5065 y Fe(2)1229 5057 y Fn(+)p Fj(p)1322 5037 y Fb(0)1322 5075 y Fe(1)1354 5057 y Fg(j)1411 4985 y Fr(\(1)32 b(+)1633 4964 y(~)1614 4985 y Fq(D)1685 4942 y Fn(\(1\))1683 5007 y(0)1774 4985 y Fr(\()p Fh(p)1859 4997 y Fn(1)1896 4985 y Fr(\)\))24 b(\(1)32 b Fo(\000)2206 4964 y Fr(~)2187 4985 y Fq(D)2258 4942 y Fn(\(1\))2256 5007 y(0)2347 4985 y Fr(\()p Fh(p)2432 4950 y Fg(0)2432 5005 y Fn(2)2488 4985 y Fo(\000)18 b Fh(p)2624 4997 y Fn(2)2680 4985 y Fr(+)g Fh(p)2816 4950 y Fg(0)2816 5005 y Fn(1)2853 4985 y Fr(\)\))2917 4843 y Fp(\))2998 4985 y Fq(:)1793 5413 y Fr(8)p eop %%Page: 9 9 9 8 bop 160 54 a Fr(F)-7 b(or)29 b(the)g(estimate)f(of)h Fq(C)949 11 y Fn(\(12\))943 76 y(1)1072 54 y Fq(;)f Fr(the)i(Lieb)e (and)h(Y)-7 b(au)29 b(form)n(ula)e(is)i(used,)g(whic)n(h)g(is)f(a)h (consequence)e(of)i(the)160 154 y(Sc)n(h)n(ur)34 b(test)i(for)e(the)h (b)r(oundedness)g(of)g(in)n(tegral)e(op)r(erators,)i(and)f(whic)n(h)h (can)g(b)r(e)g(deriv)n(ed)f(from)g(the)160 254 y(Sc)n(h)n(w)n(arz)26 b(inequalit)n(y)h([19)o(])h(\(see)g(also)e([10)o(,)i(Lemma)f(I)r (I.1]\))1279 466 y Fo(j)p Fr(\()p Fq( )s(;)14 b(C)1493 423 y Fn(\(12\))1487 488 y(1)1630 466 y Fq( )s Fr(\))p Fo(j)47 b Fr(=)e Fo(j)p Fr(\()1972 444 y(^)1954 466 y Fq( )t(;)2074 417 y Fm(\\)2049 466 y Fq(C)2114 423 y Fn(\(12\))2108 488 y(1)2236 466 y Fq( )s Fr(\))p Fo(j)662 675 y(\024)772 562 y Fp(Z)869 675 y Fq(d)p Fh(p)965 687 y Fn(1)1017 675 y Fq(d)p Fh(p)1113 687 y Fn(2)1164 675 y Fq(d)p Fh(p)1260 641 y Fg(0)1260 696 y Fn(1)1311 675 y Fq(d)p Fh(p)1407 641 y Fg(0)1407 696 y Fn(2)1468 675 y Fo(j)1508 654 y Fr(^)1491 675 y Fq( )s Fr(\()p Fh(p)1633 687 y Fn(1)1671 675 y Fq(;)14 b Fh(p)1761 687 y Fn(2)1798 675 y Fr(\))p Fo(j)23 b(j)p Fq(k)1942 687 y Fl(C)1998 675 y Fr(\()p Fh(p)2083 687 y Fn(1)2121 675 y Fq(;)14 b Fh(p)2211 687 y Fn(2)2248 675 y Fr(;)g Fh(p)2338 641 y Fg(0)2338 696 y Fn(1)2375 675 y Fq(;)g Fh(p)2465 641 y Fg(0)2465 696 y Fn(2)2502 675 y Fr(\))p Fo(j)24 b(j)2621 654 y Fr(^)2604 675 y Fq( )s Fr(\()p Fh(p)2746 641 y Fg(0)2746 696 y Fn(1)2784 675 y Fq(;)14 b Fh(p)2874 641 y Fg(0)2874 696 y Fn(2)2911 675 y Fr(\))p Fo(j)1134 908 y(\024)1245 795 y Fp(Z)1342 908 y Fq(d)p Fh(p)1438 920 y Fn(1)1489 908 y Fq(d)p Fh(p)1585 920 y Fn(2)1646 908 y Fo(j)1686 886 y Fr(^)1669 908 y Fq( )s Fr(\()p Fh(p)1811 920 y Fn(1)1848 908 y Fq(;)g Fh(p)1938 920 y Fn(2)1976 908 y Fr(\))p Fo(j)2031 874 y Fn(2)2087 908 y Fo(\001)k Fq(J)2174 920 y Fn(0)2212 908 y Fr(\()p Fh(p)2297 920 y Fn(1)2334 908 y Fq(;)c Fh(p)2424 920 y Fn(2)2461 908 y Fr(\))804 b(\(4.9\))779 1146 y Fq(J)825 1158 y Fn(0)862 1146 y Fr(\()p Fh(p)947 1158 y Fn(1)985 1146 y Fq(;)14 b Fh(p)1075 1158 y Fn(2)1112 1146 y Fr(\))46 b(=)1301 1033 y Fp(Z)1398 1146 y Fq(d)p Fh(p)1494 1112 y Fg(0)1494 1167 y Fn(1)1545 1146 y Fq(d)p Fh(p)1641 1112 y Fg(0)1641 1167 y Fn(2)1702 1146 y Fo(j)p Fq(k)1768 1158 y Fl(C)1824 1146 y Fr(\()p Fh(p)1909 1158 y Fn(1)1947 1146 y Fq(;)14 b Fh(p)2037 1158 y Fn(2)2074 1146 y Fr(;)g Fh(p)2164 1112 y Fg(0)2164 1167 y Fn(1)2201 1146 y Fq(;)g Fh(p)2291 1112 y Fg(0)2291 1167 y Fn(2)2328 1146 y Fr(\))p Fo(j)2417 1090 y Fq(f)9 b Fr(\()p Fq(p)2541 1102 y Fn(1)2577 1090 y Fr(\))p 2417 1127 194 4 v 2417 1203 a Fq(f)g Fr(\()p Fq(p)2541 1174 y Fg(0)2541 1225 y Fn(1)2577 1203 y Fr(\))2653 1090 y Fq(g)s Fr(\()p Fq(p)2770 1102 y Fn(2)2807 1090 y Fr(\))p 2653 1127 187 4 v 2653 1203 a Fq(g)s Fr(\()p Fq(p)2770 1174 y Fg(0)2770 1225 y Fn(2)2807 1203 y Fr(\))160 1336 y(since)26 b(the)f(estimate)h(of)f Fo(j)p Fq(k)991 1348 y Fl(C)1047 1336 y Fo(j)g Fr(is)h(symmetric)f(with)g(resp)r(ect)g (to)h(the)f(in)n(terc)n(hange)f(\()p Fh(p)2813 1348 y Fn(1)2851 1336 y Fq(;)14 b Fh(p)2941 1348 y Fn(2)2978 1336 y Fr(\))23 b Fo($)g Fr(\()p Fh(p)3224 1305 y Fg(0)3224 1356 y Fn(1)3262 1336 y Fq(;)14 b Fh(p)3352 1305 y Fg(0)3352 1356 y Fn(2)3389 1336 y Fr(\))p Fq(:)160 1435 y Fr(The)30 b(functions)g Fq(f)9 b Fr(\()p Fq(p)p Fr(\))27 b Fq(>)f Fr(0)k(and)f Fq(g)s Fr(\()p Fq(p)p Fr(\))e Fq(>)f Fr(0)j(for)h Fq(p)c(>)g Fr(0)k(are)e(suitably)i(c)n(hosen)f(con)n(v)n(ergence)e (generating)160 1535 y(functions.)71 b(W)-7 b(e)39 b(tak)n(e)f Fq(f)9 b Fr(\()p Fq(p)p Fr(\))42 b(=)f Fq(p)1275 1505 y Fn(2)1351 1535 y Fr(and)d Fq(g)45 b Fr(=)c(1)d(and)h(use)f(spherical) g(co)r(ordinates,)i(i.e.)3133 1468 y Fp(R)3173 1564 y Ff(R)3220 1548 y Fe(3)3265 1535 y Fq(d)p Fh(p)i Fr(=)160 1590 y Fp(R)216 1610 y Fg(1)200 1686 y Fn(0)300 1656 y Fq(p)342 1626 y Fn(2)379 1656 y Fq(dp)478 1590 y Fp(R)517 1686 y Fl(S)561 1669 y Fe(2)611 1656 y Fq(d!)34 b Fr(with)932 1590 y Fp(R)971 1686 y Fl(S)1015 1669 y Fe(2)1066 1656 y Fq(d!)d Fr(=)d(2)p Fq(\031)1390 1590 y Fp(R)1446 1610 y Fn(1)1429 1686 y Fg(\000)p Fn(1)1532 1656 y Fq(d)p Fr(\(cos)14 b Fq(#)p Fr(\))p Fq(:)59 b Fr(Estimating)2361 1624 y Fn(1)p 2335 1638 84 4 v 2335 1685 a Fl(E)2384 1693 y Fi(p)2457 1656 y Fo(\024)2560 1624 y Fn(1)p 2560 1638 35 4 v 2560 1685 a Fl(p)2635 1656 y Fr(and)30 b Fo(j)p Fr(1)20 b Fo(\006)2989 1636 y Fr(~)2969 1656 y Fq(D)3038 1668 y Fn(0)3075 1656 y Fo(j)29 b(\024)e Fr(2)p Fq(;)59 b Fr(one)160 1767 y(can)28 b(use)f(the)h(form)n(ulae)573 1853 y Fp(Z)619 2041 y Ff(R)666 2025 y Fe(3)722 1910 y Fq(d)p Fh(p)818 1880 y Fg(0)818 1930 y Fn(1)p 722 1947 134 4 v 739 2023 a Fq(p)781 1994 y Fg(0)p Fn(2)781 2045 y(1)988 1910 y Fr(1)p 898 1947 221 4 v 898 2023 a Fq(q)22 b Fr(+)c Fq(p)1082 1994 y Fg(0)1082 2045 y Fn(1)1304 1910 y Fr(1)p 1162 1947 326 4 v 1162 2023 a Fo(j)p Fh(q)h Fr(+)f Fh(p)1390 1994 y Fg(0)1390 2045 y Fn(1)1427 2023 y Fo(j)1450 1999 y Fn(2)1543 1966 y Fr(=)1664 1910 y(2)p Fq(\031)p 1664 1947 92 4 v 1690 2023 a(q)1780 1853 y Fp(Z)1863 1873 y Fg(1)1826 2041 y Fn(0)1957 1910 y Fq(dp)2042 1880 y Fg(0)2042 1930 y Fn(1)p 1957 1947 123 4 v 1978 2023 a Fq(p)2020 1994 y Fg(0)2020 2045 y Fn(1)2212 1910 y Fr(1)p 2122 1947 221 4 v 2122 2023 a Fq(q)k Fr(+)c Fq(p)2306 1994 y Fg(0)2306 2045 y Fn(1)2389 1966 y Fr(ln)2505 1910 y Fq(q)k Fr(+)c Fq(p)2689 1880 y Fg(0)2689 1930 y Fn(1)p 2482 1947 267 4 v 2482 2023 a Fo(j)p Fq(q)k Fo(\000)c Fq(p)2689 1994 y Fg(0)2689 2045 y Fn(1)2726 2023 y Fo(j)2805 1966 y Fr(=)2941 1910 y Fq(\031)2991 1880 y Fn(3)p 2926 1947 119 4 v 2926 2023 a Fr(2)p Fq(q)3008 1999 y Fn(2)1372 2130 y Fp(Z)1418 2319 y Ff(R)1465 2302 y Fe(3)1520 2187 y Fq(d)p Fh(q)p 1520 2224 94 4 v 1528 2300 a Fq(q)1568 2276 y Fn(2)1799 2187 y Fr(1)p 1657 2224 326 4 v 1657 2300 a Fo(j)p Fh(q)g Fr(+)h Fh(p)1885 2312 y Fn(1)1922 2300 y Fo(j)1945 2276 y Fn(2)2038 2243 y Fr(=)2159 2187 y Fq(\031)2209 2157 y Fn(3)p 2159 2224 88 4 v 2163 2300 a Fq(p)2205 2312 y Fn(1)3255 2243 y Fr(\(4.10\))160 2458 y(to)28 b(obtain)f Fq(J)566 2470 y Fn(0)603 2458 y Fr(\()p Fh(p)688 2470 y Fn(1)726 2458 y Fq(;)14 b Fh(p)816 2470 y Fn(2)853 2458 y Fr(\))24 b Fo(\024)1006 2421 y Fl(\015)t(e)1076 2396 y Fe(2)1109 2421 y Fl(\031)1150 2396 y Fe(2)p 1006 2439 176 4 v 1077 2487 a Fn(4)1206 2458 y Fq(p)1248 2470 y Fn(1)1335 2458 y Fr(and)k(therefore)577 2685 y Fo(j)p Fr(\()p Fq( )s(;)14 b(C)791 2651 y Fn(\(12\))927 2685 y Fq( )s Fr(\))p Fo(j)47 b Fr(=)f(2)22 b Fo(j)p Fr(\()p Fq( )s(;)14 b(C)1475 2642 y Fn(\(12\))1469 2707 y(1)1598 2685 y Fq( )s Fr(\))p Fo(j)47 b(\024)e Fq(\015)5 b(e)1954 2651 y Fn(2)2015 2629 y Fq(\031)2065 2599 y Fn(2)p 2015 2666 88 4 v 2038 2742 a Fr(2)2126 2572 y Fp(Z)2223 2685 y Fq(d)p Fh(p)2319 2697 y Fn(1)2370 2685 y Fq(d)p Fh(p)2466 2697 y Fn(2)2527 2685 y Fo(j)2567 2663 y Fr(^)2550 2685 y Fq( )s Fr(\()p Fh(p)2692 2697 y Fn(1)2729 2685 y Fq(;)14 b Fh(p)2819 2697 y Fn(2)2856 2685 y Fr(\))p Fo(j)2911 2651 y Fn(2)2972 2685 y Fq(p)3014 2697 y Fn(1)1336 2955 y Fr(=)46 b Fq(\015)5 b(e)1534 2920 y Fn(2)1594 2899 y Fq(\031)1644 2868 y Fn(2)p 1594 2936 V 1617 3012 a Fr(4)1715 2955 y(\()p Fq( )s(;)14 b Fr(\()p Fq(p)1915 2967 y Fn(1)1971 2955 y Fr(+)k Fq(p)2096 2967 y Fn(2)2133 2955 y Fr(\))c Fq( )s Fr(\))p Fq(:)964 b Fr(\(4.11\))160 3146 y(F)-7 b(or)30 b(an)f(upp)r(er)h(estimate)g(of)g(the)g(single-particle)f(p)r(oten)n (tial)h Fq(V)2199 3116 y Fn(\()p Fl(k)q Fn(\))2312 3146 y Fr(+)20 b Fq(V)2464 3103 y Fn(\()p Fl(k)q Fn(\))2445 3169 y(2)2586 3146 y Fr(w)n(e)30 b(use)g(that)g(for)f Fq(\015)j Fo(\024)26 b Fr(4)p Fq(=\031)160 3264 y Fr(the)e(massless)e (\()p Fq(m)h Fr(=)g(0\))g(v)n(ersion)f(of)h(this)h(p)r(oten)n(tial)f (ful\014ls)h Fq(V)2096 3220 y Fn(\()p Fl(k)q Fn(\))2077 3286 y Fl(m)p Fn(=0)2234 3264 y Fr(+)10 b Fq(V)2376 3220 y Fn(\()p Fl(k)q Fn(\))2357 3286 y(2)p Fl(;m)p Fn(=0)2580 3264 y Fo(\024)23 b Fr(0)g([17)o(].)35 b(In)24 b(addition)f(one)160 3392 y(has)31 b(the)g(relativ)n(e)f(form)g(b)r(oundedness)h(of)g Fq(V)1612 3362 y Fn(\()p Fl(k)q Fn(\))1735 3392 y Fr(and)g Fq(V)1967 3349 y Fn(\()p Fl(k)q Fn(\))1948 3414 y(2)2091 3392 y Fr(with)g(resp)r(ect)g(to)g(the)g Fq(m)d Fr(=)g(0)j(p)r(oten)n (tials,)160 3509 y Fo(j)p Fr(\()p Fq( )s(;)14 b Fr(\()p Fq(V)409 3479 y Fn(\()p Fl(k)q Fn(\))505 3509 y Fo(\000)s Fq(V)640 3466 y Fn(\()p Fl(k)q Fn(\))621 3531 y Fl(m)p Fn(=0)768 3509 y Fr(\))g Fq( )s Fr(\))p Fo(j)38 b(\024)1061 3477 y Fn(3)p 1061 3491 34 4 v 1061 3538 a(2)1104 3509 y Fq(m\015)5 b Fr(\()p Fq( )s(;)14 b( )s Fr(\))20 b(and)g Fo(j)p Fr(\()p Fq( )s(;)14 b Fr(\()p Fq(V)1863 3466 y Fn(\()p Fl(k)q Fn(\))1843 3531 y(2)1959 3509 y Fo(\000)s Fq(V)2093 3466 y Fn(\()p Fl(k)q Fn(\))2075 3531 y(2)p Fl(;m)p Fn(=0)2275 3509 y Fr(\))p Fq( )s Fr(\))p Fo(j)37 b(\024)23 b Fq(md)2660 3521 y Fn(0)2697 3509 y Fq(\015)2745 3479 y Fn(2)2782 3509 y Fr(\()p Fq( )s(;)14 b( )s Fr(\),)22 b(resp)r(ectiv)n(ely)160 3626 y([20)o(,)28 b(16)o(])g(with)g Fq(d)683 3638 y Fn(0)744 3626 y Fr(=)22 b(8)c(+)g(12)1058 3558 y Fo(p)p 1126 3558 42 4 v 1126 3626 a Fr(2)p Fq(:)51 b Fr(Then)28 b(one)f(obtains)288 3845 y(\()p Fq( )s(;)14 b(W)26 b( )s Fr(\))46 b Fo(\024)g Fr(\(3)p Fq(m\015)37 b Fr(+)31 b(2)p Fq(md)1245 3857 y Fn(0)1282 3845 y Fq(\015)1330 3811 y Fn(2)1367 3845 y Fr(\))23 b(\()p Fq( )s(;)14 b( )s Fr(\))43 b(+)e(\()1886 3789 y Fq(e)1925 3759 y Fn(2)p 1828 3826 192 4 v 1828 3902 a Fr(2)p Fq(\015)1913 3914 y Fl(B)s(R)2062 3845 y Fr(+)2169 3789 y Fq(\015)5 b(e)2256 3759 y Fn(2)2292 3789 y Fq(\031)2342 3759 y Fn(2)p 2169 3826 212 4 v 2254 3902 a Fr(4)2390 3845 y(\))23 b(\()p Fq( )s(;)14 b Fr(\()p Fq(E)2664 3857 y Fl(p)2698 3865 y Fe(1)2754 3845 y Fr(+)k Fq(E)2898 3857 y Fl(p)2932 3865 y Fe(2)2969 3845 y Fr(\))c Fq( )s Fr(\))p Fq(:)128 b Fr(\(4.12\))160 4080 y(F)-7 b(or)18 b(the)h(lo)n(w)n(er)e(b)r(ound)i (w)n(e)f(use)g Fq(V)1203 4050 y Fn(\(12\))1348 4080 y Fo(\025)23 b Fr(0)18 b(and)g(\()p Fq( )s(;)c(V)1842 4037 y Fn(\()p Fl(k)q Fn(\))1822 4102 y(2)1934 4080 y Fq( )s Fr(\))24 b Fo(\025)e(\000)p Fq(md)2315 4092 y Fn(0)2352 4080 y Fq(\015)5 b Fr(\()p Fq( )s(;)14 b( )s Fr(\))42 b(since)18 b(\()p Fq( )s(;)c(V)3044 4037 y Fn(\()p Fl(k)q Fn(\))3025 4102 y(2)p Fl(;m)p Fn(=0)3226 4080 y Fq( )s Fr(\))23 b Fo(\025)g Fr(0)160 4179 y([16)o(].)38 b(Then)27 b(with)h(\(4.4\))g(and)f(\(4.11\),)202 4399 y(\()p Fq( )s(;)14 b(W)26 b( )s Fr(\))46 b Fo(\025)g(\000)818 4342 y Fq(\015)p 766 4379 151 4 v 766 4455 a(\015)809 4467 y Fl(B)s(R)950 4399 y Fr(\()p Fq( )s(;)14 b Fr(\()p Fq(E)1169 4411 y Fl(p)1203 4419 y Fe(1)1244 4399 y Fr(+)s Fq(E)1373 4411 y Fl(p)1407 4419 y Fe(2)1443 4399 y Fr(\))g Fq( )s Fr(\))27 b Fo(\000)f Fr(2)p Fq(md)1854 4411 y Fn(0)1891 4399 y Fq(\015)h Fr(\()p Fq( )s(;)14 b( )s Fr(\))27 b Fo(\000)f Fq(\015)5 b(e)2381 4364 y Fn(2)2441 4342 y Fq(\031)2491 4312 y Fn(2)p 2441 4379 88 4 v 2464 4455 a Fr(4)2562 4399 y(\()p Fq( )s(;)14 b Fr(\()p Fq(E)2781 4411 y Fl(p)2815 4419 y Fe(1)2856 4399 y Fr(+)s Fq(E)2985 4411 y Fl(p)3019 4419 y Fe(2)3055 4399 y Fr(\))g Fq( )s Fr(\))p Fq(:)42 b Fr(\(4.13\))160 4609 y(Com)n(bining)27 b(\(4.12\))g(and)h(\(4.13\))f (w)n(e)g(obtain)g(the)h(\014nal)g(result)868 4774 y Fo(j)p Fr(\()p Fq( )s(;)14 b(W)26 b( )s Fr(\))p Fo(j)46 b(\024)g Fq(c)23 b Fr(\()p Fq( )s(;)14 b(T)25 b( )s Fr(\))42 b(+)f(\(3)p Fq(m\015)c Fr(+)32 b(2)p Fq(md)2368 4786 y Fn(0)2404 4774 y Fq(\015)2452 4740 y Fn(2)2489 4774 y Fr(\))24 b(\()p Fq( )s(;)14 b( )s Fr(\))495 b(\(4.14\))160 4954 y(with)41 b Fq(c)k Fr(=)f Fq(\015)5 b Fr(\()688 4922 y Fn(1)p 641 4936 127 4 v 641 4983 a Fl(\015)676 4991 y Fi(B)r(R)819 4954 y Fr(+)921 4922 y Fl(e)952 4897 y Fe(2)984 4922 y Fl(\031)1025 4897 y Fe(2)p 921 4936 138 4 v 973 4983 a Fn(4)1068 4954 y Fr(\))85 b(for)40 b Fq(\015)49 b(>)44 b(e)1565 4924 y Fn(2)1602 4954 y Fq(=)p Fr(2)p Fq(;)83 b Fr(and)41 b Fq(c)j Fo(\034)g Fr(1)c(for)g Fq(\015)49 b Fo(\024)44 b Fq(e)2636 4924 y Fn(2)2673 4954 y Fq(=)p Fr(2)p Fq(:)84 b Fr(When)41 b Fq(\015)49 b(<)44 b Fr(0)p Fq(:)p Fr(89)160 5065 y(\(corresp)r(onding)21 b(to)h Fq(Z)29 b Fo(\024)23 b Fr(122\),)f(the)h(form)f(b)r(ound)g Fq(c)h Fr(is)f(smaller)f(than)i(one)f(whic)n(h)g(pro)n(v)n(es)e(the)j (assertion)160 5164 y(\(4.2\).)1793 5413 y(9)p eop %%Page: 10 10 10 9 bop 160 51 a Fw(5)135 b(Sub)t(dominance)44 b(of)h(the)h (second-order)e(p)t(opten)l(tials)160 304 y Fr(W)-7 b(e)29 b(wish)f(to)g(establish)g(up)h(to)f(whic)n(h)g(cen)n(tral)g(\014eld)g (strength)g Fq(\015)33 b Fr(the)c(single-particle)d(as)i(w)n(ell)g(as)g (t)n(w)n(o-)160 404 y(particle)34 b(second-order)e(p)r(oten)n(tials)i (are)g(con)n(trolled)f(b)n(y)h(the)h(resp)r(ectiv)n(e)e(\014rst-order)g (p)r(oten)n(tials,)i(i.e.)160 503 y(w)n(e)28 b(w)n(an)n(t)f(to)g(pro)n (v)n(e)f(for)h Fq( )f Fo(2)e(H)1167 515 y Fn(+)p Fl(;)p Fn(2)1275 503 y Fq(;)1211 705 y Fo(j)p Fr(\()p Fq( )s(;)14 b(V)1427 662 y Fn(\()p Fl(k)q Fn(\))1408 727 y(2)1534 705 y Fq( )s Fr(\))p Fo(j)46 b(\024)g Fq(c)1839 717 y Fn(1)1899 705 y Fr(\()p Fq( )s(;)14 b Fr(\()p Fo(\000)p Fq(V)2189 671 y Fn(\()p Fl(k)q Fn(\))2282 705 y Fr(\))g Fq( )s Fr(\))880 b(\(5.1\))160 888 y(and)1247 987 y Fo(j)p Fr(\()p Fq( )s(;)14 b(C)1461 953 y Fn(\(12\))1597 987 y Fq( )s Fr(\))p Fo(j)47 b(\024)f Fq(c)1903 999 y Fn(2)1963 987 y Fr(\()p Fq( )s(;)14 b(V)2156 953 y Fn(\(12\))2292 987 y Fq( )s Fr(\))916 b(\(5.2\))160 1137 y(with)35 b(constan)n(ts)e Fq(c)766 1149 y Fn(1)837 1137 y Fr(and)h Fq(c)1041 1149 y Fn(2)1113 1137 y Fr(smaller)f(than)h(one.)56 b(In)34 b(con)n(trast)f(to)h(the)h(previous)e(section)g(where)h(w)n(e)160 1248 y(relied)24 b(on)f Fq(D)565 1205 y Fn(\()p Fl(k)q Fn(\))563 1270 y(0)681 1248 y Fr(b)r(eing)h(a)f(m)n(ultiplication)g(op) r(erator)f(in)i(momen)n(tum)f(space)g(to)g(carry)f(out)i(all)f (estimates)160 1347 y(in)f(F)-7 b(ourier)20 b(space,)i(w)n(e)e(are)g (no)n(w)h(in)g(the)h(situation)f(that)g Fq(V)1993 1317 y Fn(\()p Fl(k)q Fn(\))2107 1347 y Fr(as)f(w)n(ell)h(as)g Fq(V)2526 1317 y Fn(\(12\))2670 1347 y Fr(are)f(diagonal)g(op)r (erators)160 1459 y(in)35 b(co)r(ordinate)d(space.)56 b(This)34 b(p)r(oses)f(the)i(problem)e(th)n(t)h(the)h(k)n(ernels)e(of)h Fq(V)2596 1416 y Fn(\()p Fl(k)q Fn(\))2578 1481 y(2)2723 1459 y Fr(and)g Fq(C)2956 1428 y Fn(\(12\))3113 1459 y Fr(whic)n(h)g(are)160 1558 y(kno)n(wn)h(in)h(momen)n(tum)f(space)g (ha)n(v)n(e)f(to)h(b)r(e)h(transformed)e(to)h(co)r(ordinate)f(space.)59 b(The)36 b(metho)r(d)g(of)160 1658 y(pro)r(of)27 b(will)h(b)r(e)g (displa)n(y)n(ed)f(for)g(the)h(one-particle)e(case)h(\(5.1\).)285 1845 y(F)-7 b(rom)28 b(the)g(de\014nition)h(of)f Fq(V)1179 1802 y Fn(\()p Fl(k)q Fn(\))1160 1867 y(2)1300 1845 y Fr(in)g(\(4.1\),)h(the)f(explicit)h(expression)d(\(3.3\))i(for)g(the)h (sym)n(b)r(ol)e(of)i Fq(B)3375 1802 y Fn(\()p Fl(k)q Fn(\))3371 1867 y(1)160 1962 y Fr(and)f(the)g(F)-7 b(ourier)26 b(represen)n(tation)g(of)i(the)g(Coulom)n(b)f(\014eld)h Fq(V)2133 1932 y Fn(\()p Fl(k)q Fn(\))2225 1962 y Fr(,)g(one)f(obtains) g(for)h(the)g(k)n(ernel)e Fq(k)3280 1919 y Fn(\()p Fl(k)q Fn(\))3277 1987 y Fl(V)3401 1962 y Fr(of)160 2082 y Fq(V)227 2039 y Fn(\()p Fl(k)q Fn(\))208 2104 y(2)320 2082 y Fr(,)189 2319 y Fq(k)235 2276 y Fn(\()p Fl(k)q Fn(\))232 2343 y Fl(V)328 2319 y Fr(\()p Fh(p)p Fq(;)14 b Fh(p)503 2285 y Fg(0)526 2319 y Fr(\))47 b(=)768 2263 y Fq(\015)816 2233 y Fn(2)p 725 2300 171 4 v 725 2376 a Fr(16)p Fq(\031)859 2352 y Fn(4)920 2206 y Fp(Z)1017 2319 y Fq(d)p Fh(p)1113 2285 y Fg(00)1325 2263 y Fr(1)p 1179 2300 334 4 v 1179 2376 a Fo(j)p Fh(p)1255 2352 y Fg(00)1316 2376 y Fo(\000)18 b Fh(p)p Fo(j)1475 2352 y Fn(2)1713 2263 y Fr(1)p 1555 2300 357 4 v 1555 2376 a Fo(j)p Fh(p)1631 2352 y Fg(00)1692 2376 y Fo(\000)g Fh(p)1828 2352 y Fg(0)1852 2376 y Fo(j)1875 2352 y Fn(2)1945 2319 y Fr(\(1)g Fo(\000)2140 2298 y Fr(~)2120 2319 y Fq(D)2191 2276 y Fn(\()p Fl(k)q Fn(\))2189 2341 y(0)2284 2319 y Fr(\()p Fh(p)2369 2285 y Fg(00)2412 2319 y Fr(\)\))24 b(\()2692 2263 y(1)p 2542 2300 342 4 v 2542 2376 a Fq(E)2603 2388 y Fl(p)2637 2372 y Fb(00)2701 2376 y Fr(+)18 b Fq(E)2845 2388 y Fl(p)2926 2319 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b(=)1218 2821 y Fp(Z)1315 2934 y Fq(d)p Fh(x)1408 2946 y Fn(1)1459 2934 y Fq(d)p Fh(x)1552 2946 y Fn(2)p 1613 2862 334 4 v 1613 2934 a Fq( )s Fr(\()p Fh(x)1752 2946 y Fn(1)1790 2934 y Fq(;)14 b Fh(x)1877 2946 y Fn(2)1915 2934 y Fr(\))1975 2821 y Fp(Z)2072 2934 y Fq(d)p Fh(x)2165 2900 y Fg(0)2214 2912 y Fr(\024)2212 2934 y Fq(k)2258 2891 y Fn(\(1\))2255 2958 y Fl(V)2347 2934 y Fr(\()p Fh(x)2429 2946 y Fn(1)2467 2934 y Fq(;)g Fh(x)2554 2900 y Fg(0)2577 2934 y Fr(\))23 b Fq( )s Fr(\()p Fh(x)2771 2900 y Fg(0)2795 2934 y Fq(;)14 b Fh(x)2882 2946 y Fn(2)2920 2934 y Fr(\))345 b(\(5.4\))160 3155 y(with)773 3254 y(\024)771 3276 y Fq(k)817 3233 y Fn(\(1\))814 3300 y Fl(V)906 3276 y Fr(\()p Fh(x)988 3288 y Fn(1)1026 3276 y Fq(;)14 b Fh(x)1113 3242 y Fg(0)1136 3276 y Fr(\))47 b(=)1412 3220 y(1)p 1336 3257 194 4 v 1336 3333 a(\(2)p Fq(\031)s Fr(\))1492 3309 y Fn(3)1553 3163 y Fp(Z)1650 3276 y Fq(d)p Fh(p)23 b Fq(e)1808 3242 y Fl(i)p Fj(p)n(x)1911 3250 y Fe(1)1961 3163 y Fp(Z)2058 3276 y Fq(d)p 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4854 y Fe(2)259 4842 y Fr(\()p Fq(E)352 4854 y Fl(p)386 4838 y Fb(00)458 4842 y Fr(+)25 b Fq(E)609 4857 y Fg(j)p Fj(p)671 4840 y Fb(00)711 4857 y Fg(\000)p Fj(q)p Fg(j)827 4842 y Fr(\))859 4812 y Fg(\000)p Fn(1)948 4842 y Fq(:)80 b Fr(This)39 b(enhances)f(small)g(v)-5 b(alues)39 b(of)f Fq(q)k Fr(and)c(w)n(e)h(therefore)e(drop)h Fh(q)h Fr(in)g(the)160 4977 y(energy)24 b(denominator.)34 b(Additionally)25 b(w)n(e)f(set)g(the)h(mass)e(to)h(zero)g(and)g(use)g(that)g Fo(k)p Fr(1)12 b Fo(\000)2936 4956 y Fr(~)2918 4977 y Fq(D)2989 4934 y Fn(\(1\))2987 4999 y(0)3077 4977 y Fr(\()p Fh(p)3162 4947 y Fg(00)3205 4977 y Fr(\))p Fo(k)36 b Fr(=)23 b(2)160 5094 y(to)31 b(replace)e(\(1)20 b Fo(\000)747 5073 y Fr(~)728 5094 y Fq(D)799 5051 y Fn(\(1\))797 5116 y(0)888 5094 y Fr(\()p Fh(p)973 5064 y Fg(00)1016 5094 y Fr(\)\))31 b(b)n(y)f(2.)45 b(The)30 b(con)n(v)n(ergence)e(prop)r (erties)h(of)i(the)f(in)n(tegral)f(remain)h(thereb)n(y)1772 5413 y(10)p eop %%Page: 11 11 11 10 bop 160 51 a Fr(unc)n(hanged,)26 b(but)g(one)g(pic)n(ks)f(up)h (an)f(additional)h(m)n(ultiplicativ)n(e)f(constan)n(t)g Fq(C)32 b Fr(\(of)26 b(order)f(unit)n(y\))h(in)g(the)160 166 y(estimate)i(of)g Fo(j)613 144 y Fr(\024)611 166 y Fq(k)657 123 y Fn(\(1\))654 191 y Fl(V)746 166 y Fr(\()p Fh(x)828 178 y Fn(1)865 166 y Fq(;)14 b Fh(x)952 136 y Fg(0)976 166 y Fr(\))p Fo(j)p Fq(:)51 b Fr(The)28 b(result)f(is)202 406 y Fo(j)p Fq(I)261 418 y Fn(1)299 406 y Fr(\()p Fh(x)381 418 y Fn(1)419 406 y Fq(;)14 b Fh(x)506 372 y Fg(0)529 406 y Fr(\))p Fo(j)47 b(\024)e Fq(C)844 289 y Fp(\022)905 293 y(Z)1002 406 y Fq(d)p Fh(q)23 b Fq(e)1157 372 y Fg(\000)p Fl(i)p Fj(qx)1312 380 y Fe(1)1399 350 y Fr(1)p 1381 387 78 4 v 1381 463 a Fq(q)1421 439 y Fn(2)1468 289 y Fp(\023)14 b(\022)1604 293 y(Z)1701 406 y Fq(d)p Fh(p)1797 372 y Fg(00)1863 406 y Fq(e)1902 372 y Fl(i)p Fj(p)1967 347 y Fb(00)2008 372 y Fn(\()p Fj(x)2074 380 y Fe(1)2106 372 y Fg(\000)p Fj(x)2198 347 y Fb(0)2220 372 y Fn(\))2304 350 y Fr(1)p 2283 387 85 4 v 2283 463 a Fq(p)2325 439 y Fg(00)2377 289 y Fp(\023)2484 406 y Fr(=)46 b Fq(C)2693 350 y Fr(2)p Fq(\031)2785 320 y Fn(2)p 2693 387 130 4 v 2715 463 a Fq(x)2762 475 y Fn(1)2838 406 y Fo(\001)3003 350 y Fr(4)p Fq(\031)p 2876 387 347 4 v 2876 463 a Fo(j)p Fh(x)2949 475 y Fn(1)3005 463 y Fo(\000)18 b Fh(x)3138 439 y Fg(0)3162 463 y Fo(j)3185 439 y Fn(2)3232 406 y Fq(:)42 b Fr(\(5.8\))285 734 y(Going)29 b(bac)n(k)f(to)h(the)g(desired) g(estimate)g(of)g Fq(V)1755 691 y Fn(\(1\))1736 756 y(2)1844 734 y Fq(;)h Fr(w)n(e)e(apply)h(the)h(Lieb)f(and)g(Y)-7 b(au)29 b(form)n(ula)g(for)f(the)160 834 y(one-particle)f(case)f(to)i (\(5.4\))f(and)h(obtain)645 1064 y Fo(j)p Fr(\()p Fq( )s(;)14 b(V)861 1021 y Fn(\(1\))842 1087 y(2)964 1064 y Fq( )s Fr(\))p Fo(j)47 b(\024)1234 951 y Fp(Z)1331 1064 y Fq(d)p Fh(x)1424 1076 y Fn(1)1475 1064 y Fq(d)p Fh(x)1568 1076 y Fn(2)1629 1064 y Fo(j)p Fq( )s Fr(\()p Fh(x)1791 1076 y Fn(1)1829 1064 y Fq(;)14 b Fh(x)1916 1076 y Fn(2)1954 1064 y Fr(\))p Fo(j)2009 1030 y Fn(2)2060 951 y Fp(Z)2157 1064 y Fq(d)p Fh(x)2250 1030 y Fg(0)2297 1064 y Fo(j)2322 1042 y Fr(\024)2320 1064 y Fq(k)2366 1021 y Fn(\(1\))2363 1089 y Fl(V)2455 1064 y Fr(\()p Fh(x)2537 1076 y Fn(1)2575 1064 y Fq(;)g Fh(x)2662 1030 y Fg(0)2685 1064 y Fr(\))p Fo(j)2774 1008 y Fq(f)9 b Fr(\()p Fq(x)2903 1020 y Fn(1)2940 1008 y Fr(\))p 2774 1045 199 4 v 2781 1121 a Fq(f)g Fr(\()p Fq(x)2910 1097 y Fg(0)2933 1121 y Fr(\))1137 1357 y Fo(\024)1247 1244 y Fp(Z)1344 1357 y Fq(d)p Fh(x)1437 1369 y Fn(1)1489 1357 y Fq(d)p Fh(x)1582 1369 y Fn(2)1643 1357 y Fo(j)p Fq( )s Fr(\()p Fh(x)1805 1369 y Fn(1)1843 1357 y Fq(;)14 b Fh(x)1930 1369 y Fn(2)1967 1357 y Fr(\))p Fo(j)2022 1323 y Fn(2)2079 1357 y Fo(\001)k Fq(C)2218 1301 y(\031)s(\015)2316 1271 y Fn(2)p 2218 1338 136 4 v 2265 1414 a Fr(4)2418 1301 y(1)p 2396 1338 85 4 v 2396 1414 a Fq(x)2443 1426 y Fn(1)3297 1357 y Fr(\(5.9\))160 1573 y(where)27 b Fq(f)9 b Fr(\()p Fq(x)p Fr(\))24 b(=)f Fq(x)28 b Fr(and)f(\(4.10\))g(w)n(as)g (used.)37 b(The)27 b(same)g(estimate)h(holds)f(for)g Fq(V)2653 1530 y Fn(\(2\))2634 1596 y(2)2742 1573 y Fq(:)h Fr(Hence)g(w)n(e)f(get)1211 1774 y Fo(j)p Fr(\()p Fq( )s(;)14 b(V)1427 1731 y Fn(\()p Fl(k)q Fn(\))1408 1796 y(2)1534 1774 y Fq( )s Fr(\))p Fo(j)46 b(\024)g Fq(c)1839 1786 y Fn(1)1899 1774 y Fr(\()p Fq( )s(;)14 b Fr(\()p Fo(\000)p Fq(V)2189 1739 y Fn(\()p Fl(k)q Fn(\))2282 1774 y Fr(\))g Fq( )s Fr(\))838 b(\(5.10\))160 1956 y(with)26 b Fq(c)383 1968 y Fn(1)443 1956 y Fr(=)d Fq(C)d(\015)5 b(\031)s(=)p Fr(4)47 b(whic)n(h)25 b(is)f(less)h(than)g(one)g(for)f Fq(\015)k(<)1968 1924 y Fn(4)p 1938 1938 93 4 v 1938 1985 a Fl(\031)r(C)2041 1956 y Fq(:)d Fr(This)g(pro)n(v)n(es)e(that)i (the)g(total)g(one-particle)160 2056 y(in)n(teraction)i(is)h(negativ)n (e,)933 2250 y(\()p Fq( )s(;)14 b Fr(\()p Fq(V)1158 2216 y Fn(\()p Fl(k)q Fn(\))1269 2250 y Fr(+)k Fq(V)1419 2207 y Fn(\()p Fl(k)q Fn(\))1400 2272 y(2)1512 2250 y Fr(\))c Fq( )s Fr(\))46 b Fo(\024)g Fr(\(1)18 b Fo(\000)g Fq(c)2015 2262 y Fn(1)2053 2250 y Fr(\))23 b(\()p Fq( )s(;)14 b(V)2301 2216 y Fn(\()p Fl(k)q Fn(\))2408 2250 y Fq( )s Fr(\))46 b Fq(<)g Fr(0)559 b(\(5.11\))160 2433 y(if)26 b Fq(c)270 2445 y Fn(1)330 2433 y Fq(<)d Fr(1)p Fq(:)48 b Fr(W)-7 b(e)25 b(recall)f(that)h(in)h(the)f(massless)f(case,)h(\(5.11\))f (holds)h(for)f Fq(\015)k(<)22 b Fr(4)p Fq(=\031)28 b Fr(i.e.)36 b Fq(C)29 b Fr(=)23 b(1.)35 b(Ho)n(w)n(ev)n(er,)160 2532 y(the)24 b(pro)r(of)f(for)g Fq(m)g Fr(=)g(0)g(is)h(done)f(in)h(a)f (v)n(ery)f(di\013eren)n(t)i(w)n(a)n(y)-7 b(,)23 b(in)n(v)n(olving)g (the)h(partial)e(w)n(a)n(v)n(e)g(represen)n(tation)160 2632 y(of)29 b(the)h(op)r(erators)c(and)j(their)g(Mellin)g(transforms)f ([16)o(,)h(17)o(].)41 b(This)29 b(metho)r(d)g(is)g(not)g(applicable)f (in)h(the)160 2732 y Fq(m)23 b Fo(6)p Fr(=)g(0)k(case.)285 2903 y(In)h(a)f(similar)g(w)n(a)n(y)-7 b(,)26 b(the)i(t)n(w)n (o-particle)e(estimate)i(is)f(obtained,)1183 3085 y Fo(j)p Fr(\()p Fq( )s(;)14 b(C)1397 3051 y Fn(\(12\))1534 3085 y Fq( )s Fr(\))p Fo(j)46 b(\024)1822 3064 y Fr(~)1803 3085 y Fq(C)20 b(\015)5 b(\031)26 b Fr(\()p Fq( )s(;)14 b(V)2196 3051 y Fn(\(12\))2332 3085 y Fq( )s Fr(\))p Fq(;)811 b Fr(\(5.12\))160 3268 y(again)33 b(with)602 3247 y(~)583 3268 y Fq(C)40 b Fr(a)34 b(constan)n(t)f(of)h(order)e (unit)n(y)-7 b(.)56 b(Th)n(us)33 b Fq(c)1936 3280 y Fn(2)2007 3268 y Fr(=)2124 3247 y(~)2105 3268 y Fq(C)6 b(\015)f(\031)37 b Fr(is)d(smaller)e(than)i(one)g(for)f Fq(\015)38 b(<)3395 3235 y Fn(1)p 3365 3249 V 3365 3306 a Fl(\031)3420 3291 y Fn(~)3406 3306 y Fl(C)160 3391 y Fr(whic)n(h)e(for)559 3370 y(~)540 3391 y Fq(C)42 b Fr(=)36 b(1)e(giv)n(es)g Fq(\015)41 b(<)1227 3358 y Fn(1)p 1223 3372 41 4 v 1223 3419 a Fl(\031)1309 3391 y Fr(\(corresp)r(onding)34 b(to)h Fq(Z)41 b Fo(\024)35 b Fr(43\).)59 b(W)-7 b(e)36 b(note)f(that)h(this)f (rather)f(lo)n(w)160 3501 y(estimate)26 b(for)f(the)h(critical)f Fq(\015)31 b Fr(is)25 b(not)h(sharp:)35 b(W)-7 b(e)26 b(ha)n(v)n(e)f(estimated)h(the)g(factors)e(in)i(the)h(k)n(ernel)d(of)i Fq(C)3345 3471 y Fn(\(12\))160 3613 y Fr(of)h(the)g(t)n(yp)r(e)g(\(1)17 b Fo(\000)774 3592 y Fr(~)755 3613 y Fq(D)826 3569 y Fn(\(1\))824 3635 y(0)915 3613 y Fr(\()p Fh(q)h Fr(+)e Fh(p)1149 3582 y Fg(0)1172 3613 y Fr(\)\))e(\(1)k(+)1442 3592 y(~)1423 3613 y Fq(D)1494 3569 y Fn(\(1\))1492 3635 y(0)1583 3613 y Fr(\()p Fh(p)1668 3582 y Fg(0)1692 3613 y Fr(\)\))27 b(b)n(y)g(4)f(while)h(the)h(p)r(eaking)e(v)-5 b(alue)27 b(\()p Fh(q)c Fr(=)g(0\))k(w)n(ould)f(b)r(e)160 3730 y(\(1)19 b Fo(\000)356 3709 y Fr(~)336 3730 y Fq(D)407 3687 y Fn(\(1\))405 3752 y(0)496 3730 y Fr(\()p Fh(p)581 3700 y Fg(0)605 3730 y Fr(\)\))14 b(\(1)19 b(+)879 3709 y(~)859 3730 y Fq(D)930 3687 y Fn(\(1\))928 3752 y(0)1019 3730 y Fr(\()p Fh(p)1104 3700 y Fg(0)1128 3730 y Fr(\)\))38 b(=)24 b(0)p Fq(:)51 b Fr(Therefore,)27 b(w)n(e)h(conjecture)f(that)i (the)f(v)-5 b(alue)28 b(for)f Fq(c)3059 3742 y Fn(2)3125 3730 y Fr(is)h(smaller)160 3829 y(than)g Fq(\015)5 b(\031)s(:)160 4175 y Fw(6)135 b(P)l(ositivit)l(y)47 b(of)f(the)f(pseudo-relativistic) i(op)t(erator)e Fa(H)3124 4132 y Fr(\(2\))160 4429 y(The)33 b(op)r(erator)d(whic)n(h)i(w)n(e)g(ha)n(v)n(e)f(selected)h(b)n(y)f (means)h(of)g(restriction)f(to)h(the)h(p)r(ositiv)n(e)e(sp)r(ectral)h (sub-)160 4528 y(space)40 b Fo(H)465 4540 y Fn(+)p Fl(;)p Fn(2)614 4528 y Fr(of)h(the)g(free)g(Dirac)f(op)r(erator,)j(\003)1722 4540 y Fn(++)1827 4528 y Fq(H)1903 4498 y Fn(\(2\))1992 4528 y Fr(\003)2050 4540 y Fn(++)2156 4528 y Fq(;)e Fr(is)f(not)h(from) g(the)g(outset)g(a)f(p)r(ositiv)n(e)160 4647 y(op)r(erator.)66 b(Whereas)36 b(in)i(the)g(single-particle)e(case)h(one)g(has)g Fq(P)2279 4604 y Fn(\()p Fl(k)q Fn(\))2267 4668 y(+)2372 4647 y Fq(D)2443 4604 y Fn(\()p Fl(k)q Fn(\))2441 4672 y Fl(V)2536 4647 y Fq(P)2601 4604 y Fn(\()p Fl(k)q Fn(\))2589 4668 y(+)2733 4647 y Fo(\025)i Fr(0)f(and)f(hence)h(also)160 4770 y(\003)218 4727 y Fn(\()p Fl(k)q Fn(\))218 4790 y(+)325 4770 y Fq(U)391 4740 y Fn(\()p Fl(k)q Fn(\))p Fg(\000)p Fn(1)568 4770 y Fq(D)639 4727 y Fn(\()p Fl(k)q Fn(\))637 4794 y Fl(V)732 4770 y Fq(U)798 4740 y Fn(\()p Fl(k)q Fn(\))904 4770 y Fr(\003)962 4727 y Fn(\()p Fl(k)q Fn(\))962 4790 y(+)1078 4770 y Fo(\025)22 b Fr(0)50 b(for)26 b(an)g(exact)g(c)n(hoice)g(of)h Fq(U)35 b Fr(\(see)27 b(section)f(2\),)h(the)g(neglect)f(of)h(the)g(in-)160 4869 y(terelectronic)e(in)n(teraction)f(in)i(the)g(t)n(w)n(o-particle)d (pro)5 b(jectors)24 b(as)g(w)n(ell)i(as)e(the)i(p)r(erturbativ)n(e)f (treatmen)n(t)160 4969 y(of)j(the)g(unitary)f(transformation)e Fq(U)37 b Fr(can)27 b(in)g(principle)h(destro)n(y)e(this)h(prop)r(ert)n (y)-7 b(.)36 b(Therefore)26 b(w)n(e)h(giv)n(e)g(a)160 5069 y(pro)r(of)g(of)h(p)r(ositivit)n(y)f(b)n(y)h(using)f(the)h (explicit)g(represen)n(tation)e(of)h Fq(H)2323 5038 y Fn(\(2\))2412 5069 y Fq(:)1772 5413 y Fr(11)p eop %%Page: 12 12 12 11 bop 285 51 a Fr(Again)27 b(w)n(e)g(mak)n(e)g(use)h(of)f(a)g (single-particle)f(estimate)i([21)o(])720 251 y(\()p Fq( )s(;)14 b Fr(\()p Fq(D)949 208 y Fn(\()p Fl(k)q Fn(\))947 273 y(0)1061 251 y Fr(+)k Fq(V)1211 217 y Fn(\()p Fl(k)q Fn(\))1322 251 y Fr(+)g Fq(V)1472 208 y Fn(\()p Fl(k)q Fn(\))1453 273 y(2)1564 251 y Fr(\))c Fq( )s Fr(\))47 b Fo(\025)f Fr(\(1)32 b Fo(\000)2121 195 y Fq(\015)p 2070 232 151 4 v 2070 308 a(\015)2113 320 y Fl(B)s(R)2262 251 y Fo(\000)g Fq(d\015)2450 217 y Fn(2)2487 251 y Fr(\))24 b(\()p Fq( )s(;)14 b(E)2730 263 y Fl(p)2764 272 y Fi(k)2819 251 y Fq( )s Fr(\))389 b(\(6.1\))160 485 y(where)27 b Fq(d)c Fr(=)563 453 y Fn(1)p 563 467 34 4 v 563 514 a(8)620 485 y Fr(\()662 453 y Fl(\031)p 663 467 41 4 v 667 514 a Fn(2)730 485 y Fo(\000)825 453 y Fn(2)p 821 467 V 821 514 a Fl(\031)872 485 y Fr(\))904 455 y Fn(2)942 485 y Fq(:)49 b Fr(Estimating)26 b Fq(V)1506 455 y Fn(\(12\))1652 485 y Fo(\025)c Fr(0)27 b(as)f(b)r(efore)g(and)g(applying)g(the)i (estimate)e(\(4.11\))g(for)160 596 y Fq(C)225 566 y Fn(\(12\))348 596 y Fr(,)i(one)f(gets)687 824 y(\()p Fq( )s(;)14 b(H)889 789 y Fn(\(2\))992 824 y Fq( )s Fr(\))46 b Fo(\025)g Fr(\(1)32 b Fo(\000)1502 768 y Fq(\015)p 1451 805 151 4 v 1451 881 a(\015)1494 893 y Fl(B)s(R)1643 824 y Fo(\000)g Fq(d\015)1831 789 y Fn(2)1868 824 y Fr(\))24 b(\()p Fq( )s(;)14 b(T)25 b( )s Fr(\))42 b Fo(\000)f Fq(\015)5 b(e)2448 789 y Fn(2)2508 768 y Fq(\031)2558 737 y Fn(2)p 2508 805 88 4 v 2531 881 a Fr(4)2629 824 y(\()p Fq( )s(;)14 b(T)25 b( )s Fr(\))p Fq(:)356 b Fr(\(6.2\))160 1055 y(The)40 b(r.h.s.)73 b(is)40 b(p)r(ositiv)n(e)f(if)h Fq(c)1136 1067 y Fn(0)1217 1055 y Fr(:=)j(1)26 b Fo(\000)g Fq(\015)5 b Fr(\()1644 1022 y Fn(1)p 1597 1036 127 4 v 1597 1083 a Fl(\015)1632 1091 y Fi(B)r(R)1774 1055 y Fr(+)26 b Fq(e)1904 1024 y Fn(2)1941 1055 y Fq(\031)1991 1024 y Fn(2)2029 1055 y Fq(=)p Fr(4\))40 b Fo(\000)26 b Fq(d\015)2367 1024 y Fn(2)2447 1055 y Fq(>)43 b Fr(0)83 b(whic)n(h)39 b(is)h(the)g(case)f(for)160 1177 y Fq(\015)28 b(<)23 b Fr(0)p Fq(:)p Fr(825)44 b(\()p Fq(Z)29 b Fo(\024)23 b Fr(113\).)36 b(W)-7 b(e)28 b(conjecture)f(that)h(this)g(b)r(ound)g (for)f Fq(H)2320 1146 y Fn(\(2\))2437 1177 y Fr(is)g(not)h(sharp,)f(b)r (ecause)g(for)g(the)160 1276 y(massless)f(single-particle)f(op)r (erator)g(one)h(obtains)g(\(1)k Fo(\000)2033 1239 y Fl(\015)p 1988 1257 V 1988 1305 a(\015)2023 1313 y Fi(B)r(R)2142 1276 y Fr(+)16 b Fq(d\015)2314 1246 y Fn(2)2351 1276 y Fr(\))e(\()p Fq( )s(;)g(p)2565 1288 y Fl(k)2606 1276 y Fq( )s Fr(\))50 b(in)26 b(place)g(of)h(the)g(r.h.s.)160 1387 y(of)i(\(6.1\))f(\(this)i(result)e(is)g(again)g(found)h(with)g (the)g(help)g(of)f(Mellin)h(transform)f(tec)n(hniques)g([16)o(]\).)41 b(This)160 1486 y(b)r(ound)25 b(for)f Fq(m)f Fr(=)g(0)h(leads)g(to)h(p) r(ositivit)n(y)f(of)h Fq(H)1625 1456 y Fn(\(2\))1738 1486 y Fr(for)f Fq(\015)k(<)23 b Fr(0)p Fq(:)p Fr(986)p Fq(;)f Fr(whic)n(h)j(is)f(v)n(ery)g(close)g(to)g(the)h(limiting)160 1586 y(v)-5 b(alue)28 b Fq(\015)g Fr(=)22 b(1)27 b(from)h(the)g(exact)f (one-electron)f(Dirac)h(theory)-7 b(.)160 1931 y Fw(7)135 b(Concluding)45 b(remarks)160 2185 y Fr(W)-7 b(e)31 b(ha)n(v)n(e)e (established)h(a)g(pseudo-relativistic)e(op)r(erator)h Fq(H)2102 2155 y Fn(\(2\))2221 2185 y Fr(whic)n(h)h(describ)r(es)g(t)n (w)n(o)f(electrons)h(in)g(a)160 2284 y(cen)n(tral)d(Coulom)n(b)g(p)r (oten)n(tial)h(of)g(strength)f Fq(\015)5 b Fr(.)37 b(This)27 b(op)r(erator,)g(whic)n(h)g(includes)h(terms)g(up)g(to)f(second)160 2384 y(order)d(in)g(the)h(coupling)f(constan)n(t)g Fq(e)1308 2354 y Fn(2)1345 2384 y Fr(,)h(is)f(w)n(ell-de\014ned)h(for)f(p)r(oten) n(tial)g(strengths)g Fq(\015)j(<)c Fr(0)p Fq(:)p Fr(89)p Fq(:)g Fr(Moreo)n(v)n(er,)160 2484 y(the)35 b(terms)e(of)h Fq(H)724 2453 y Fn(\(2\))847 2484 y Fr(describing)f(in)n(teraction)g(p) r(oten)n(tials)h(of)f(second)h(order)e(in)i Fq(e)2800 2453 y Fn(2)2871 2484 y Fr(are)f(sho)n(wn)g(to)h(b)r(e)160 2583 y(smaller)f(than)h(the)g(corresp)r(onding)e(\014rst-order)f(terms) i(for)h(at)f(least)g Fq(\015)38 b(<)33 b Fr(1)p Fq(=\031)s(;)g Fr(whic)n(h)g(assures)f(that)160 2683 y(the)c(single-particle)f(p)r (oten)n(tial)g(of)h Fq(H)1356 2653 y Fn(\(2\))1472 2683 y Fr(is)g(attractiv)n(e)f(and)g(the)h(t)n(w)n(o-particle)e(p)r(oten)n (tial)i(repulsiv)n(e)e(as)160 2782 y(is)g(the)h(case)e(for)h(the)h (Coulom)n(b-Dirac)d(op)r(erator.)35 b(The)26 b(sp)r(ectrum)g(of)g Fq(H)2468 2752 y Fn(\(2\))2584 2782 y Fr(is)g(p)r(ositiv)n(e)f(for)h Fq(\015)i(<)22 b Fr(0)p Fq(:)p Fr(825)p Fq(:)285 2954 y Fr(It)45 b(is)g(straigh)n(tforw)n(ard)d(to)j(deriv)n(e)f(a)g (pseudo-relativistic)f(op)r(erator)h(for)g(the)h Fq(N)9 b Fr(-electron)44 b(ion)160 3053 y(with)d Fq(N)51 b Fo(\024)43 b Fq(Z)6 b Fr(,)42 b(applying)d(the)h(same)f(transformation)f(sc)n (heme.)73 b(It)40 b(can)f(b)r(e)h(sho)n(wn)f(that)h(to)f(sec-)160 3153 y(ond)g(order)f(in)h Fq(e)709 3123 y Fn(2)746 3153 y Fq(;)g Fr(p)r(oten)n(tials)g(a\013ecting)f(more)g(than)h(t)n(w)n(o)g (electrons)f(sim)n(ultaneously)g(do)g(not)h(o)r(c-)160 3252 y(cur.)46 b(Rather,)30 b(w)n(e)g(get)h(a)f(simple)g (generalisation)f(of)h Fq(H)1952 3222 y Fn(\(2\))2041 3252 y Fq(;)h Fr(if)g(acting)f(on)g(the)h(p)r(ositiv)n(e)f(sp)r(ectral) g(sub-)160 3352 y(space)38 b Fo(H)463 3364 y Fn(+)p Fl(;N)638 3352 y Fr(=)j(\003)802 3364 y Fn(+)p Fl(;N)949 3352 y Fr(\()p Fo(A)p Fr(\()p Fq(H)1148 3364 y Fn(1)1186 3352 y Fr(\()p Fm(R)1272 3322 y Fn(3)1315 3352 y Fr(\))26 b Fo(\012)g Fm(C)1518 3322 y Fn(4)1561 3352 y Fr(\))1593 3322 y Fl(N)1656 3352 y Fr(\))80 b(of)38 b(an)n(tisymmetrised)g Fq(N)9 b Fr(-electron)37 b(spinors,)j(where)160 3471 y(\003)218 3483 y Fn(+)p Fl(;N)389 3471 y Fr(=)22 b(\003)534 3428 y Fn(\(1\))534 3492 y(+)623 3471 y Fr(\003)681 3428 y Fn(\(2\))681 3492 y(+)784 3471 y Fo(\001)14 b(\001)g(\001)f Fr(\003)952 3428 y Fn(\()p Fl(N)6 b Fn(\))952 3492 y(+)1116 3471 y Fr(is)25 b(the)i(pro)r(duct)e(of)h(the)g(free)g(single-particle) e(pro)5 b(jectors.)34 b(The)26 b(op)r(erator)160 3570 y(is)i(giv)n(en)f(b)n(y)324 3826 y Fq(H)400 3783 y Fn(\(2\))393 3850 y Fl(N)535 3826 y Fr(=)677 3722 y Fl(N)646 3747 y Fp(X)646 3926 y Fl(k)q Fn(=1)767 3826 y Fr(\()p Fq(D)870 3783 y Fn(\()p Fl(k)q Fn(\))868 3848 y(0)981 3826 y Fr(+)18 b Fq(V)1131 3792 y Fn(\()p Fl(k)q Fn(\))1242 3826 y Fr(+)g Fq(V)1392 3783 y Fn(\()p Fl(k)q Fn(\))1373 3848 y(2)1485 3826 y Fr(\))42 b(+)1725 3722 y Fl(N)1694 3747 y Fp(X)1675 3914 y Fi(n;k)q Fe(=1)1697 3962 y Fi(n)c Fr(0)k(for)f Fq(c)805 5077 y Fl(N)893 5065 y Fr(=)c(1)33 b Fo(\000)1193 5028 y Fl(\015)p 1149 5046 V 1149 5093 a(\015)1184 5101 y Fi(B)r(R)1319 5065 y Fo(\000)18 b Fq(d\015)1493 5035 y Fn(2)1549 5065 y Fo(\000)h Fq(\015)5 b(e)1720 5035 y Fn(2)1780 5032 y Fl(\031)1821 5007 y Fe(2)p 1780 5046 74 4 v 1800 5093 a Fn(4)1863 5065 y Fr(\()p Fq(N)28 b Fo(\000)19 b Fr(1\))38 b Fq(>)24 b Fr(0)p Fq(;)52 b Fr(requiring)27 b Fq(\015)i(<)24 b Fr(0)p Fq(:)p Fr(446)46 b(\()p Fq(Z)30 b Fo(\024)24 b Fr(61\))160 5164 y(in)32 b(the)f(case)g(of)g(a)g(neutral)g(atom,)g Fq(N)38 b Fr(=)29 b Fq(Z)q(:)60 b Fr(Again,)32 b(this)f(estimate)h(is)f (not)g(sharp)f(b)r(ecause)h(w)n(e)g(ha)n(v)n(e)1772 5413 y(12)p eop %%Page: 13 13 13 12 bop 160 51 a Fr(used)33 b(the)g(rather)f(crude)g(\(in)h(the)g (large-)p Fq(N)40 b Fr(case\))32 b(estimate)h Fq(V)2201 20 y Fn(\()p Fl(nk)q Fn(\))2366 51 y Fo(\025)e Fr(0)p Fq(:)64 b Fr(Ho)n(w)n(ev)n(er,)32 b(the)h(additional)160 150 y(scaling)38 b(with)h(\()p Fq(N)34 b Fo(\000)25 b Fr(1\))p Fq(=)p Fr(2)38 b(of)g(the)h(sum)f(of)g(electron-electron)f(in) n(teraction)g(terms,)k(as)c(w)n(ell)i(as)e(the)160 250 y(marginal)26 b(dominance)g(of)g Fq(V)1080 220 y Fn(\()p Fl(nk)q Fn(\))1240 250 y Fr(with)h(resp)r(ect)g(to)f Fq(C)1877 220 y Fn(\()p Fl(nk)q Fn(\))2038 250 y Fr(when)h Fq(Z)32 b Fr(approac)n(hes)24 b(43,)i(mak)n(e)g(clear)f(that)160 349 y(the)i(second-order)d(t)n(w)n(o-particle)h(in)n(teraction)h(terms) g(should)g(not)g(b)r(e)h(neglected)f(in)h(atomic)f(structure)160 449 y(calculations)h(for)g(high-)p Fq(N)36 b Fr(ions.)160 856 y Ft(I)26 b(should)f(lik)n(e)h(to)g(thank)f(H.Sieden)n(top,)g (H.Kalf)h(and)g(E.Sto)r(c)n(kmey)n(er)e(for)j(fruitful)f(discussions.) 160 1201 y Fw(References)202 1415 y Fr([1])41 b(W.P)n(auli,)27 b(in)h Fd(Handbuch)i(der)h(Physik)p Fr(,)e(V)-7 b(ol.)28 b(V)g(\(Springer-V)-7 b(erlag,)25 b(Berlin,)i(1958\),)f(P)n(art)h(I,)g (p.1)202 1577 y([2])41 b(Ph.Durand)31 b(and)g(J.-P)-7 b(.Malrieu,)31 b(in)h Fd(A)n(b)h(Initio)h(Metho)l(ds)g(in)g(Quantum)d (Chemistry)p Fr(,)j(edited)e(b)n(y)331 1677 y(I.K.P)-7 b(.La)n(wley)26 b(\(John)h(Wiley)h(and)g(Sons,)f(New)h(Y)-7 b(ork,)27 b(1987\))202 1840 y([3])41 b(J.Dolb)r(eault,)28 b(M.J.Esteban)f(and)g(E.S)n(\023)-39 b(er)n(\023)g(e,)25 b(Calc.)i(V)-7 b(ar.)27 b Fh(10)p Fr(,)g(321)f(\(2000\))202 2002 y([4])41 b(M.Douglas)27 b(and)g(N.M.Kroll,)g(Ann.)h(Ph)n(ys.)f (\(N.Y.\))i Fh(82)p Fr(,)e(89)g(\(1974\))202 2165 y([5])41 b(B.A.Hess,)28 b(Ph)n(ys.)e(Rev.)i(A)g Fh(33)p Fr(,)f(3742)f(\(1986\)) 202 2328 y([6])41 b(A.W)-7 b(olf,)28 b(M.Reiher)g(and)f(B.A.Hess,)h (J.Chem.)g(Ph)n(ys.)e Fh(117)p Fr(,)h(9215)f(\(2002\))202 2491 y([7])41 b(E.M.Rose,)27 b Fd(R)l(elativistic)k(Ele)l(ctr)l(on)e (The)l(ory)h Fr(\(John)d(Wiley)h(and)f(Sons,)g(New)h(Y)-7 b(ork,)27 b(1961\))202 2654 y([8])41 b(P)-7 b(.M.Morse)24 b(and)h(H.F)-7 b(esh)n(bac)n(h,)26 b Fd(Metho)l(ds)j(of)f(The)l(or)l (etic)l(al)i(Physics)d Fr(\(Mc)f(Gra)n(w)e(Hill,)j(New)e(Y)-7 b(ork,)331 2753 y(1953\),)26 b(P)n(art)g(I)r(I,)i Fo(x)p Fr(9)202 2916 y([9])41 b(T.Kato,)22 b Fd(Perturb)l(ation)i(The)l(ory)i (for)f(Line)l(ar)g(Op)l(er)l(ators)p Fr(,)e(V)-7 b(ol.)22 b Fh(132)f Fr(of)h Fd(Grund)t(lehr)l(en)i(der)h(mathe-)331 3016 y(matischen)31 b(Wissenschaften)d Fr(\(Springer-V)-7 b(erlag,)26 b(Berlin,)h(1966\))160 3178 y([10])41 b(D.H.Jakuba\031a-Am) n(undsen,)26 b(PhD)i(Thesis,)f(Univ)n(ersit)n(y)g(of)h(Munic)n(h)f (\(2004\))331 3278 y(h)n(ttp://www.mathematik.uni-m)n(uenc)n (hen.de/~dj/pubmath.h)n(tml)160 3441 y([11])41 b(J.Suc)n(her,)27 b(Ph)n(ys.)g(Rev.)g(A)h Fh(22)p Fr(,)f(348)f(\(1980\))160 3604 y([12])41 b(J.Suc)n(her,)27 b(PhD)h(Thesis,)f(Colum)n(bia)g(Univ)n (ersit)n(y)-7 b(,)27 b(USA)h(\(1958\))160 3766 y([13])41 b(L.L.F)-7 b(oldy)27 b(and)h(S.A.W)-7 b(outh)n(uysen,)28 b(Ph)n(ys.)f(Rev.)g Fh(78)p Fr(,)g(29)g(\(1950\))160 3929 y([14])41 b(A.Sob)r(olev,)27 b(Revista)h(Matematica)f(Ib)r (eroamericana)e(\(2004\),)h(in)i(prin)n(t)160 4092 y([15])41 b(G.Jansen)27 b(and)g(B.A.Hess,)h(Ph)n(ys.)f(Rev.)g(A)h Fh(39)p Fr(,)f(6016)f(\(1989\))160 4255 y([16])41 b(R.Brummelh)n(uis,) 28 b(H.Sieden)n(top)f(and)h(E.Sto)r(c)n(kmey)n(er,)e(Do)r(c.)i(Math.)f Fh(7)p Fr(,)h(167)e(\(2002\))160 4418 y([17])41 b(D.H.Jakuba\031a-Am)n (undsen,)26 b(Math.)i(Ph)n(ys.)f(Electron.)f(J.)i Fh(8)p Fr(\(3\),)f(1)g(\(2002\))160 4580 y([18])41 b(V.I.Burenk)n(o)n(v)26 b(and)i(W.D.Ev)-5 b(ans,)27 b(Pro)r(c.)g(Ro)n(y)-7 b(.)27 b(So)r(c.)g(\(Edin)n(burgh\))g Fh(128)p Fr(A,)h(993)e(\(1998\))160 4743 y([19])41 b(E.H.Lieb)28 b(and)f(H.-T.Y)-7 b(au,)28 b(Comm)n(un.)f(Math.)h(Ph)n(ys.)f Fh(118)p Fr(,)g(177)f(\(1988\))160 4906 y([20])41 b(C.Tix,)28 b(Preprin)n(t)e(mp-arc/97-441)d(\(1997\))160 5069 y([21])41 b(A.Ian)n(tc)n(henk)n(o)26 b(and)i(D.H.Jakuba\031a-Am)n (undsen,)e(Ann.)i(Henri)g(P)n(oincar)n(\023)-39 b(e)24 b Fh(4)p Fr(,)k(1)f(\(2003\))1772 5413 y(13)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF