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Fp(has)h(b)s(een)g(a)f(source)i(of)e(con)m(tro) m(v)m(ersy)j(in)257 4625 y(ph)m(ysics,)35 b(as)e(it)f(seems)i(to)f(b)s (e)g(inconsisten)m(t)h(with)e(the)i(existence)g(of)f(an)g(energy)h (mini-)257 4745 y(mizing)d(ground)j(state.)46 b(Indeed,)35 b(in)e(his)g(original)d(prop)s(osal)i(Dirac)g(himself)g(declared)257 4866 y(all)39 b(negativ)m(e)i(energy)g(lev)m(els)g(to)f(b)s(e)h (\014lled,)h(and)e(this)h(lead)f(to)g(imp)s(ortan)m(t)f(concepts)257 4986 y(suc)m(h)47 b(as)e(normal-ordering)d(and)j(an)m(ti-particles.)78 b(The)46 b(lac)m(k)f(of)g(semib)s(oundedness)257 5107 y(is)39 b(the)h(price)g(to)f(pa)m(y)h(for)f(ha)m(ving)g(a)h(square)g (ro)s(ot)f(of)g(the)h(Klein-Gordon)d(op)s(erator,)257 5227 y Fl(\000)p Fp(\001)23 b(+)f(1)33 b(whic)m(h)g(is)f(a)g(\\lo)s (cal",)e(i.e.,)i(di\013eren)m(tial)f(op)s(erator.)p eop %%Page: 2 3 2 2 bop 257 266 a Fq(BM-1,)33 b(Septem)m(b)s(er)g(22,)f(2000)2054 b Fp(2)404 573 y(One)37 b(w)m(a)m(y)h(to)f(restore)g(the)h(semib)s (oundedness)g(of)e(the)i(electron)e(Hamiltonian)e(is)257 693 y(to)42 b(study)h(a)e(simpli\014ed)f(op)s(erator)h(obtained)h(from) e(restricting)h Fn(D)2812 708 y Fm(0)2893 693 y Fp(to)h(its)f(p)s (ositiv)m(e)257 814 y(sp)s(ectral)33 b(subspace)h Fl(H)1115 829 y Fm(+)1202 814 y Fp(:=)28 b(\003)1401 829 y Fm(+)1460 814 y Fl(H)q Fp(,)k(where)1155 992 y(\003)1223 1007 y Fm(+)1342 992 y Fp(:=)60 b Fn(\037)1566 1008 y Fm([0)p Fj(;)p Ff(1)p Fm(\))1743 992 y Fp([)p Fn(D)1851 1007 y Fm(0)1890 992 y Fp(])h(=)f Fn(\037)2175 1008 y Fm([1)p Fj(;)p Ff(1)p Fm(\))2352 992 y Fp([)p Fn(D)2460 1007 y Fm(0)2499 992 y Fp(])28 b Fn(;)727 b Fp(\(I.6\))257 1171 y(and,)33 b(in)f(F)-8 b(ourier)31 b(represen)m(tation,)1135 1341 y Fk(\000)1181 1421 y Fl(F)g Fp(\003)1352 1436 y Fm(+)1433 1421 y Fl(F)1515 1380 y Ff(\000)p Fm(1)1609 1341 y Fk(\001)1655 1421 y Fp([)1684 1395 y Fn(~)1682 1421 y(\030)t Fp(])60 b(=)1962 1354 y(1)p 1962 1398 49 4 v 1962 1490 a(2)2066 1421 y(+)2198 1354 y Fn(~)-51 b(\013)22 b Fl(\001)2333 1328 y Fn(~)2330 1354 y(\030)27 b Fp(+)22 b Fn(\014)p 2196 1398 363 4 v 2249 1490 a(\030)2297 1461 y Fm(2)2358 1490 y Fp(+)g(1)2591 1421 y Fn(:)690 b Fp(\(I.7\))257 1645 y(As)34 b(a)e(semib)s(ounded)h(relativistic)e (electron)i(mo)s(del)e(that)i(accommo)s(dates)f(for)g(b)s(ound)257 1765 y(states,)52 b(Bethe)47 b(and)g(Salp)s(eter)f(prop)s(osed)i([3])e (to)h(add)g(a)f(p)s(oten)m(tial)f Fn(V)69 b Fp(to)46 b Fn(D)3252 1780 y Fm(0)3338 1765 y Fp(and)257 1885 y(restrict)32 b(it)e(again)f(to)i(the)h(p)s(ositiv)m(e)f(subspace)i Fl(H)2082 1900 y Fm(+)2172 1885 y Fp(of)e(the)g(free)h(Dirac)e(op)s (erator.)42 b(The)257 2006 y(t)m(w)m(o-electron)27 b(analogon)f(of)g (this)h(pro)s(cedure)g(w)m(as)h(in)m(tro)s(duced)f(earlier)f(b)m(y)i (Bro)m(wn)f(and)257 2126 y(Ra)m(v)m(enhall)32 b([4].)404 2246 y(W)-8 b(e)33 b(assume)g Fn(V)54 b Fp(to)32 b(satisfy)h(the)g (follo)m(wing)c(quadratic)k(form)e(b)s(ound,)1120 2425 y Fl(h)17 b Fn( )t(;)g(V)1365 2384 y Fm(2)1421 2425 y Fn( )k Fl(i)60 b(\024)g Fn(\014)1802 2384 y Fm(2)1858 2425 y Fl(h)17 b Fn( )t(;)g Fp(\()p Fl(\000)p Fp(\001)22 b(+)g(1\))17 b Fn( )j Fl(i)i Fn(;)692 b Fp(\(I.8\))257 2604 y(for)32 b(all)f Fn( )h Fl(2)c Fn(H)820 2567 y Fm(2)859 2604 y Fp(\()p Fh(R)962 2567 y Fm(3)1008 2604 y Fp(;)17 b Fh(C)1118 2567 y Fm(4)1163 2604 y Fp(\))32 b(and)h(some)g Fn(\014)g(<)27 b Fp(1,)33 b(and)f(furthermore)1049 2782 y Fl(k)p Fn(V)21 b Fp(\()p Fn(x)p Fp(\))p Fl(k)1358 2799 y Ff(L)p Fm(\()p Fe(C)1477 2780 y Fd(4)1517 2799 y Fm(\))1593 2782 y Fl(\000)-16 b(!)44 b Fp(0)17 b Fn(;)113 b Fp(as)98 b Fl(j)p Fn(x)p Fl(j)44 b(\000)-16 b(!)44 b(1)27 b Fn(:)605 b Fp(\(I.9\))257 2961 y(Then)34 b(the)f(op)s(erator)1336 3139 y Fn(B)1410 3154 y Fj(V)1531 3139 y Fp(=)60 b(\003)1735 3154 y Fm(+)1811 3139 y Fp(\()p Fn(D)1930 3154 y Fm(0)1991 3139 y Fl(\000)23 b Fn(V)e Fp(\))c(\003)2292 3154 y Fm(+)2373 3139 y Fn(;)860 b Fp(\(I.10\))257 3318 y(resulting)45 b(from)f(this)i(pro)s(cedure,)j(is)c(de\014ned)j(on)d(the)h(same)f (domain)f(as)i Fn(B)3228 3333 y Fm(0)3268 3318 y Fp(,)i(i.e.,)257 3438 y(dom[)p Fn(B)543 3453 y Fj(V)604 3438 y Fp(])h(=)f(dom[)p Fn(B)1090 3453 y Fm(0)1129 3438 y Fp(])h(=)g(\003)1398 3453 y Fm(+)1457 3438 y Fp(dom)o([)p Fn(D)1749 3453 y Fm(0)1788 3438 y Fp(])17 b Fn(;)45 b Fp(since)g Fn(V)67 b Fp(is)44 b Fn(B)2463 3453 y Fm(0)2503 3438 y Fp(-b)s(ounded)h(with)g (relativ)m(e)257 3558 y(b)s(ound)25 b Fn(\014)6 b Fp(.)41 b(Moreo)m(v)m(er,)28 b Fn(B)1202 3573 y Fj(V)1287 3558 y Fp(is)d(b)s(ounded)g(from)e(b)s(elo)m(w)i(b)m(y)h(1)6 b Fl(\000)g Fn(\014)33 b(>)28 b Fp(0)c(and)h(in)f(particular)257 3679 y(in)m(v)m(ertible.)42 b(T)-8 b(o)29 b(lo)s(cate)f(its)h(essen)m (tial)g(sp)s(ectrum)g(w)m(e)h(use)g(a)f(standard)g(argumen)m(t)g (\(see,)257 3799 y(e.g.,)38 b([15)o(]\).)54 b(Condition)34 b(\(I.9\))i(implies)e(that)i Fn(B)2048 3758 y Ff(\000)p Fm(1)2043 3824 y(0)2167 3799 y Fl(\000)24 b Fn(B)2347 3758 y Ff(\000)p Fm(1)2342 3826 y Fj(V)2478 3799 y Fp(is)35 b(the)i(norm)e(limit)d(of)k(the)257 3920 y(op)s(erators)e 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y Fj(\037)2436 4257 y Fc(R)2487 4246 y Fj(V)2570 4219 y Fp(\003)2638 4234 y Fm(+)2697 4219 y Fn(V)38 b(\037)2853 4234 y Fj(R)2927 4219 y Fl(j)p Fn(D)3036 4234 y Fm(0)3075 4219 y Fl(j)3103 4177 y Ff(\000)p Fm(1)3214 4219 y Fp(\003)3282 4234 y Fm(+)3368 4219 y Fn(:)3260 4351 y Fp(\(I.11\))257 4538 y(No)m(w)24 b Fn(B)549 4496 y Ff(\000)p Fm(1)544 4565 y Fj(\037)588 4576 y Fc(R)639 4565 y Fj(V)716 4538 y Fp(\003)784 4553 y Fm(+)843 4538 y Fn(V)45 b Fp(is)23 b(b)s(ounded)h(b)s(ecause)h(of)e(\(I.8\))g(and)h Fn(\037)2329 4553 y Fj(R)2403 4538 y Fl(j)p Fn(D)2512 4553 y Fm(0)2551 4538 y Fl(j)2579 4501 y Ff(\000)p Fm(1)2697 4538 y Fp(is)f(compact)g(b)s(ecause)257 4670 y(of)37 b(the)g(lo)s(cal)d(compactness)k(prop)s(ert)m(y)f(of)f(the)i(free)f (Dirac)e(op)s(erator.)55 b(Th)m(us,)39 b Fn(B)3298 4629 y Ff(\000)p Fm(1)3293 4694 y(0)3418 4670 y Fl(\000)257 4790 y Fn(B)336 4749 y Ff(\000)p Fm(1)331 4817 y Fj(\037)375 4828 y Fc(R)426 4817 y Fj(V)524 4790 y Fp(is)f(compact,)h(for)e(eac)m (h)i Fn(R)f(>)e Fp(0,)j(and)f(so)h(is)e(its)g(norm)g(limit)e Fn(B)2913 4749 y Ff(\000)p Fm(1)2908 4815 y(0)3033 4790 y Fl(\000)26 b Fn(B)3215 4749 y Ff(\000)p Fm(1)3210 4817 y Fj(V)3310 4790 y Fn(:)38 b Fp(By)257 4911 y(W)-8 b(eyl's)33 b(essen)m(tial)g(sp)s(ectrum)g(theorem,)f(w)m(e)i(therefore)f(obtain) 1209 5089 y Fn(\033)1264 5104 y Fm(ess)1355 5089 y Fp([)p Fn(B)1456 5104 y Fj(V)1517 5089 y Fp(])56 b(=)f Fn(\033)1786 5104 y Fm(ess)1878 5089 y Fp([)p Fn(B)1979 5104 y Fm(0)2018 5089 y Fp(])h(=)f([1)p Fn(;)17 b Fl(1)p Fp(\))27 b Fn(:)716 b Fp(\(I.12\))257 5268 y(Our)24 b(concern)g(in)f(this)g(pap)s(er)g(is)g (to)g(pro)m(v)m(e)i(the)f(follo)m(wing)c(exp)s(onen)m(tial)j(deca)m(y)i (estimate)257 5388 y(for)32 b(the)h(eigenfunctions)g(of)f Fn(B)1397 5403 y Fj(V)1490 5388 y Fp(including)f(an)i(estimate)e(on)i (their)f(deca)m(y)i(rate.)p eop %%Page: 3 4 3 3 bop 257 266 a Fq(BM-1,)33 b(Septem)m(b)s(er)g(22,)f(2000)2054 b Fp(3)257 573 y Fg(Theorem)37 b(I.1)49 b Fi(Supp)-5 b(ose)32 b Fn(V)55 b Fi(satis\014es)32 b(\(I.8\))g(and)h(\(I.9\).)43 b(F)-7 b(urthermor)i(e,)32 b(let)h Fp(0)28 b Fn(<)f(b)h(<)257 693 y Fp(1)22 b Fl(\000)h Fn(a)28 b(<)f Fp(1)p Fi(.)45 b(Then)1298 782 y Fk(\015)1298 841 y(\015)1298 901 y(\015)1353 896 y Fn(e)1398 855 y Fj(a)p Ff(j)p Fj(x)p Ff(j)1519 896 y Fn(\037)1580 912 y Fm(\()p Fj(b)12 b(;)k Fm(1)p Ff(\000)p Fj(a)p Fm(\))1844 896 y Fp([)p Fn(B)1945 911 y Fj(V)2006 896 y Fp(])2033 782 y Fk(\015)2033 841 y(\015)2033 901 y(\015)2151 896 y Fn(<)62 b Fl(1)22 b Fn(;)822 b Fp(\(I.13\))257 1116 y Fi(wher)-5 b(e)46 b Fn(\037)605 1131 y Fm(\()p Fj(b)12 b(;)j Fm(1)p Ff(\000)p Fj(a)p Fm(\))869 1116 y Fp([)p Fn(B)970 1131 y Fj(V)1031 1116 y Fp(])46 b Fi(denotes)f(the)h(sp)-5 b(e)g(ctr)g(al)45 b(pr)-5 b(oje)g(ction)46 b(of)f Fn(B)2660 1131 y Fj(V)2767 1116 y Fi(onto)h(the)g(interval)257 1236 y Fp(\()p Fn(b)17 b(;)39 b Fp(1)22 b Fl(\000)h Fn(a)p Fp(\))p Fi(.)257 1416 y Fp(Although)35 b(its)g(sp)s(ectrum)h(resem)m(bles)g(the)g (standard)g(sp)s(ectrum)g(of)f(Sc)m(hr\177)-49 b(odinger)36 b(op-)257 1536 y(erators,)29 b Fn(B)680 1551 y Fj(V)769 1536 y Fp(is)f(not)f(a)h(di\013eren)m(tial)e(op)s(erator,)i(and)g(the)g (exp)s(onen)m(tial)f(deca)m(y)j(of)d(eigen-)257 1656 y(functions)c(do)s(es)g(not)g Fi(dir)-5 b(e)g(ctly)23 b Fp(follo)m(w)e(from)g(a)i(standard)g(Com)m(b)s(es-Thomas)f(argumen)m (t.)257 1777 y(W)-8 b(e)26 b(succeed,)i(ho)m(w)m(ev)m(er,)h(in)24 b(adapting)g(a)g(Com)m(b)s(es-Thomas)h(argumen)m(t)f(to)h(pro)m(v)m(e)h (The-)257 1897 y(orem)32 b(I.1,)h(ab)s(o)m(v)m(e.)404 2017 y(F)-8 b(or)32 b(the)h(Klein-Gordon)d(op)s(erator)i(with)g (external)h(p)s(oten)m(tial)e(of)i(the)g(form)e Fn(K)k Fp(:=)257 2056 y Fl(p)p 340 2056 404 4 v 82 x(\000)p Fp(\001)23 b(+)f Fn(m)704 2109 y Fm(2)760 2138 y Fl(\000)16 b Fn(V)22 b Fp(\()p Fn(x)p Fp(\),)30 b(exp)s(onen)m(tial)f(deca)m(y)i (of)e(eigenfunctions)h(w)m(as)g(pro)m(v)m(ed)h(in)e([5,)g(6],)257 2258 y(using)f(path)h(in)m(tegral)e(tec)m(hniques.)44 b(Later,)29 b(in)f(a)g(study)h([10])f(of)g(the)h(tunnel)g(e\013ect)g (in)e(a)257 2379 y(t)m(w)m(o-w)m(ell)h(p)s(oten)m(tial)e(in)i(the)g (semiclassical)e(limit,)g(a)i(Com)m(b)s(es-Thomas)g(argumen)m(t,)h(or) 257 2499 y(Agmon)39 b(estimate,)h(w)m(as)g(applied)e(to)g(pro)m(v)m(e)j (exp)s(onen)m(tial)d(deca)m(y)j(of)d(eigenfunctions)257 2619 y(of)28 b Fn(K)7 b Fp(.)42 b(The)30 b(op)s(erator)d Fn(B)1182 2634 y Fj(V)1243 2619 y Fp(,)i(whic)m(h)g(w)m(e)g(study)h(in) d(the)i(presen)m(t)h(pap)s(er,)f(can)g(b)s(e)f(written)257 2740 y(as)35 b(\003)447 2755 y Fm(+)522 2740 y Fn(K)24 b Fp(\003)697 2755 y Fm(+)756 2740 y Fp(,)35 b(and,)g(in)e(fact,)i(the) g(main)d(problem)h(addressed)k(here)e(is)f(to)g(pro)m(v)m(e)h(that)257 2860 y(the)c(insertion)d(of)i(\003)998 2875 y Fm(+)1086 2860 y Fp(do)s(es)g(not)g(e\013ect)g(the)g(exp)s(onen)m(tial)g(deca)m (y)h(in)e(an)g(essen)m(tial)h(w)m(a)m(y)-8 b(.)404 2980 y(The)32 b(case)g(of)e(the)i(relativistic)d(h)m(ydrogen-lik)m(e)i (atom,)f(for)h(whic)m(h)g Fn(V)22 b Fp(\()p Fn(x)p Fp(\))28 b(=)f Fn(\013)q(Z)r(=)p Fl(j)p Fn(x)p Fl(j)257 3101 y Fp(is)36 b(a)h(Coulom)m(b)e(p)s(oten)m(tial,)h(has)h(receiv)m(ed)h(sp)s (ecial)d(atten)m(tion.)55 b(In)37 b([9])g(the)f(quadratic)257 3221 y(form)24 b(asso)s(ciated)h(to)f Fn(B)1122 3236 y Fj(V)1208 3221 y Fp(is)g(sho)m(wn)i(to)f(b)s(e)g(b)s(ounded)g(from)f (b)s(elo)m(w)g(on)h(smo)s(oth)f(spinors,)257 3342 y(pro)m(vided)31 b Fn(\013)q(Z)j Fl(\024)29 b Fn(\013)q(Z)1056 3357 y Fj(c)1117 3342 y Fp(:=)f(2)p Fn(=)p Fp(\()p Fn(\031)t(=)p Fp(2)16 b(+)i(2)p Fn(=\031)t Fp(\))29 b(and)i(un)m(b)s(ounded)g(from)e (b)s(elo)m(w)i(otherwise.)257 3462 y(The)24 b(corresp)s(onding)e(F)-8 b(riedric)m(hs)22 b(extension)h(in)e(case)j Fn(Z)34 b Fl(\024)28 b Fn(Z)2497 3477 y Fj(c)2554 3462 y Fp(is)22 b(denoted)h(b)m(y)h(the)e(same)257 3582 y(sym)m(b)s(ol,)31 b(and)g(it)f(is)g(sho)m(wn)i(in)e([9])h(and)g(later)e(re\014ned)j(in)e ([16])h(that)f Fn(\033)2827 3597 y Fm(ess)2947 3582 y Fp(=)d([1)p Fn(;)17 b Fl(1)p Fp(\))30 b(and)257 3703 y Fn(\033)312 3718 y Fm(sc)411 3703 y Fp(=)35 b Fl(;)p Fp(.)57 b(Due)37 b(to)f(a)h(result)g(in)f([17],)i Fn(B)1759 3718 y Fj(V)1855 3703 y Fl(\025)e Fp(1)25 b Fl(\000)g Fn(\013)q(Z)44 b Fp(is)36 b(ev)m(en)j(strictly)d(p)s(ositiv)m(e,)i(for) 257 3823 y Fn(Z)d Fl(\024)28 b Fn(Z)531 3838 y Fj(c)566 3823 y Fp(.)43 b(F)-8 b(urthermore,)32 b(for)g Fn(Z)j(<)27 b(Z)1649 3838 y Fj(c)1684 3823 y Fp(,)33 b(there)g(is)f(an)g (in\014nite)g(n)m(um)m(b)s(er)h(of)f(eigen)m(v)-5 b(alues)257 3944 y(of)38 b Fn(B)448 3959 y Fj(V)547 3944 y Fp(b)s(elo)m(w)g(1,)h (accum)m(ulating)d(only)i(at)f(1)h([17].)59 b(These)40 b(results)f(co)m(v)m(er)g(all)d(kno)m(wn)257 4064 y(elemen)m(ts)d(b)s (ecause)h Fn(Z)1084 4079 y Fj(c)1146 4064 y Fl(\031)28 b Fp(124.)404 4184 y(In)41 b(order)f(to)h(apply)f(Theorem)h(I.1)g(to)f (Coulom)m(b)f(p)s(oten)m(tials,)j Fn(V)21 b Fp(\()p Fn(x)p Fp(\))42 b(=)f Fn(\013)q(Z)r(=)p Fl(j)p Fn(x)p Fl(j)p Fp(,)257 4305 y(w)m(e)d(ha)m(v)m(e)f(to)f(require)g(that)g Fn(Z)41 b Fl(\024)34 b Fp(68,)j(b)s(ecause)g(otherwise)g Fn(\013)q(Z)r(=)p Fl(j)p Fn(x)p Fl(j)e Fp(is)g(relativ)m(ely)g Fn(D)3423 4320 y Fm(0)3463 4305 y Fp(-)257 4425 y(b)s(ounded)j(with)e (relativ)m(e)f(b)s(ound)i Fl(\025)e Fp(1,)i(due)g(to)f(the)h(fact)f (that)g(Kato's)g(inequalit)m(y)g(is)257 4545 y(sharp)d([12].)404 4666 y(Our)i(in)m(terest)g(in)g(the)g(exp)s(onen)m(tial)g(deca)m(y)h (of)f(eigenfunctions)g(of)g Fn(B)2987 4681 y Fj(V)3083 4666 y Fp(originates)257 4786 y(from)d(the)i(question)g(of)e(existence) j(of)e(a)g(ground)g(state)h(of)f(a)g(pseudorelativistic)f(elec-)257 4907 y(tron)45 b(mo)s(delled)e(b)m(y)j Fn(B)1130 4922 y Fj(V)1236 4907 y Fp(and)f(coupled)h(to)e(the)i(quan)m(tized)g (electromagnetic)d(\014eld,)257 5027 y(similar)30 b(to)i(a)g(result)h (for)f(nonrelativistic)e(quan)m(tum)j(electro)s(dynamics)f(in)g([1,)g (2].)44 b(W)-8 b(e)257 5147 y(ha)m(v)m(e)28 b(th)m(us)g(arranged)e(the) h(material)c(in)j(this)g(pap)s(er)h(in)e(suc)m(h)j(a)f(w)m(a)m(y)g (that)f(Theorem)h(I.1)257 5268 y(is)33 b(a)g(consequence)j(of)d (Corollary)e(I)s(I)s(I.2)i(and)h(Lemma)d(I)s(I)s(I.1.)45 b(In)34 b(a)f(forthcoming)e(pub-)257 5388 y(lication)j(on)i (pseudorelativistic)f(quan)m(tum)h(electro)s(dynamics)f([13],)i (Corollary)d(I)s(I)s(I.2)p eop %%Page: 4 5 4 4 bop 257 266 a Fq(BM-1,)33 b(Septem)m(b)s(er)g(22,)f(2000)2054 b Fp(4)257 573 y(and)33 b(Lemma)e(I)s(I)s(I.1)i(will)d(serv)m(e)35 b(as)e(basic)f(input)h(for)f(the)h(pro)s(of)f(of)g(the)h(existence)h (of)e(a)257 693 y(ground)h(state.)257 1025 y Fo(I)t(I)162 b(Exp)t(onen)l(tial)55 b(deca)l(y)257 1244 y Fi(Pr)-5 b(o)g(of)33 b(of)g(The)-5 b(or)g(em)32 b(I.1:)41 b Fp(Our)31 b(pro)s(of)f(pro)s(ceeds)h(along)f(the)h(lines)e(of)i(Theorem)f(I)s (I.1)h(in)257 1364 y([1].)53 b(W)-8 b(e)36 b(sho)m(w)g(that)g(the)g (assertion)f(holds)g(on)h Fl(H)g Fp(and)g(hence)h(also)d(on)i Fl(H)3056 1379 y Fm(+)3148 1364 y Fp(=)c(\003)3324 1379 y Fm(+)3383 1364 y Fl(H)q Fp(.)257 1484 y(Considering)45 b Fn(B)881 1499 y Fj(V)988 1484 y Fp(as)h(an)g(op)s(erator)e(on)i Fl(H)q Fp(,)j(zero)d(is)f(an)h(additional)d(eigen)m(v)-5 b(alue)45 b(of)257 1605 y(in\014nite)35 b(m)m(ultiplicit)m(y)-8 b(.)48 b(Our)36 b(requiremen)m(t)g Fn(b)d(>)f Fp(0)j(insures)h(that)g (this)f(sp)s(ectral)g(p)s(oin)m(t)257 1725 y(is)h(a)m(v)m(oided)g(and)h (that)f(dim)15 b(Ran)h Fn(\037)1557 1740 y Fm(\001)1620 1725 y Fp([)p Fn(B)1721 1740 y Fj(V)1782 1725 y Fp(])34 b Fn(<)g Fl(1)p Fp(,)i(where)i(\001)33 b(:=)h(\()p Fn(b)17 b(;)41 b Fp(1)25 b Fl(\000)g Fn(a)p Fp(\))p Fn(:)36 b Fp(W)-8 b(e)36 b(ma)m(y)257 1846 y(assume)24 b(that)f Fn(\033)t Fp([)p Fn(B)947 1861 y Fj(V)1008 1846 y Fp(])s Fl(\\)s Fp(\001)29 b Fl(6)p Fp(=)e Fl(;)p Fp(.)41 b(Let)23 b Fn(b)1644 1809 y Ff(0)q(0)1715 1846 y Fp(:=)k(min)o Fl(f)p Fn(\033)t Fp([)p Fn(B)2218 1861 y Fj(V)2279 1846 y Fp(])s Fl(\\)s Fp(\001)p Fl(g)p Fp(,)f(1)s Fl(\000)s Fn(a)2745 1809 y Ff(0)q(0)2815 1846 y Fp(:=)i(max)p Fl(f)p Fn(\033)t Fp([)p Fn(B)3338 1861 y Fj(V)3399 1846 y Fp(])s Fl(\\)257 1966 y Fp(\001)p Fl(g)p Fp(,)37 b(and)e(set)i(\001)881 1930 y Ff(00)956 1966 y Fp(:=)c([)p Fn(b)1160 1930 y Ff(00)1219 1966 y Fn(;)41 b Fp(1)24 b Fl(\000)h Fn(a)1513 1930 y Ff(00)1555 1966 y Fp(])33 b Fl(\032)g Fp(\001.)53 b(W)-8 b(e)36 b(pic)m(k)g Fn(b)d(<)f(b)2489 1930 y Ff(0)2546 1966 y Fn(<)g(b)2695 1930 y Ff(0)q(0)2774 1966 y Fp(and)j Fn(a)e(<)g(a)3210 1930 y Ff(0)3266 1966 y Fn(<)f(a)3425 1930 y Ff(0)q(0)3468 1966 y Fp(,)257 2086 y(and)h(w)m(e)h(set)f(\001) 824 2050 y Ff(0)875 2086 y Fp(:=)28 b(\()p Fn(b)1085 2050 y Ff(0)1125 2086 y Fn(;)39 b Fp(1)22 b Fl(\000)h Fn(a)1413 2050 y Ff(0)1436 2086 y Fp(\),)33 b(so)f(w)m(e)i(ha)m(v)m(e) 1348 2297 y(\001)1429 2256 y Ff(00)1532 2297 y Fl(\032)60 b Fp(\001)1750 2256 y Ff(0)1834 2297 y Fl(\032)p 1972 2217 105 4 v 61 w Fp(\001)2053 2269 y Ff(0)2137 2297 y Fl(\032)g Fp(\001)23 b Fn(:)865 b Fp(\(I)s(I.1\))257 2508 y(W)-8 b(e)52 b(ma)m(y)e(replace)h Fn(\037)1083 2523 y Fm(\001)1197 2508 y Fp(b)m(y)60 b Fk(e)-64 b Fn(\037)1411 2523 y Fm(\001)1470 2504 y Fb(0)1555 2508 y Fl(2)60 b Fn(C)1758 2472 y Ff(1)1751 2533 y Fm(0)1832 2508 y Fp(\(\001)1951 2472 y Ff(0)1975 2508 y Fp(;)17 b([0)p Fn(;)g Fp(1]\),)54 b(whic)m(h)d(is)g(c)m(hosen)h(suc)m(h)g(that)266 2629 y Fk(e)-64 b Fn(\037)318 2644 y Fm(\001)377 2625 y Fb(0)404 2629 y Fp([)p Fn(B)505 2644 y Fj(V)566 2629 y Fp(])28 b Fl(\021)g Fp(1)k(on)h(\001)1024 2593 y Ff(00)1066 2629 y Fp(,)g(implying)d(that)40 b Fk(e)-63 b Fn(\037)1802 2644 y Fm(\001)1861 2625 y Fb(0)1888 2629 y Fp([)p Fn(B)1989 2644 y Fj(V)2050 2629 y Fp(])27 b(=)h Fn(\037)2269 2644 y Fm(\001)2332 2629 y Fp([)p Fn(B)2433 2644 y Fj(V)2494 2629 y Fp(].)404 2749 y(De\014ne)49 b Fn(B)796 2764 y Fj(\037)840 2775 y Fc(R)890 2764 y Fj(V)1006 2749 y Fp(:=)54 b(\003)1231 2764 y Fm(+)1290 2749 y Fp(\()p Fn(D)1409 2764 y Fm(0)1481 2749 y Fl(\000)34 b Fn(\037)1653 2764 y Fj(R)1711 2749 y Fn(V)21 b Fp(\)\003)1895 2764 y Fm(+)1954 2749 y Fp(,)52 b(where)e Fn(\037)2392 2764 y Fj(R)2498 2749 y Fp(is)e(a)g(smo)s(oth)g(c)m(haracter-)257 2870 y(istic)i(function)h(of)g(the)g(set)h Fl(fj)p Fn(x)p Fl(j)59 b(\025)g Fn(R)q Fl(g)p Fp(,)d(e.g.,)g Fn(\037)2224 2885 y Fj(R)2282 2870 y Fp(\()p Fn(x)p Fp(\))k(:=)f Fn(\037)2696 2885 y Fm(1)2735 2870 y Fp(\()p Fn(x=R)q Fp(\),)d(1)35 b Fl(\000)g Fn(\037)3330 2885 y Fm(1)3429 2870 y Fl(2)257 2990 y Fn(C)334 2954 y Ff(1)327 3015 y Fm(0)409 2990 y Fp(\()p Fn(B)521 3005 y Fm(2)561 2990 y Fp(\(0\);)17 b([0)p Fn(;)g Fp(1]\),)47 b(and)e(supp)18 b Fn(\037)1519 3005 y Fm(1)1608 2990 y Fl(\022)49 b(fj)p Fn(x)p Fl(j)g(\025)h Fp(1)p Fl(g)p Fp(.)81 b(Then,)50 b(for)44 b(an)m(y)i Fn(")j(>)g Fp(0,)f(there)257 3110 y(is)32 b(an)h Fn(R)c(<)e Fl(1)32 b Fp(suc)m(h)i(that)f Fl(k)p Fn(\037)1372 3125 y Fj(R)1429 3110 y Fn(V)22 b Fl(k)1558 3125 y Ff(1)1660 3110 y Fn(<)28 b(")p Fp(,)k(and)h(w)m(e)g(ha)m(v)m(e)417 3321 y Fl(h)p Fn( )t Fl(j)p Fn(B)625 3336 y Fj(\037)669 3347 y Fc(R)719 3336 y Fj(V)780 3321 y Fn( )t Fl(i)59 b Fp(=)h Fl(h)p Fn( )t Fl(j)28 b(j)p Fn(D)1352 3336 y Fm(0)1391 3321 y Fl(j)p Fn( )t 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b(=)f(0,)33 b(or)1137 4103 y Fn(\037)1198 4118 y Fm(\001)1261 4103 y Fp([)p Fn(B)1362 4118 y Fj(V)1423 4103 y Fp(])60 b(=)69 b Fk(e)-64 b Fn(\037)1707 4118 y Fm(\001)1766 4099 y Fb(0)1793 4103 y Fp([)p Fn(B)1894 4118 y Fj(V)1955 4103 y Fp(])39 b Fl(\000)47 b Fk(e)-63 b Fn(\037)2198 4118 y Fm(\001)2257 4099 y Fb(0)2283 4103 y Fp([)p Fn(B)2384 4118 y Fj(\037)2428 4129 y Fc(R)2479 4118 y Fj(V)2540 4103 y Fp(])22 b Fn(:)654 b Fp(\(I)s(I.3\))257 4314 y(Since)45 b Fk(e)-63 b Fn(\037)577 4329 y Fm(\001)636 4310 y Fb(0)696 4314 y Fl(2)34 b Fn(C)873 4278 y Ff(1)866 4339 y Fm(0)947 4314 y Fp(\(\001)1066 4278 y Ff(0)1090 4314 y Fp(;)17 b([0)p Fn(;)g Fp(1]\),)36 b(it)f(has)h(an)g(almost)f(analytic)f(extension)j Fn(f)45 b Fl(2)34 b Fn(C)3246 4278 y Ff(1)3239 4339 y Fm(0)3320 4314 y Fp(\()p Fh(C)20 b Fp(\).)257 4435 y(That)38 b(is,)g(giv)m(en)f (an)g(arbitrary)g Fn(\016)i(>)c Fp(0)i(and)h Fn(N)46 b Fl(2)36 b Fh(N)9 b Fp(,)44 b(one)38 b(can)f(construct)h(\(see,)i (e.g.,)257 4555 y([11]\))33 b(a)f(smo)s(oth)g(function)g Fn(f)43 b Fp(on)32 b Fh(C)59 b Fp(suc)m(h)34 b(that)1066 4766 y Fl(U)10 b Fp(\(\001)1257 4725 y Ff(0)1281 4766 y Fp(\))83 b(:=)g(\001)1669 4725 y Ff(0)1714 4766 y Fp(+)22 b Fl(fj)p Fn(\020)8 b Fl(j)26 b(\024)i Fn(\016)t Fl(g)88 b(\033)g Fp(supp)q Fn(f)33 b(;)511 b Fp(\(I)s(I.4\))1169 4911 y Fn(f)11 b Fl(j)1256 4926 y Fm(\001)1415 4911 y Fp(=)105 b Fk(e)-63 b Fn(\037)1649 4926 y Fm(\001)1708 4907 y Fb(0)1756 4911 y Fn(;)1487 b Fp(\(I)s(I.5\))977 4972 y Fk(\014)977 5032 y(\014)1010 5057 y Fn(@)p 1061 5034 40 3 v 15 x Fj(z)1101 5057 y Fn(f)11 b Fp(\()p Fn(z)t Fp(\))1285 4972 y Fk(\014)1285 5032 y(\014)1415 5057 y Fp(=)97 b Fl(O)1670 4976 y Fk(\000)1715 5057 y Fl(j)p Fp(Im)16 b Fn(z)t Fl(j)1953 5016 y Fj(N)2021 4976 y Fk(\001)2089 5057 y Fn(:)1154 b Fp(\(I)s(I.6\))257 5268 y(F)-8 b(or)23 b(our)g(application)e(b)s(elo)m(w,)26 b(w)m(e)e(c)m(ho)s(ose)h Fn(N)38 b Fp(=)27 b(1)c(and)h Fn(\016)32 b Fp(:=)27 b(min)o Fl(f)p Fp(\()p Fn(a)2793 5232 y Ff(0)2820 5268 y Fl(\000)t Fn(a)p Fp(\))17 b Fn(;)j Fp(\()p Fn(b)3133 5232 y Ff(0)3160 5268 y Fl(\000)t Fn(b)p Fp(\))p Fl(g)p Fn(=)p Fp(2.)257 5388 y(An)34 b(almost)e(analytic)h(extension)h(can)g(b)s(e)g(used)h(to) f(de\014ne)g(a)g(functional)e(calculus)h(b)m(y)p eop %%Page: 5 6 5 5 bop 257 266 a Fq(BM-1,)33 b(Septem)m(b)s(er)g(22,)f(2000)2054 b Fp(5)257 573 y(means)33 b(of)f(the)h(follo)m(wing)d(inden)m(tit)m(y) -8 b(,)989 837 y Fk(e)-63 b Fn(\037)1042 852 y Fm(\001)1101 833 y Fb(0)1127 837 y Fp(\()p Fn(x)p Fp(\))61 b(=)1502 769 y Fn(i)p 1465 814 108 4 v 1465 905 a Fp(2)p Fn(\031)1616 701 y Fk(Z)1671 927 y Ff(U)7 b Fm(\(\001\))1840 837 y Fp(\()p Fn(z)26 b Fl(\000)d Fn(x)p Fp(\))2142 796 y Ff(\000)p Fm(1)2263 769 y Fn(@)5 b(f)p 2263 814 116 4 v 2267 905 a(@)p 2323 850 50 4 v 5 w(z)2411 837 y(dz)27 b Fl(^)22 b Fn(d)p 2673 782 V(z)27 b(:)498 b Fp(\(I)s(I.7\))257 1141 y(Using)22 b(the)h(second)h(resolv)m(en)m(t)f(equation)f(and)g (the)h(abreviations)e Fn(d)-16 b(f)11 b Fp(\()p Fn(z)t Fp(\))28 b(:=)3062 1102 y Fj(i)p 3035 1118 79 4 v 3035 1175 a Fm(2)p Fj(\031)3133 1097 y(@)t(f)p 3133 1118 83 4 v 3136 1175 a(@)p 3177 1138 36 3 v 4 w(z)3226 1141 y Fn(dz)6 b Fl(^)q Fn(d)p 3446 1086 50 4 v(z)257 1262 y Fp(and)p 447 1207 119 4 v 33 w Fn(\037)508 1277 y Fj(R)594 1262 y Fp(:=)27 b(1)22 b Fl(\000)h Fn(\037)956 1277 y Fj(R)1014 1262 y Fp(,)32 b(w)m(e)i(can)f(write)633 1524 y Fk(e)-63 b Fn(\037)686 1539 y Fm(\001)749 1524 y Fp(\()p Fn(B)5 b Fp(\))83 b(=)1146 1389 y Fk(Z)1262 1524 y Fn(d)-16 b(f)11 b Fp(\()p Fn(z)t Fp(\))1498 1414 y Fk(h)1544 1524 y Fp(\()p Fn(B)1656 1539 y Fj(V)1739 1524 y Fl(\000)23 b Fn(z)t Fp(\))1926 1483 y Ff(\000)p Fm(1)2065 1524 y Fl(\000)45 b Fp(\()p Fn(B)2299 1539 y Fj(\037)2343 1550 y Fc(R)2393 1539 y Fj(V)2476 1524 y Fl(\000)23 b Fn(z)t Fp(\))2663 1483 y Ff(\000)p Fm(1)2758 1414 y Fk(i)3270 1524 y Fp(\(I)s(I.8\))987 1779 y(=)1146 1643 y Fk(Z)1262 1779 y Fn(d)-16 b(f)11 b Fp(\()p Fn(z)t Fp(\))17 b(\()p Fn(B)1610 1794 y Fj(\037)1654 1805 y Fc(R)1703 1794 y Fj(V)1786 1779 y Fl(\000)23 b Fn(z)t Fp(\))1973 1738 y Ff(\000)p Fm(1)2085 1779 y Fp(\003)2153 1794 y Fm(+)p 2228 1724 V 2228 1779 a Fn(\037)2289 1794 y Fj(R)2363 1779 y Fn(V)39 b Fp(\003)2527 1794 y Fm(+)2602 1779 y Fp(\()p Fn(B)2714 1794 y Fj(V)2797 1779 y Fl(\000)23 b Fn(z)t Fp(\))2984 1738 y Ff(\000)p Fm(1)3101 1779 y Fn(:)404 2042 y Fp(No)m(w,)i(w)m(e)g(pic)m(k)e(a)g(smo)s(oth,)h(con)m (v)m(ex)i(function)c Fn(\036)28 b Fp(:)g Fh(R)2301 2057 y Fm(+)2394 2042 y Fl(!)f Fh(R)2587 2057 y Fm(+)2652 2042 y Fp(,)f(satisfying)c Fn(\036)27 b Fl(\021)h Fp(0)23 b(on)257 2162 y(\(0)p Fn(;)17 b Fp(2])29 b(and)g Fn(\036)p Fp(\()p Fn(r)s Fp(\))e(=)g Fn(r)18 b Fl(\000)d Fp(3,)31 b(for)d Fn(r)i Fl(\025)f Fp(4,)g(so)h Fl(j)p Fn(\036)1884 2126 y Ff(0)1906 2162 y Fl(j)e(\024)g Fp(1.)42 b(De\014ning)28 b Fn(F)42 b Fp(:)27 b Fh(R)2795 2126 y Fm(3)2869 2162 y Fl(!)g Fh(R)3062 2177 y Fm(+)3127 2162 y Fp(,)j Fn(F)14 b Fp(\()p Fn(x)p Fp(\))27 b(=)257 2283 y Fn(aR)18 b(\036)p Fp(\()p Fl(j)p Fn(x)p Fl(j)p Fn(=R)q Fp(\),)34 b(w)m(e)g(infer)f(that)h Fn(F)43 b Fl(\021)31 b Fp(0)i(on)h Fl(fj)p Fn(x)p Fl(j)29 b(\024)h Fp(2)p Fn(R)q Fl(g)j Fp(and)h Fn(F)14 b Fp(\()p Fn(x)p Fp(\))30 b(=)g Fn(a)p Fl(j)p Fn(x)p Fl(j)23 b(\000)g Fp(3)p Fn(aR)35 b Fp(for)257 2403 y Fl(j)p Fn(x)p Fl(j)28 b(\025)g Fp(4)p Fn(R)q Fp(.)43 b(F)-8 b(urthermore,)32 b Fl(jr)p Fn(F)14 b Fp(\()p Fn(x)p Fp(\))p Fl(j)27 b(\024)i Fn(a)p Fp(,)j(for)g(all)f Fn(x)d Fl(2)g Fh(R)2405 2367 y Fm(3)2450 2403 y Fp(.)404 2523 y(Note)k(that,)h(since)g Fn(a)p Fl(j)p Fn(x)p Fl(j)22 b(\000)h Fp(const)28 b Fl(\024)g Fn(F)14 b Fp(\()p Fn(x)p Fp(\))28 b Fl(\024)g Fn(a)p Fl(j)p Fn(x)p Fl(j)p Fp(,)33 b(Eq.)g(\(I.13\))f(is)g(equiv)-5 b(alen)m(t)33 b(to)1420 2659 y Fk(\015)1420 2719 y(\015)1476 2743 y Fn(e)1521 2702 y Fj(F)1604 2743 y Fk(e)-63 b Fn(\037)1657 2758 y Fm(\001)1716 2739 y Fb(0)1743 2743 y Fp([)p Fn(B)1844 2758 y Fj(V)1905 2743 y Fp(])1932 2659 y Fk(\015)1932 2719 y(\015)2047 2743 y Fn(<)60 b Fl(1)22 b Fn(:)938 b Fp(\(I)s(I.9\))404 2963 y(W)-8 b(e)33 b(de\014ne)807 3183 y Fn(B)881 3198 y Fj(R;F)1073 3183 y Fp(:=)60 b Fn(e)1281 3142 y Fj(F)1357 3183 y Fn(B)1431 3198 y Fj(\037)1475 3209 y Fc(R)1525 3198 y Fj(V)1603 3183 y Fn(e)1648 3142 y Ff(\000)p Fj(F)1856 3183 y Fp(and)94 b(\003)2175 3198 y Fj(F)2294 3183 y Fp(:=)60 b Fn(e)2502 3142 y Fj(F)2578 3183 y Fp(\003)2646 3198 y Fm(+)2721 3183 y Fn(e)2766 3142 y Ff(\000)p Fj(F)2902 3183 y Fn(;)293 b Fp(\(I)s(I.10\))257 3403 y(so)37 b(that)g Fn(e)642 3367 y Fj(F)701 3403 y Fp(\()p Fn(B)813 3418 y Fj(\037)857 3429 y Fc(R)907 3418 y Fj(V)993 3403 y Fl(\000)26 b Fn(z)t Fp(\))1183 3367 y Ff(\000)p Fm(1)1278 3403 y Fn(e)1323 3367 y Ff(\000)p Fj(F)1471 3403 y Fp(=)35 b(\()p Fn(B)1694 3418 y Fj(R;F)1851 3403 y Fl(\000)26 b Fn(z)t Fp(\))2041 3367 y Ff(\000)p Fm(1)2136 3403 y Fp(.)56 b(F)-8 b(urthermore,)38 b(w)m(e)g(observ)m(e)g (that)257 3524 y Fn(e)302 3488 y Fj(F)p 361 3469 V 361 3524 a Fn(\037)422 3539 y Fj(R)508 3524 y Fp(=)p 611 3469 V 27 w Fn(\037)672 3539 y Fj(R)730 3524 y Fp(,)33 b(and)f(hence)i(w)m(e)g(ha)m(v)m(e)257 3659 y Fk(\015)257 3719 y(\015)313 3744 y Fn(e)358 3703 y Fj(F)433 3744 y Fn(\037)494 3759 y Fm(\001)557 3744 y Fp([)p Fn(B)658 3759 y Fj(V)719 3744 y Fp(])746 3659 y Fk(\015)746 3719 y(\015)886 3744 y Fp(=)1045 3659 y Fk(\015)1045 3719 y(\015)1101 3744 y Fn(e)1146 3703 y Fj(F)1229 3744 y Fk(e)-63 b Fn(\037)1282 3759 y Fm(\001)1341 3740 y Fb(0)1368 3744 y Fp([)p Fn(B)1469 3759 y Fj(V)1530 3744 y Fp(])p Fn(\037)1618 3759 y Fm(\001)1681 3744 y Fp([)p Fn(B)1782 3759 y Fj(V)1843 3744 y Fp(])1870 3659 y Fk(\015)1870 3719 y(\015)3249 3744 y Fp(\(I)s(I.11\))886 3956 y(=)1045 3812 y Fk(\015)1045 3872 y(\015)1045 3932 y(\015)1045 3991 y(\015)1101 3956 y Fn(e)1146 3915 y Fj(F)1221 3821 y Fk(Z)1337 3956 y Fn(d)-16 b(f)11 b Fp(\()p Fn(z)t Fp(\))17 b(\()p Fn(B)1685 3971 y Fj(\037)1729 3982 y Fc(R)1779 3971 y Fj(V)1862 3956 y Fl(\000)23 b Fn(z)t Fp(\))2049 3915 y Ff(\000)p Fm(1)2160 3956 y Fp(\003)2228 3971 y Fm(+)p 2304 3902 V 2304 3956 a Fn(\037)2365 3971 y Fj(R)2439 3956 y Fn(V)38 b Fp(\003)2602 3971 y Fm(+)2677 3956 y Fp(\()p Fn(B)2789 3971 y Fj(V)2873 3956 y Fl(\000)22 b Fn(z)t Fp(\))3059 3915 y Ff(\000)p Fm(1)3154 3956 y Fn(\037)3215 3971 y Fm(\001)3278 3956 y Fp([)p Fn(B)3379 3971 y Fj(V)3440 3956 y Fp(])3467 3812 y Fk(\015)3467 3872 y(\015)3467 3932 y(\015)3467 3991 y(\015)885 4220 y Fl(\024)1045 4084 y Fk(Z)1162 4220 y Fl(j)p Fn(d)-16 b(f)11 b Fp(\()p Fn(z)t Fp(\))p Fl(j)1458 4105 y Fk(\015)1458 4165 y(\015)1458 4225 y(\015)1513 4220 y Fp(\()p Fn(B)1625 4235 y Fj(R;F)1779 4220 y Fl(\000)23 b Fn(z)t Fp(\))1966 4179 y Ff(\000)p Fm(1)2077 4220 y Fp(\003)2145 4235 y Fj(F)p 2220 4165 V 2220 4220 a Fn(\037)2281 4235 y Fj(R)2356 4220 y Fn(V)38 b Fp(\003)2519 4235 y Fm(+)2594 4220 y Fn(\037)2655 4235 y Fm(\001)2718 4220 y Fp([)p Fn(B)2819 4235 y Fj(V)2880 4220 y Fp(])17 b(\()p Fn(B)27 b Fl(\000)c Fn(z)t Fp(\))3250 4179 y Ff(\000)p Fm(1)3345 4105 y Fk(\015)3345 4165 y(\015)3345 4225 y(\015)885 4474 y Fl(\024)1045 4339 y Fk(Z)1162 4474 y Fl(j)p Fn(d)-16 b(f)11 b Fp(\()p Fn(z)t Fp(\))p Fl(j)1458 4360 y Fk(\015)1458 4420 y(\015)1458 4479 y(\015)1513 4474 y Fp(\()p Fn(B)1625 4489 y Fj(R;F)1779 4474 y Fl(\000)23 b Fn(z)t Fp(\))1966 4433 y Ff(\000)p Fm(1)2077 4474 y Fp(\003)2145 4489 y Fj(F)2204 4360 y Fk(\015)2204 4420 y(\015)2204 4479 y(\015)2276 4360 y(\015)2276 4420 y(\015)2276 4479 y(\015)2331 4474 y Fn(V)38 b Fp(\()p Fn(D)2545 4489 y Fm(0)2607 4474 y Fp(+)22 b Fn(i)p Fp(\))2776 4433 y Ff(\000)p Fm(1)2870 4360 y Fk(\015)2870 4420 y(\015)2870 4479 y(\015)1518 4593 y(\015)1518 4653 y(\015)1518 4713 y(\015)1573 4708 y Fp(\()p Fn(D)1692 4723 y Fm(0)1754 4708 y Fp(+)g Fn(i)p Fp(\))17 b(\003)2008 4723 y Fm(+)2083 4708 y Fn(\037)2144 4723 y Fm(\001)2207 4708 y Fp([)p Fn(B)2308 4723 y Fj(V)2369 4708 y Fp(])2396 4593 y Fk(\015)2396 4653 y(\015)2396 4713 y(\015)2468 4593 y(\015)2468 4653 y(\015)2468 4713 y(\015)2523 4708 y Fp(\()p Fn(B)28 b Fl(\000)22 b Fn(z)t Fp(\))2849 4667 y Ff(\000)p Fm(1)2944 4593 y Fk(\015)2944 4653 y(\015)2944 4713 y(\015)3022 4708 y Fn(:)257 4960 y Fp(In)31 b(order)g(to)g(estimate)e(the)i(norm)f (of)g(\()p Fn(B)1776 4975 y Fj(R;F)1927 4960 y Fl(\000)18 b Fn(z)t Fp(\))2109 4924 y Ff(\000)p Fm(1)2204 4960 y Fp(\003)2272 4975 y Fj(F)2361 4960 y Fp(in)30 b(the)h(in)m(tegral)e(on) i(the)g(righ)m(t)257 5080 y(side)38 b(of)e(\(I)s(I.11\),)i(w)m(e)g (\014rst)g(remark)f(that)g(\003)1888 5095 y Fj(F)1981 5080 y Fp(=)f(\003)2161 5044 y Fm(2)2161 5105 y Fj(F)2256 5080 y Fp(is)h(a)f(b)s(ounded)i(pro)5 b(jection,)38 b(due)257 5200 y(to)c(Corollary)f(I)s(I)s(I.2.)48 b(Hence)36 b(its)e(image,)f (\003)1882 5215 y Fj(F)1940 5200 y Fl(H)q Fp(,)i(is)f(closed)h(and)f (left)f(in)m(v)-5 b(arian)m(t)33 b(under)p eop %%Page: 6 7 6 6 bop 257 266 a Fq(BM-1,)33 b(Septem)m(b)s(er)g(22,)f(2000)2054 b Fp(6)257 573 y(the)32 b(action)f(of)g(the)h(resolv)m(en)m(t)g(\()p Fn(B)1513 588 y Fj(R;F)1665 573 y Fl(\000)20 b Fn(z)t Fp(\))1849 537 y Ff(\000)p Fm(1)1944 573 y Fp(,)32 b(since)g Fn(B)2315 588 y Fj(R;F)2478 573 y Fp(comm)m(utes)g(with)f(\003)3225 588 y Fj(F)3283 573 y Fp(.)43 b(F)-8 b(or)257 693 y Fn(\036)28 b Fp(=)f(\003)514 708 y Fj(F)573 693 y Fn(\036)g Fl(2)h Fp(\003)820 708 y Fj(F)878 693 y Fl(H)q Fp(,)33 b(w)m(e)h(ha)m(v)m(e) 417 810 y Fk(\015)417 870 y(\015)472 895 y Fp(\003)540 910 y Fj(F)615 895 y Fp(\()p Fn(B)727 910 y Fj(R;F)881 895 y Fl(\000)23 b Fn(z)t Fp(\))17 b(\003)1153 910 y Fj(F)1228 895 y Fn(\036)1286 810 y Fk(\015)1286 870 y(\015)501 1050 y Fp(=)660 965 y Fk(\015)660 1025 y(\015)716 1050 y Fp(\003)784 1065 y Fj(F)859 1050 y Fn(e)904 1008 y Fj(F)979 1050 y Fp(\003)1047 1065 y Fm(+)1122 1050 y Fp(\()p Fn(D)1241 1065 y Fm(0)1303 1050 y Fl(\000)23 b Fn(\037)1464 1065 y Fj(R)1521 1050 y Fn(V)f Fp(\))17 b(\003)1723 1065 y Fm(+)1798 1050 y Fn(e)1843 1008 y Ff(\000)p Fj(F)1973 1050 y Fp(\003)2041 1065 y Fj(F)2116 1050 y Fn(\036)44 b Fl(\000)h Fn(z)21 b(\036)2464 965 y Fk(\015)2464 1025 y(\015)501 1204 y Fp(=)660 1120 y Fk(\015)660 1179 y(\015)716 1204 y Fn(e)761 1163 y Fj(F)836 1204 y Fn(D)917 1219 y Fm(0)973 1204 y Fp(\003)1041 1219 y Fm(+)1117 1204 y Fn(e)1162 1163 y Ff(\000)p Fj(F)1292 1204 y Fn(\036)49 b Fl(\000)i Fp(\003)1595 1219 y Fj(F)1670 1204 y Fn(e)1715 1163 y Fj(F)1790 1204 y Fp(\003)1858 1219 y Fm(+)1933 1204 y Fn(e)1978 1163 y Ff(\000)p Fj(F)2109 1204 y Fn(\037)2170 1219 y Fj(R)2244 1204 y Fn(V)38 b(e)2384 1163 y Fj(F)2460 1204 y Fp(\003)2528 1219 y Fm(+)2603 1204 y Fn(e)2648 1163 y Ff(\000)p Fj(F)2778 1204 y Fp(\003)2846 1219 y Fj(F)2921 1204 y Fn(\036)50 b Fl(\000)g Fn(z)22 b(\036)3281 1120 y Fk(\015)3281 1179 y(\015)501 1368 y Fp(=)660 1284 y Fk(\015)660 1344 y(\015)716 1368 y Fp(\()p Fn(D)835 1383 y Fm(0)896 1368 y Fp(+)g Fn(i)r(~)-51 b(\013)23 b Fl(\001)1172 1343 y Fn(~)1162 1368 y Fl(r)p Fn(F)14 b Fp(\))j(\003)1445 1383 y Fj(F)1519 1368 y Fn(\036)50 b Fl(\000)g Fp(\003)1822 1383 y Fj(F)1897 1368 y Fn(\037)1958 1383 y Fj(R)2033 1368 y Fn(V)38 b(\036)49 b Fl(\000)i Fn(z)21 b(\036)2487 1284 y Fk(\015)2487 1344 y(\015)501 1533 y Fp(=)660 1448 y Fk(\015)660 1508 y(\015)716 1533 y Fn(D)797 1548 y Fm(0)853 1533 y Fn(\036)49 b Fp(+)h Fn(i)r(~)-51 b(\013)23 b Fl(\001)1263 1508 y Fn(~)1254 1533 y Fl(r)p Fn(F)30 b(\036)50 b Fl(\000)g Fp(\003)1733 1548 y Fj(F)1808 1533 y Fn(\037)1869 1548 y Fj(R)1943 1533 y Fn(V)39 b(\036)49 b Fl(\000)h Fn(z)22 b(\036)2398 1448 y Fk(\015)2398 1508 y(\015)500 1715 y Fl(\025)660 1605 y Fk(\020)720 1715 y Fp(1)g Fl(\000)g Fn(a)h Fl(\000)f(k)p Fn(\037)1174 1730 y Fj(R)1232 1715 y Fn(V)g Fl(k)17 b(k)p Fp(\003)1496 1730 y Fj(F)1553 1715 y Fl(k)22 b(\000)43 b Fp(sup)1725 1801 y Ff(U)7 b Fm(\(\001)1862 1782 y Fb(0)1884 1801 y Fm(\))1928 1715 y Fl(j)p Fn(z)t Fl(j)2033 1605 y Fk(\021)2109 1715 y Fl(k)p Fn(\036)p Fl(k)22 b Fn(:)906 b Fp(\(I)s(I.12\))257 1992 y(W)-8 b(e)38 b(observ)m(e)g(that)f(up)s(on) g(c)m(ho)s(osing)f Fn(\016)j Fp(:=)c(\()p Fn(a)1957 1956 y Ff(0)2006 1992 y Fl(\000)25 b Fn(a)p Fp(\))p Fn(=)p Fp(3,)38 b(w)m(e)g(ha)m(v)m(e)g(sup)2884 2016 y Ff(U)7 b Fm(\(\001)3021 1997 y Fb(0)3043 2016 y Fm(\))3091 1992 y Fl(j)p Fn(z)t Fl(j)35 b(\024)h Fp(1)25 b Fl(\000)257 2113 y Fn(a)308 2076 y Ff(0)360 2113 y Fp(+)i Fn(\016)46 b Fp(=)41 b(1)28 b Fl(\000)g Fp(2)p Fn(a)951 2076 y Ff(0)974 2113 y Fn(=)p Fp(3)f Fl(\000)i Fn(a=)p Fp(3.)67 b(By)41 b(Corollary)f(I)s(I)s(I.2,)i Fl(k)p Fp(\003)2423 2128 y Fj(F)2481 2113 y Fl(k)g Fp(=)f(1)28 b(+)f Fn(O)s Fp(\()p Fn(a)p Fp(\))40 b(and)h(th)m(us)257 2233 y Fl(k)p Fn(\037)368 2248 y Fj(R)426 2233 y Fn(V)22 b Fl(k)17 b(k)p Fp(\003)690 2248 y Fj(F)747 2233 y Fl(k)53 b(\024)g Fp(\()p Fn(a)1069 2197 y Ff(0)1125 2233 y Fl(\000)33 b Fn(a)p Fp(\))p Fn(=)p Fp(3,)51 b(pro)m(vided)c Fn(R)54 b(<)f Fl(1)47 b Fp(is)g(c)m(hosen)i (su\016cien)m(tly)f(large.)257 2353 y(Hence)1070 2499 y Fk(\015)1070 2559 y(\015)1125 2583 y Fp(\003)1193 2598 y Fj(F)1268 2583 y Fp(\()p Fn(B)1380 2598 y Fj(R;F)1534 2583 y Fl(\000)23 b Fn(z)t Fp(\))17 b(\003)1806 2598 y Fj(F)1881 2583 y Fn(\036)1939 2499 y Fk(\015)1939 2559 y(\015)2054 2583 y Fl(\025)2202 2516 y Fn(a)2253 2480 y Ff(0)2299 2516 y Fl(\000)22 b Fn(a)p 2202 2560 248 4 v 2301 2652 a Fp(3)2476 2583 y Fl(k)p Fn(\036)p Fl(k)g Fn(:)539 b Fp(\(I)s(I.13\))257 2816 y(Using)33 b(again)e Fl(k)p Fp(\003)910 2831 y Fj(F)968 2816 y Fl(k)c Fp(=)h(1)22 b(+)g Fn(O)s Fp(\()p Fn(a)p Fp(\),)32 b(w)m(e)h(th)m(us)h(obtain)d (that)1011 3073 y Fl(k)p Fp(\()p Fn(B)1173 3088 y Fj(R;F)1326 3073 y Fl(\000)23 b Fn(z)t Fp(\))1513 3032 y Ff(\000)p Fm(1)1625 3073 y Fp(\003)1693 3088 y Fj(F)1751 3073 y Fl(k)60 b(\024)2009 3005 y Fp(3)22 b(+)g Fl(O)s Fp(\()p Fn(a)p Fp(\))p 2009 3050 379 4 v 2074 3141 a Fn(a)2125 3112 y Ff(0)2171 3141 y Fl(\000)g Fn(a)2457 3073 y(<)60 b Fl(1)22 b Fn(:)480 b Fp(\(I)s(I.14\))404 3314 y(Next,)33 b Fn(V)54 b Fp(is)32 b(relativ)m(ely)g Fn(D)1385 3329 y Fm(0)1424 3314 y Fp(-b)s(ounded,)h(b)m(y)h(\(I.8\),)e(hence)1370 3431 y Fk(\015)1370 3491 y(\015)1426 3516 y Fn(V)38 b Fp(\()p Fn(D)1640 3531 y Fm(0)1701 3516 y Fp(+)23 b Fn(i)p Fp(\))1871 3475 y Ff(\000)p Fm(1)1965 3431 y Fk(\015)1965 3491 y(\015)2081 3516 y Fn(<)60 b Fl(1)21 b Fn(;)857 b Fp(\(I)s(I.15\))257 3717 y(and)31 b(furthermore)g Fn(D)1064 3732 y Fm(0)1120 3717 y Fp(\003)1188 3732 y Fm(+)1278 3717 y Fp(is)f(relativ)m(ely)g Fn(B)1872 3732 y Fj(V)1964 3717 y Fp(b)s(ounded.)44 b(T)-8 b(ogether)31 b(with)g(the)g(b)s(ound-) 257 3838 y(edness)k(of)d(\001)c Fl(\032)g Fp([0)p Fn(;)17 b Fp(1],)32 b(this)g(implies)527 3952 y Fk(\015)527 4012 y(\015)527 4072 y(\015)582 4067 y Fp(\()p Fn(D)701 4082 y Fm(0)763 4067 y Fp(+)22 b Fn(i)p Fp(\))17 b(\003)1017 4082 y Fm(+)1092 4067 y Fn(\037)1153 4082 y Fm(\001)1216 4067 y Fp([)p Fn(B)1317 4082 y Fj(V)1378 4067 y Fp(])1405 3952 y Fk(\015)1405 4012 y(\015)1405 4072 y(\015)3222 4067 y Fp(\(I)s(I.16\))610 4279 y Fl(\024)771 4165 y Fk(\015)771 4225 y(\015)771 4284 y(\015)826 4279 y Fp(\()p Fn(D)945 4294 y Fm(0)1006 4279 y Fp(+)22 b Fn(i)p Fp(\))17 b(\003)1260 4294 y Fm(+)1336 4279 y Fp(\()p Fn(B)1448 4294 y Fj(V)1531 4279 y Fp(+)22 b Fn(iC)7 b Fp(\))1777 4238 y Ff(\000)p Fm(1)1871 4165 y Fk(\015)1871 4225 y(\015)1871 4284 y(\015)1949 4279 y Fl(\001)1999 4165 y Fk(\015)1999 4225 y(\015)1999 4284 y(\015)2054 4279 y Fp(\()p Fn(B)2166 4294 y Fj(V)2249 4279 y Fp(+)22 b Fn(iC)7 b Fp(\))17 b Fn(\037)2573 4294 y Fm(\001)2636 4279 y Fp([)p Fn(B)2737 4294 y Fj(V)2798 4279 y Fp(])2825 4165 y Fk(\015)2825 4225 y(\015)2825 4284 y(\015)2941 4279 y Fn(<)60 b Fl(1)21 b Fn(;)257 4503 y Fp(for)32 b(0)c Fn(<)f(C)35 b(<)28 b Fl(1)k Fp(su\016cien)m(tly)h(large.)404 4623 y(Finally)-8 b(,)36 b(our)h(c)m(hoice)h Fn(N)46 b Fp(=)36 b(1)h(for)g(the)h(almost)e (analytic)g(con)m(tiun)m(uation)g(insures)257 4744 y(that)i Fl(j)p Fn(d)-16 b(f)11 b Fp(\()p Fn(z)t Fp(\))p Fl(j)37 b Fp(=)g Fl(O)s Fp(\(Im)16 b Fn(z)t Fp(\))39 b(comp)s(ensates)g(for)f (the)h(singularit)m(y)d(arising)h(from)g Fl(k)p Fp(\()p Fn(B)31 b Fl(\000)257 4864 y Fn(z)t Fp(\))344 4828 y Ff(\000)p Fm(1)439 4864 y Fl(k)d(\024)g(j)p Fp(Im)16 b Fn(z)t Fl(j)860 4828 y Ff(\000)p Fm(1)955 4864 y Fp(.)43 b(Hence)1074 5066 y Fl(j)p Fn(d)-16 b(f)11 b Fp(\()p Fn(z)t Fp(\))p Fl(j)1366 4981 y Fk(\015)1366 5041 y(\015)1420 5066 y Fp(\()p Fn(B)27 b Fl(\000)c Fn(z)t Fp(\))1746 5025 y Ff(\000)p Fm(1)1841 4981 y Fk(\015)1841 5041 y(\015)1957 5066 y Fl(\024)60 b(O)s Fp(\(1\))17 b Fn(dz)26 b Fl(^)d Fn(d)p 2580 5011 50 4 v(z)j(:)544 b Fp(\(I)s(I.17\))257 5268 y(The)46 b(desired)f(estimate)f(\(I)s(I.9\))h(no)m(w)g(follo)m(ws) e(from)h(inserting)f(\(I)s(I.14\){\(I)s(I.17\))h(in)m(to)257 5388 y(\(I)s(I.11\).)2863 b Fa(2)p eop %%Page: 7 8 7 7 bop 257 266 a Fq(BM-1,)33 b(Septem)m(b)s(er)g(22,)f(2000)2054 b Fp(7)257 573 y Fo(I)t(I)t(I)163 b(Estimates)53 b(on)g(Comm)l(utators) 257 792 y Fp(The)28 b(purp)s(ose)f(of)f(this)g(section)g(is)g(to)g (estimate)g(the)h(norms)f(of)g(v)-5 b(arious)25 b(op)s(erators)h(that) 257 912 y(in)m(v)m(olv)m(e)37 b(the)f(resolv)m(en)m(t)i(of)d Fl(j)p Fn(D)1397 927 y Fm(0)1436 912 y Fl(j)h Fp(and)g(an)h(exp)s(onen) m(tial)e(w)m(eigh)m(t)i(function)e Fn(e)3102 876 y Fj(F)3197 912 y Fp(\(acting)257 1033 y(as)e(\()p Fn(e)460 997 y Fj(F)519 1033 y Fn( )t Fp(\)\()p Fn(x)p Fp(\))28 b(:=)f Fn(e)958 997 y Fj(F)10 b Fm(\()p Fj(x)p Fm(\))1112 1033 y Fn( )t Fp(\()p Fn(x)p Fp(\)\),)785 1253 y Fn(K)868 1268 y Fm(1)990 1253 y Fp(:=)83 b Fn(e)1221 1212 y Fj(F)1297 1253 y Fl(j)p Fn(D)1406 1268 y Fm(0)1445 1253 y Fl(j)1473 1212 y Ff(\000)p Fm(1)1584 1253 y Fn(e)1629 1212 y Ff(\000)p Fj(F)1764 1253 y Fn(;)1441 b Fp(\(I)s(I)s(I.1\))785 1398 y Fn(K)868 1413 y Fm(2)990 1398 y Fp(:=)83 b Fn(e)1221 1357 y Fj(F)1297 1317 y Fk(\002)1355 1398 y Fl(j)p Fn(D)1464 1413 y Fm(0)1503 1398 y Fl(j)1531 1357 y Ff(\000)p Fm(1)1642 1398 y Fn(;)44 b(e)1758 1357 y Ff(\000)p Fj(F)1872 1317 y Fk(\003)2001 1398 y Fp(=)2165 1317 y Fk(\002)2206 1398 y Fn(e)2251 1357 y Fj(F)2327 1398 y Fn(;)g Fl(j)p Fn(D)2507 1413 y Fm(0)2546 1398 y Fl(j)2574 1357 y Ff(\000)p Fm(1)2685 1317 y Fk(\003)2743 1398 y Fn(e)2788 1357 y Ff(\000)p Fj(F)2924 1398 y Fn(;)281 b Fp(\(I)s(I)s(I.2\))785 1553 y Fn(K)868 1568 y Fm(3)990 1553 y Fp(:=)1176 1472 y Fk(\002)1235 1553 y Fl(j)p Fn(D)1344 1568 y Fm(0)1383 1553 y Fl(j)1411 1511 y Ff(\000)p Fm(1)1521 1553 y Fn(;)45 b(e)1638 1511 y Ff(\000)p Fj(F)1751 1472 y Fk(\003)1809 1553 y Fn(e)1854 1511 y Fj(F)2001 1553 y Fp(=)88 b Fn(e)2210 1511 y Ff(\000)p Fj(F)2340 1472 y Fk(\002)2382 1553 y Fn(e)2427 1511 y Fj(F)2502 1553 y Fn(;)45 b Fl(j)p Fn(D)2683 1568 y Fm(0)2722 1553 y Fl(j)2750 1511 y Ff(\000)p Fm(1)2860 1472 y Fk(\003)2924 1553 y Fn(;)281 b Fp(\(I)s(I)s(I.3\))1485 1872 y Fn(D)1566 1887 y Fm(0)1622 1872 y Fn(K)1705 1887 y Fm(2)1766 1872 y Fn(;)166 b(D)2040 1887 y Fm(0)2096 1872 y Fn(K)2179 1887 y Fm(3)2241 1872 y Fn(:)964 b Fp(\(I)s(I)s(I.4\))257 2092 y(Our)26 b(norm)f(estimates)g(b)s(elo)m(w)g(on)h(\(I)s(I)s (I.1\){\(I)s(I)s(I.4\))f(serv)m(e)i(as)f(a)f(main)f(input)h(for)g(pro)m (ving)257 2213 y(Theorem)33 b(I.1,)g(esp)s(ecially)e(the)i(b)s (oundedness)i(of)d(\003)2199 2228 y Fj(F)2258 2213 y Fp(.)257 2416 y Fg(Lemma)37 b(I)s(I)s(I.1)49 b Fi(Assume)g(that)g Fn(F)67 b Fl(2)55 b Fn(C)1841 2380 y Fm(1)1880 2416 y Fp(\()p Fh(R)1984 2380 y Fm(3)2029 2416 y Fp(;)17 b Fh(R)2139 2431 y Fm(+)2204 2416 y Fp(\))49 b Fi(is)f(such)h(that)h Fn(F)14 b Fp(\(0\))53 b(=)g(0)c Fi(and)257 2545 y Fn(a)36 b Fp(:=)f(sup)629 2569 y Fj(x)689 2545 y Fl(j)726 2520 y Fn(~)717 2545 y Fl(r)p Fn(F)14 b Fp(\()p Fn(x)p Fp(\))p Fl(j)35 b Fn(<)g Fp(1)p Fi(.)57 b(Then)38 b(the)h(op)-5 b(er)g(ators)38 b Fn(K)2252 2560 y Fm(1)2292 2545 y Fi(,)i Fn(K)2445 2560 y Fm(2)2484 2545 y Fi(,)g Fn(K)2637 2560 y Fm(3)2676 2545 y Fi(,)g Fn(D)2827 2560 y Fm(0)2867 2545 y Fn(K)2950 2560 y Fm(2)2989 2545 y Fi(,)g(and)e Fn(D)3333 2560 y Fm(0)3373 2545 y Fn(K)3456 2560 y Fm(3)257 2666 y Fi(ar)-5 b(e)34 b(de\014ne)-5 b(d)32 b(on)h Fl(S)7 b Fp(\()p Fh(R)1059 2630 y Fm(3)1104 2666 y Fp(;)17 b Fh(C)1214 2630 y Fm(4)1260 2666 y Fp(\))p Fi(,)33 b(the)h(sp)-5 b(ac)g(e)32 b(of)i(spinor-value)-5 b(d)32 b(Schwartz)h(test)h (functions,)257 2786 y(and)41 b(extend)g(to)h(b)-5 b(ounde)g(d)41 b(op)-5 b(er)g(ators)41 b(on)h Fl(H)q Fi(,)h(denote)-5 b(d)41 b(by)g(the)h(same)f(symb)-5 b(ols,)42 b(with)257 2907 y(norms)34 b(b)-5 b(ounde)g(d)35 b(by)1574 3127 y Fl(k)p Fn(K)1707 3142 y Fm(1)1746 3127 y Fl(k)83 b Fp(=)g Fl(O)2120 3046 y Fk(\000)2166 3127 y Fp(\(1)21 b Fl(\000)i Fn(a)p Fp(\))2463 3085 y Ff(\000)p Fm(2)2558 3046 y Fk(\001)2625 3127 y Fn(;)580 b Fp(\(I)s(I)s(I.5\))1257 3272 y Fl(k)p Fn(K)1390 3287 y Fm(2)1429 3272 y Fl(k)17 b Fn(;)51 b Fl(k)p Fn(K)1707 3287 y Fm(3)1746 3272 y Fl(k)83 b Fp(=)g Fl(O)2120 3191 y Fk(\000)2166 3272 y Fn(a)17 b Fp(\(1)k Fl(\000)i Fn(a)p Fp(\))2531 3231 y Ff(\000)p Fm(2)2625 3191 y Fk(\001)2693 3272 y Fn(;)512 b Fp(\(I)s(I)s(I.6\))1016 3417 y Fl(k)p Fn(D)1147 3432 y Fm(0)1186 3417 y Fn(K)1269 3432 y Fm(2)1308 3417 y Fl(k)17 b Fn(;)51 b Fl(k)p Fn(D)1584 3432 y Fm(0)1624 3417 y Fn(K)1707 3432 y Fm(3)1746 3417 y Fl(k)83 b Fp(=)g Fl(O)2120 3336 y Fk(\000)2166 3417 y Fn(a)17 b Fp(\(1)k Fl(\000)i Fn(a)p Fp(\))2531 3376 y Ff(\000)p Fm(2)2625 3336 y Fk(\001)2693 3417 y Fn(:)512 b Fp(\(I)s(I)s(I.7\))257 3637 y Fi(Pr)-5 b(o)g(of:)42 b Fp(W)-8 b(e)31 b(\014rst)g(remark)f (that)h Fl(j)p Fn(D)1571 3652 y Fm(0)1610 3637 y Fl(j)1638 3601 y Ff(\000)p Fm(1)1732 3637 y Fn( )g Fp(=)d Fn(G)18 b Fl(\003)f Fn( )t Fp(,)31 b(where)h Fn(G)p Fl(\003)e Fp(denotes)i(con)m(v)m(olution)257 3758 y(with)619 4003 y Fn(G)p Fp(\()p Fn(x)p Fp(\))60 b(:=)1287 3936 y(1)p 1061 3980 501 4 v 1061 4072 a(4)p Fn(\031)1169 4043 y Fm(3)p Fj(=)p Fm(2)1278 4072 y Fp(\000\(1)p Fn(=)p Fp(2\))1588 3868 y Fk(Z)1688 3894 y Ff(1)1643 4093 y Fm(0)1779 4003 y Fp(exp)1945 3893 y Fk(h)2014 4003 y Fl(\000)22 b Fn(r)s(=)p Fp(2)g Fl(\000)g Fn(x)2434 3962 y Fm(2)2474 4003 y Fn(=)p Fp(2)p Fn(r)2619 3893 y Fk(i)2698 3936 y Fn(dr)p 2698 3980 98 4 v 2704 4072 a(r)2751 4043 y Fm(2)2827 4003 y Fn(;)378 b Fp(\(I)s(I)s(I.8\))257 4283 y(the)40 b(in)m(v)m(erse)h(F) -8 b(ourier)38 b(transform)g(of)h(\()p Fn(\030)1764 4247 y Fm(2)1829 4283 y Fp(+)27 b(1\))2019 4247 y Ff(\000)p Fm(1)2152 4283 y Fp([14)o(].)64 b(Hence,)42 b Fn(K)2802 4298 y Fm(1)2880 4283 y Fp(:)d Fl(S)7 b Fp(\()p Fh(R)3118 4247 y Fm(3)3164 4283 y Fp(;)17 b Fh(C)3273 4247 y Fm(4)3319 4283 y Fp(\))39 b Fl(!)257 4404 y(S)7 b Fp(\()p Fh(R)429 4368 y Fm(3)474 4404 y Fp(;)17 b Fh(C)584 4368 y Fm(4)630 4404 y Fp(\))32 b(can)h(b)s(e)g(written)f(as)h(an)f(in)m(tegral)f(op)s (erator)h(with)g(Sc)m(h)m(w)m(artz)j(k)m(ernel)338 4675 y Fn(K)421 4690 y Fm(1)461 4675 y Fp(\()p Fn(x;)17 b(y)t Fp(\))59 b(=)h Fn(\015)972 4539 y Fk(Z)1072 4565 y Ff(1)1027 4765 y Fm(0)1163 4675 y Fp(exp)1329 4564 y Fk(h)1398 4675 y Fl(\000)23 b Fn(r)s(=)p Fp(2)e Fl(\000)i Fp(\()p Fn(x)f Fl(\000)h Fn(y)t Fp(\))2069 4633 y Fm(2)2107 4675 y Fn(=)p Fp(2)p Fn(r)i Fp(+)d Fn(F)14 b Fp(\()p Fn(x)p Fp(\))22 b Fl(\000)g Fn(F)14 b Fp(\()p Fn(y)t Fp(\))2906 4564 y Fk(i)2979 4607 y Fn(dr)p 2979 4652 V 2985 4743 a(r)3032 4714 y Fm(2)3108 4675 y Fn(;)97 b Fp(\(I)s(I)s(I.9\))257 4970 y(where)34 b Fn(\015)595 4933 y Ff(\000)p Fm(1)717 4970 y Fp(:=)28 b(4)p Fn(\031)956 4933 y Fm(3)p Fj(=)p Fm(2)1066 4970 y Fp(\000\(1)p Fn(=)p Fp(2\).)42 b(Note)33 b(that)f(0)c Fl(\024)g Fn(K)2131 4985 y Fm(1)2170 4970 y Fp(\()p Fn(x;)17 b(y)t Fp(\))27 b Fl(\024)2552 4944 y Fk(e)2529 4970 y Fn(K)2612 4985 y Fm(1)2652 4970 y Fp(\()p Fn(x)22 b Fl(\000)h Fn(y)t Fp(\),)31 b(where)681 5215 y Fk(e)658 5240 y Fn(K)741 5255 y Fm(1)780 5240 y Fp(\()p Fn(x)p Fp(\))61 b(:=)f Fn(\015)1224 5105 y Fk(Z)1324 5131 y Ff(1)1279 5330 y Fm(0)1415 5240 y Fp(exp)1581 5130 y Fk(h)1650 5240 y Fl(\000)22 b Fn(r)s(=)p Fp(2)g Fl(\000)g Fn(x)2070 5199 y Fm(2)2110 5240 y Fn(=)p Fp(2)p Fn(r)j Fp(+)d Fn(a)p Fl(j)p Fn(x)p Fl(j)2537 5130 y Fk(i)2610 5173 y Fn(dr)p 2610 5217 V 2616 5309 a(r)2663 5280 y Fm(2)2740 5240 y Fn(;)417 b Fp(\(I)s(I)s(I.10\))p eop %%Page: 8 9 8 8 bop 257 266 a Fq(BM-1,)33 b(Septem)m(b)s(er)g(22,)f(2000)2054 b Fp(8)257 573 y(since)33 b Fl(j)533 548 y Fn(~)524 573 y Fl(r)p Fn(F)14 b Fl(j)27 b(\024)h Fn(a)p Fp(.)44 b(F)-8 b(urthermore,)32 b(observ)m(e)i(that)554 686 y Fk(Z)693 796 y(e)670 821 y Fn(K)753 836 y Fm(1)793 821 y Fp(\()p Fn(x)p Fp(\))17 b Fn(d)992 780 y Fm(3)1031 821 y Fn(x)83 b Fl(\024)h Fn(\015)1419 686 y Fk(Z)1519 712 y Ff(1)1474 911 y Fm(0)1620 754 y Fn(dr)p 1620 799 98 4 v 1626 890 a(r)1673 861 y Fm(2)1744 686 y Fk(Z)1860 821 y Fn(d)1911 780 y Fm(3)1950 821 y Fn(x)34 b Fp(exp)2205 681 y Fk(\024)2279 821 y Fl(\000)2389 754 y Fp(\(1)22 b Fl(\000)g Fn(a)p Fp(\))p Fn(r)p 2389 799 345 4 v 2537 890 a Fp(2)2743 711 y Fk(\020)2803 821 y Fp(1)g(+)2981 754 y Fn(x)3036 718 y Fm(2)p 2981 799 95 4 v 2985 890 a Fn(r)3032 861 y Fm(2)3086 711 y Fk(\021)3146 681 y(\025)1169 1034 y Fl(\024)84 b(O)1412 953 y Fk(\000)1458 1034 y Fp(\(1)21 b Fl(\000)i Fn(a)p Fp(\))1755 993 y Ff(\000)p Fm(2)1850 953 y Fk(\001)1917 1034 y Fn(;)1240 b Fp(\(I)s(I)s(I.11\))257 1223 y(whic)m(h)36 b(implies)d(that)h Fn(K)1169 1238 y Fm(1)1244 1223 y Fp(extends)j(to)e(an)g(op)s(erator)f Fl(2)e(B)s Fp(\()p Fl(H)q Fp(\),)37 b(with)d(norm)g(b)s(ounded)257 1344 y(b)m(y)574 1608 y Fl(k)p Fn(K)707 1623 y Fm(1)746 1608 y Fl(k)83 b(\024)1039 1468 y Fk(\022)1129 1608 y Fp(sup)1183 1688 y Fj(x)1293 1473 y Fk(Z)1409 1608 y Fn(K)1492 1623 y Fm(1)1531 1608 y Fp(\()p Fn(x;)17 b(y)t Fp(\))g Fn(d)1826 1567 y Fm(3)1864 1608 y Fn(y)1916 1468 y Fk(\023)1989 1490 y Fm(1)p Fj(=)p Fm(2)2115 1468 y Fk(\022)2205 1608 y Fp(sup)2260 1688 y Fj(y)2369 1473 y Fk(Z)2485 1608 y Fn(K)2568 1623 y Fm(1)2607 1608 y Fp(\()p Fn(x;)g(y)t Fp(\))g Fn(d)2902 1567 y Fm(3)2940 1608 y Fn(x)2995 1468 y Fk(\023)3069 1490 y Fm(1)p Fj(=)p Fm(2)879 1879 y Fl(\024)1039 1744 y Fk(Z)1178 1854 y(e)1155 1879 y Fn(K)1238 1894 y Fm(1)1278 1879 y Fp(\()p Fn(x)p Fp(\))g Fn(d)1477 1838 y Fm(3)1516 1879 y Fn(x)61 b Fp(=)f Fl(O)s Fp(\(1\))21 b Fn(:)1161 b Fp(\(I)s(I)s(I.12\))404 2121 y(Lik)m(ewise,)44 b(w)m(e)e(de\014ne)g Fn(K)1359 2136 y Fm(2)1440 2121 y Fp(and)f Fn(K)1721 2136 y Fm(3)1802 2121 y Fp(\014rst)g(as)h(in)m(tegral)d(op)s(erators)i Fl(S)7 b Fp(\()p Fh(R)3115 2085 y Fm(3)3160 2121 y Fp(;)17 b Fh(C)3270 2085 y Fm(4)3315 2121 y Fp(\))43 b Fl(!)257 2242 y(S)7 b Fp(\()p Fh(R)429 2206 y Fm(3)474 2242 y Fp(;)17 b Fh(C)584 2206 y Fm(4)630 2242 y Fp(\),)32 b(with)g(Sc)m(h)m (w)m(artz)j(k)m(ernel)624 2431 y Fn(K)707 2446 y Fm(2)747 2431 y Fp(\()p Fn(x;)17 b(y)t Fp(\))59 b(=)h Fl(\000)p Fn(K)1329 2446 y Fm(3)1369 2431 y Fp(\()p Fn(y)t(;)17 b(x)p Fp(\))59 b(:=)1426 b(\(I)s(I)s(I.13\))790 2627 y Fn(\015)880 2492 y Fk(Z)979 2518 y Ff(1)935 2717 y Fm(0)1071 2627 y Fp(exp)1236 2517 y Fk(h)1305 2627 y Fl(\000)23 b Fn(r)s(=)p Fp(2)e Fl(\000)i Fp(\()p Fn(x)f Fl(\000)h Fn(y)t Fp(\))1976 2586 y Fm(2)2015 2627 y Fn(=)p Fp(2)p Fn(r)2160 2517 y Fk(i)15 b(\020)2282 2627 y Fn(e)2327 2586 y Fj(F)10 b Fm(\()p Fj(x)p Fm(\))p Ff(\000)p Fj(F)g Fm(\()p Fj(y)r Fm(\))2721 2627 y Fl(\000)39 b Fp(1)2886 2517 y Fk(\021)2972 2560 y Fn(dr)p 2972 2604 98 4 v 2978 2696 a(r)3025 2667 y Fm(2)3101 2627 y Fn(:)257 2864 y Fp(Then,)34 b(using)e(that)482 2968 y Fk(\014)482 3028 y(\014)516 3053 y Fn(e)561 3012 y Fj(F)10 b Fm(\()p Fj(x)p Fm(\))p Ff(\000)p Fj(F)g Fm(\()p Fj(y)r Fm(\))938 3053 y Fl(\000)22 b Fp(1)1086 2968 y Fk(\014)1086 3028 y(\014)1202 3053 y Fl(\024)84 b Fn(e)1408 3012 y Ff(j)p Fj(F)10 b Fm(\()p Fj(x)p Fm(\))p Ff(\000)p Fj(F)g Fm(\()p Fj(y)r Fm(\))p Ff(j)1824 3053 y Fl(\000)23 b Fp(1)87 b Fl(\024)i(j)p Fn(F)14 b Fp(\()p Fn(x)p Fp(\))21 b Fl(\000)i Fn(F)14 b Fp(\()p Fn(y)t Fp(\))p Fl(j)j Fn(e)2878 3012 y Ff(j)p Fj(F)10 b Fm(\()p Fj(x)p Fm(\))p Ff(\000)p Fj(F)g Fm(\()p Fj(y)r Fm(\))p Ff(j)1202 3212 y Fl(\024)84 b Fn(a)p Fl(j)p Fn(x)22 b Fl(\000)h Fn(y)t Fl(j)17 b Fn(e)1761 3171 y Fj(a)p Ff(j)p Fj(x)p Ff(\000)p Fj(y)r Ff(j)1994 3212 y Fn(;)1163 b Fp(\(I)s(I)s(I.14\))257 3411 y(w)m(e)42 b(\014nd)f(that)g Fl(j)p Fn(K)943 3426 y Fm(2)982 3411 y Fp(\()p Fn(x;)17 b(y)t Fp(\))p Fl(j)p Fn(;)g Fl(j)p Fn(K)1392 3426 y Fm(3)1430 3411 y Fp(\()p Fn(y)t(;)g(x)p Fp(\))p Fl(j)40 b(\024)1867 3386 y Fk(e)1844 3411 y Fn(K)1927 3426 y Fm(2)1966 3411 y Fp(\()p Fn(x)29 b Fl(\000)f Fn(y)t Fp(\))40 b(:=)i Fn(a)p Fl(j)p Fn(x)28 b Fl(\000)g Fn(y)t Fl(j)2853 3386 y Fk(e)2832 3411 y Fn(K)2915 3426 y Fm(1)2953 3411 y Fp(\()p Fn(x)g Fl(\000)g Fn(y)t Fp(\),)42 b(and)257 3532 y(similarly)29 b(to)j(\(I)s(I)s(I.11\){\(I)s(I)s(I.12\),)g(this)g (yields)1001 3770 y Fl(k)p Fn(K)1134 3785 y Fm(2)1174 3770 y Fl(k)22 b Fn(;)72 b Fl(k)p Fn(K)1478 3785 y Fm(3)1517 3770 y Fl(k)60 b(\024)1764 3634 y Fk(Z)1904 3744 y(e)1881 3770 y Fn(K)1964 3785 y Fm(2)2003 3770 y Fp(\()p Fn(x)p Fp(\))17 b Fn(d)2202 3728 y Fm(3)2241 3770 y Fn(x)61 b Fp(=)f Fl(O)s Fp(\()p Fn(a)p Fp(\))22 b Fn(:)433 b Fp(\(I)s(I)s(I.15\))404 4012 y(Finally)-8 b(,)27 b(the)i(Sc)m(h)m(w)m (artz)i(k)m(ernel)e(of)g Fn(D)1799 4027 y Fm(0)1838 4012 y Fn(K)1921 4027 y Fm(2)1988 4012 y Fp(:)f Fl(S)7 b Fp(\()p Fh(R)2215 3976 y Fm(3)2260 4012 y Fp(;)17 b Fh(C)2370 3976 y Fm(4)2415 4012 y Fp(\))28 b Fl(!)f(S)7 b Fp(\()p Fh(R)2780 3976 y Fm(3)2825 4012 y Fp(;)17 b Fh(C)2935 3976 y Fm(4)2980 4012 y Fp(\))29 b(is)f(giv)m(en)h(b)m(y)658 4129 y Fk(\002)699 4210 y Fn(D)780 4225 y Fm(0)820 4210 y Fn(K)903 4225 y Fm(2)942 4129 y Fk(\003)984 4210 y Fp(\()p Fn(x;)17 b(y)t Fp(\))59 b(=)h Fl(\000)p Fn(i)r(~)-51 b(\013)24 b Fl(\001)1661 4185 y Fn(~)1652 4210 y Fl(r)1735 4225 y Fj(x)1795 4210 y Fn(K)1878 4225 y Fm(2)1918 4210 y Fp(\()p Fn(x;)17 b(y)t Fp(\))43 b(+)h Fn(\014)22 b(K)2468 4225 y Fm(2)2508 4210 y Fp(\()p Fn(x;)17 b(y)t Fp(\))k Fn(:)401 b Fp(\(I)s(I)s(I.16\))257 4400 y(F)-8 b(urthermore,)32 b(Leibnitz')g(rule)g(yields)257 4504 y Fk(\014)257 4564 y(\014)10 b Fn(~)291 4589 y Fl(r)374 4604 y Fj(x)434 4589 y Fn(K)517 4604 y Fm(2)557 4589 y Fp(\()p Fn(x;)17 b(y)t Fp(\))784 4504 y Fk(\014)784 4564 y(\014)3210 4589 y Fp(\(I)s(I)s(I.17\))341 4801 y(=)84 b Fn(\015)574 4657 y Fk(\014)574 4717 y(\014)574 4777 y(\014)574 4836 y(\014)623 4666 y(Z)723 4692 y Ff(1)679 4891 y Fm(0)824 4734 y Fn(dr)p 824 4779 V 830 4870 a(r)877 4841 y Fm(2)965 4801 y Fp(exp)1130 4691 y Fk(h)1200 4801 y Fl(\000)22 b Fn(r)s(=)p Fp(2)g Fl(\000)g Fp(\()p Fn(x)h Fl(\000)f Fn(y)t Fp(\))1870 4760 y Fm(2)1909 4801 y Fn(=)p Fp(2)p Fn(r)2054 4691 y Fk(i)1210 4963 y(n)1298 5074 y Fl(\000)1408 5006 y Fn(x)g Fl(\000)h Fn(y)p 1408 5051 229 4 v 1499 5142 a(r)1646 4963 y Fk(\020)1706 5074 y Fn(e)1751 5033 y Fj(F)10 b Fm(\()p Fj(x)p Fm(\))p Ff(\000)p Fj(F)g Fm(\()p Fj(y)r Fm(\))2144 5074 y Fl(\000)39 b Fp(1)2309 4963 y Fk(\021)2413 5074 y Fp(+)2543 5049 y Fn(~)2533 5074 y Fl(r)p Fn(F)14 b Fp(\()p Fn(x)p Fp(\))j Fn(e)2886 5033 y Fj(F)10 b Fm(\()p Fj(x)p Fm(\))p Ff(\000)p Fj(F)g Fm(\()p Fj(y)r Fm(\))3241 4963 y Fk(o)3307 4929 y(\014)3307 4989 y(\014)3307 5049 y(\014)3307 5109 y(\014)340 5348 y Fl(\024)84 b Fn(a)17 b(\015)658 5212 y Fk(Z)758 5239 y Ff(1)713 5438 y Fm(0)859 5281 y Fn(dr)p 859 5325 98 4 v 865 5416 a(r)912 5388 y Fm(2)999 5348 y Fp(exp)1165 5237 y Fk(h)1234 5348 y Fl(\000)23 b Fn(r)s(=)p Fp(2)e Fl(\000)i Fp(\()p Fn(x)f Fl(\000)h Fn(y)t Fp(\))1905 5307 y Fm(2)1943 5348 y Fn(=)p Fp(2)p Fn(r)i Fp(+)d Fn(a)p Fl(j)p Fn(x)g Fl(\000)h Fn(y)t Fl(j)2544 5237 y Fk(i)15 b(\020)2676 5281 y Fl(j)p Fn(x)22 b Fl(\000)h Fn(y)t Fl(j)2961 5244 y Fm(2)p 2676 5325 324 4 v 2814 5416 a Fn(r)3031 5348 y Fp(+)f Fl(j)p Fn(x)g Fl(\000)h Fn(y)t Fl(j)3414 5237 y Fk(\021)3494 5348 y Fn(;)p eop %%Page: 9 10 9 9 bop 257 266 a Fq(BM-1,)33 b(Septem)m(b)s(er)g(22,)f(2000)2054 b Fp(9)257 573 y(from)31 b(whic)m(h)h(w)m(e)h(conclude)f Fl(k)p Fn(D)1439 588 y Fm(0)1478 573 y Fn(K)1561 588 y Fm(2)1600 573 y Fl(k)c Fp(=)f Fl(O)s Fp(\()p Fn(a)p Fp(\))32 b(and,)g(analogously)-8 b(,)30 b Fl(k)p Fn(D)2916 588 y Fm(0)2956 573 y Fn(K)3039 588 y Fm(3)3078 573 y Fl(k)d Fp(=)h Fl(O)s Fp(\()p Fn(a)p Fp(\).)3421 693 y Fa(2)257 922 y Fg(Corollary)36 b(I)s(I)s(I.2)49 b Fp(\003)1091 937 y Fj(F)1149 922 y Fi(,)31 b Fp(1)11 b Fl(\000)g Fp(\003)1426 937 y Fj(F)1515 922 y Fi(ar)-5 b(e)30 b(b)-5 b(ounde)g(d)29 b(\(non-ortho)-5 b(gonal\))28 b(pr)-5 b(oje)g(ctions)29 b(on)h Fl(H)257 1042 y Fi(with)35 b Fl(k)p Fp(\003)587 1057 y Fj(F)645 1042 y Fl(k)p Fn(;)17 b Fl(k)p Fg(1)22 b Fl(\000)g Fp(\003)1034 1057 y Fj(F)1093 1042 y Fl(k)27 b Fp(=)h(1)22 b(+)g Fl(O)s Fp(\()p Fn(a)p Fp(\))p Fi(,)35 b(and)f Fl(k)p Fp(\003)2024 1057 y Fm(+)2105 1042 y Fl(\000)22 b Fp(\003)2272 1057 y Fj(F)2330 1042 y Fl(k)28 b Fp(=)g Fl(O)s Fp(\()p Fn(a)p Fp(\))p Fi(.)257 1270 y(Pr)-5 b(o)g(of:)41 b Fp(The)30 b(idemp)s(otency)f(of)f(\003)1498 1285 y Fj(F)1585 1270 y Fp(and)h Fg(1)14 b Fl(\000)g Fp(\003)2000 1285 y Fj(F)2089 1270 y Fp(is)28 b(trivial,)f(and)i(the)h(norm)e (estimates)257 1391 y(on)i(b)s(oth)f(\003)685 1406 y Fj(F)773 1391 y Fp(and)h Fg(1)16 b Fl(\000)g Fp(\003)1193 1406 y Fj(F)1281 1391 y Fp(follo)m(w)28 b(from)h Fl(k)p Fp(\003)1906 1406 y Fm(+)1980 1391 y Fl(\000)16 b Fp(\003)2141 1406 y Fj(F)2200 1391 y Fl(k)28 b Fp(=)f Fl(O)s Fp(\()p Fn(a)p Fp(\))j(whic)m(h)g(hence)h(remains)257 1511 y(to)i(b)s(e)f(pro)m (v)m(en.)404 1631 y(T)-8 b(o)32 b(this)h(end,)g(w)m(e)h(observ)m(e)g (that)e(on)h Fl(S)7 b Fp(\()p Fh(R)1958 1595 y Fm(3)2004 1631 y Fp(;)17 b Fh(C)2114 1595 y Fm(4)2159 1631 y Fp(\),)32 b(the)i(follo)m(wing)29 b(iden)m(tit)m(y)k(holds,)446 1851 y(\003)514 1866 y Fm(+)595 1851 y Fl(\000)23 b Fp(\003)763 1866 y Fj(F)904 1851 y Fp(=)83 b(\003)1131 1866 y Fm(+)1234 1851 y Fl(\000)45 b Fn(e)1401 1810 y Fj(F)1476 1851 y Fp(\003)1544 1866 y Fm(+)1620 1851 y Fn(e)1665 1810 y Ff(\000)p Fj(F)1866 1851 y Fp(=)88 b Fn(D)2111 1866 y Fm(0)2167 1851 y Fl(j)p Fn(D)2276 1866 y Fm(0)2315 1851 y Fl(j)2343 1810 y Ff(\000)p Fm(1)2481 1851 y Fl(\000)45 b Fn(e)2648 1810 y Fj(F)2724 1851 y Fn(D)2805 1866 y Fm(0)2861 1851 y Fl(j)p Fn(D)2970 1866 y Fm(0)3009 1851 y Fl(j)3037 1810 y Ff(\000)p Fm(1)3147 1851 y Fn(e)3192 1810 y Ff(\000)p Fj(F)904 1997 y Fp(=)83 b Fn(D)1144 2012 y Fm(0)1200 1997 y Fn(e)1245 1956 y Fj(F)1321 1997 y Fn(e)1366 1956 y Ff(\000)p Fj(F)1496 1997 y Fl(j)p Fn(D)1605 2012 y Fm(0)1644 1997 y Fl(j)1672 1956 y Ff(\000)p Fm(1)1816 1997 y Fl(\000)50 b Fn(e)1988 1956 y Fj(F)2064 1997 y Fn(D)2145 2012 y Fm(0)2201 1997 y Fl(j)p Fn(D)2310 2012 y Fm(0)2349 1997 y Fl(j)2377 1956 y Ff(\000)p Fm(1)2487 1997 y Fn(e)2532 1956 y Ff(\000)p Fj(F)904 2142 y Fp(=)83 b Fn(D)1144 2157 y Fm(0)1200 2142 y Fn(e)1245 2101 y Fj(F)1321 2061 y Fk(\002)1362 2142 y Fn(e)1407 2101 y Ff(\000)p Fj(F)1521 2142 y Fn(;)17 b Fl(j)p Fn(D)1674 2157 y Fm(0)1713 2142 y Fl(j)1741 2101 y Ff(\000)p Fm(1)1835 2061 y Fk(\003)1926 2142 y Fl(\000)2053 2061 y Fk(\002)2095 2142 y Fn(e)2140 2101 y Fj(F)2199 2142 y Fn(;)g(D)2324 2157 y Fm(0)2363 2061 y Fk(\003)2421 2142 y Fl(j)p Fn(D)2530 2157 y Fm(0)2569 2142 y Fl(j)2597 2101 y Ff(\000)p Fm(1)2708 2142 y Fn(e)2753 2101 y Ff(\000)p Fj(F)904 2306 y Fp(=)83 b Fl(\000)p Fn(D)1221 2321 y Fm(0)1278 2306 y Fn(K)1361 2321 y Fm(2)1450 2306 y Fl(\000)50 b Fn(i)19 b(~)-51 b(\013)23 b Fl(\001)1771 2281 y Fn(~)1762 2306 y Fl(r)p Fn(F)30 b(e)1983 2265 y Fj(F)2059 2306 y Fl(j)p Fn(D)2168 2321 y Fm(0)2207 2306 y Fl(j)2235 2265 y Ff(\000)p Fm(1)2345 2306 y Fn(e)2390 2265 y Ff(\000)p Fj(F)904 2461 y Fp(=)83 b Fl(\000)p Fn(D)1221 2476 y Fm(0)1278 2461 y Fn(K)1361 2476 y Fm(2)1450 2461 y Fl(\000)50 b Fn(i)19 b(~)-51 b(\013)23 b Fl(\001)1771 2435 y Fn(~)1762 2461 y Fl(r)p Fn(F)30 b(K)2021 2476 y Fm(1)2083 2461 y Fn(:)1074 b Fp(\(I)s(I)s(I.18\))257 2681 y(Hence,)34 b(due)g(to)e(Lemma)f(I)s(I)s (I.1,)708 2901 y Fl(k)p Fp(\003)826 2916 y Fm(+)906 2901 y Fl(\000)23 b Fp(\003)1074 2916 y Fj(F)1132 2901 y Fl(k)60 b(\024)h(k)p Fn(D)1511 2916 y Fm(0)1567 2901 y Fn(K)1650 2916 y Fm(2)1689 2901 y Fl(k)50 b Fp(+)g Fn(a)17 b Fl(k)p Fn(K)2116 2916 y Fm(1)2155 2901 y Fl(k)60 b Fp(=)g Fl(O)s Fp(\()p Fn(a)p Fp(\))22 b Fn(:)p Fa(2)451 b Fp(\(I)s(I)s(I.19\))257 3233 y Fo(References)306 3452 y Fp([1])49 b(V.)e(Bac)m(h,)k(J.)c(F)-8 b(r\177)-49 b(ohlic)m(h,)49 b(and)d(I.)h(M.)h(Sigal.)83 b(Quan)m(tum)47 b(electro)s(dynamics)f(of)458 3573 y(con\014ned)34 b(non-relativistic)c(particles.)42 b Fi(A)-5 b(dv.)35 b(in)f(Math.)i Fp(,)c(137:299{395,)f(1998.)306 3776 y([2])49 b(V.)37 b(Bac)m(h,)j(J.)d(F)-8 b(r\177)-49 b(ohlic)m(h,)37 b(and)h(I.)f(M.)h(Sigal.)56 b(Sp)s(ectral)36 b(analysis)h(for)g (systems)i(of)458 3897 y(atoms)44 b(and)g(molecules)f(coupled)i(to)f (the)h(quan)m(tized)g(radiation)e(\014eld.)78 b Fi(Com-)458 4017 y(mun.)34 b(Math.)i(Phys.)p Fp(,)c(207\(2\):249{290,)e(1999.)306 4220 y([3])49 b(H.)28 b(Bethe)i(and)e(E.)h(Salp)s(eter.)36 b(Quan)m(tum)28 b(mec)m(hanics)h(of)f(one-)g(and)h(t)m(w)m(o-electron) 458 4341 y(atoms.)g(In)c(S.)g(Fl)s(\177)-51 b(ugge,)23 b(editor,)j Fi(Handbuch)h(der)g(Physik,)h(XXXV)p Fp(,)f(pages)d (88{436.)458 4461 y(Springer,)32 b(Berlin,)g(1957.)306 4665 y([4])49 b(G.)32 b(E.)g(Bro)m(wn)h(and)f(D.)g(G.)g(Ra)m(v)m (enhall.)42 b(On)32 b(the)h(in)m(teraction)e(of)h(t)m(w)m(o)h (electrons.)458 4785 y Fi(Pr)-5 b(o)g(c.)69 b(R)-5 b(oy.)70 b(So)-5 b(c.)69 b(L)-5 b(ondon)34 b(A)p Fp(,)f(208:552{559,)e(1951.)306 4988 y([5])49 b(R.)43 b(Carmona.)76 b(P)m(ath)44 b(in)m(tegral)e(for)h (relativistic)e(Sc)m(hr\177)-49 b(odinger)44 b(op)s(erators.)76 b(In)458 5109 y(H.)39 b(Holden)g(and)g(A.)g(Jensen,)j(editors,)f Fi(Schr\177)-50 b(odinger)39 b(Op)-5 b(er)g(ators)p Fp(,)40 b(v)m(olume)f(345)458 5229 y(of)i Fi(L)-5 b(e)g(ctur)g(e)43 b(Notes)h(in)e(Physics)p Fp(,)i(pages)e(65{92,)g(Heidelb)s(erg,)h (1988.)e(Springer-)458 5349 y(V)-8 b(erlag.)p eop %%Page: 10 11 10 10 bop 257 266 a Fq(BM-1,)33 b(Septem)m(b)s(er)g(22,)f(2000)2006 b Fp(10)306 573 y([6])49 b(R.)38 b(Carmona,)h(W.)f(Masters,)j(and)e(B.) f(Simon.)59 b(Relativistic)35 b(Sc)m(hr\177)-49 b(odinger)39 b(op-)458 693 y(erators:)55 b(Asymptotic)38 b(b)s(eha)m(viour)h(of)f (the)g(eigenfunctions.)61 b Fi(J.)40 b(F)-7 b(unct.)40 b(A)n(nal.)p Fp(,)458 814 y(1992.)306 1017 y([7])49 b(P)-8 b(.A.M.)57 b(Dirac.)112 b(The)57 b(quan)m(tum)f(theory)h(of)e(the)i (electron.)113 b Fi(Pr)-5 b(o)g(c.)56 b(R)-5 b(oyal)458 1137 y(So)g(c.)34 b(L)-5 b(ondon)p Fp(,)32 b(117\(A\):610{624,)e(1928.) 306 1341 y([8])49 b(P)-8 b(.A.M.)48 b(Dirac.)84 b(The)47 b(quan)m(tum)h(theory)f(of)f(the)h(electron)g(I)s(I.)86 b Fi(Pr)-5 b(o)g(c.)47 b(R)-5 b(oyal)458 1461 y(So)g(c.)34 b(L)-5 b(ondon)p Fp(,)32 b(118\(A\):351{361,)e(1928.)306 1665 y([9])49 b(D.)30 b(Ev)-5 b(ans,)33 b(P)-8 b(.)31 b(P)m(erry)-8 b(,)32 b(and)g(H.)f(Sieden)m(top.)41 b(The)32 b(sp)s(ectrum)f(of)g(relativistic)d(one-)458 1785 y(electron)e(atoms)f (according)g(to)h(Bethe)h(and)f(Salp)s(eter.)32 b Fi(Commun.)c(Math.)h (Phys.)p Fp(,)458 1905 y(178\(3\):733{746,)h(July)i(1996.)257 2109 y([10])49 b(B.)35 b(Hel\013er)f(and)h(B)g(.P)m(arissee.)51 b(Comparaison)33 b(en)m(tre)j(la)d(d)m(\023)-46 b(ecroissance)36 b(de)g(fonc-)458 2229 y(tions)28 b(propres)i(p)s(our)f(les)f(op)m(\023) -46 b(erateurs)30 b(de)f(Dirac)f(et)h(de)g(Klein-Gordon.)e(Applica-)458 2350 y(tion)34 b(\022)-49 b(a)35 b(l')m(\023)-46 b(etude)35 b(de)h(l'e\013et)f(tunnel.)51 b Fi(A)n(nn.)37 b(Inst.)g(H.)g(Poinc)-5 b(ar)n(\023)-47 b(e)p Fp(,)34 b(60\(2\):147{187,)458 2470 y(1994.)257 2673 y([11])49 b(B.)25 b(Hel\013er)g(and)g(J.)h (Sj\177)-49 b(ostrand.)31 b(Semi-classical)22 b(analysis)j(for)g(Harp)s (er's)g(equation)458 2794 y(iii:)50 b(Can)m(tor)37 b(structure)i(of)d (the)i(sp)s(ectrum.)58 b Fi(Suppl.)38 b(Bul)5 b(l.)39 b(So)-5 b(c.)38 b(Math.)i(F)-7 b(r)i(anc)g(e)p Fp(,)458 2914 y(39\(117\),)31 b(1989.)257 3117 y([12])49 b(T.)f(Kato.)87 b(Smo)s(oth)47 b(op)s(erators)g(and)h(comm)m(utators.)87 b Fi(Stud.)49 b(Math.)g(Appl.)f Fp(,)458 3238 y(31:535{546,)31 b(1968.)257 3441 y([13])49 b(O.)37 b(Matte.)59 b(One-electron)38 b(atoms)f(in)g(the)h(Bro)m(wn-Ra)m(v)m(enhall)f(mo)s(del)f(coupled)458 3562 y(to)49 b(the)i(quan)m(tized)f(radiation)e(\014eld.)94 b(Master's)51 b(thesis,)k(Univ)m(ersit\177)-49 b(at)49 b(Mainz)458 3682 y(\(Diploma)29 b(Thesis\),)34 b(In)f(Preparation.)257 3885 y([14])49 b(G.)31 b(Sc)m(hleink)m(ofer.)43 b(In)m(tro)s(duction)31 b(to)g(pseudo)s(di\013eren)m(tial)g(op)s(erators.)42 b(P)m(osgrado)458 4006 y(en)33 b(Matematicas)f(Univ)m(ersidad)g (Nacional)f(de)i(Colom)m(bia,)e(1996.)257 4209 y([15])49 b(B.)24 b(Thaller.)k Fi(The)f(Dir)-5 b(ac)26 b(Equation)p Fp(.)j(Springer-V)-8 b(erlag,)24 b(Heidelb)s(erg,)h(New)g(Y)-8 b(ork,)458 4330 y(1992.)257 4533 y([16])49 b(C.)35 b(Tix.)48 b(Self-adjoin)m(tness)33 b(and)h(sp)s(ectral)g(prop)s(erties)g(of)g(a)g (pseudo-relativistic)458 4653 y(Hamiltonian)44 b(due)49 b(to)e(Bro)m(wn)i(and)f(Ra)m(v)m(enhall.)88 b Fi(Pr)-5 b(eprint)49 b(mp-ar)-5 b(c)48 b(97-441)p Fp(,)458 4774 y(1997.)257 4977 y([17])h(C.)31 b(Tix.)40 b(Strict)31 b(p)s(ositivit)m(y)e(of)h(a)h(relativistic)d(Hamiltonian)f(due)k(to)g (Bro)m(wn)g(and)458 5098 y(Ra)m(v)m(enhall.)43 b Fi(Bul)5 b(l.)34 b(L)-5 b(ondon)34 b(Math.)h(So)-5 b(c.)p Fp(,)32 b(30\(3\):283{290,)e(1998.)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF