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33 w(n) p Fm 29 w(2) p Fo 29 w(\000) f(and) h(\() p Fl(V) 21 b( ) p Fo 4 w(\)\() p Fl(n) p Fo(\)) 29 b(=) g(0) k(for) p Fl 33 w(n) 41 b(=) p Fm -61 w(2) p Fo 29 w(\000.\)) 46 b(W) -8 b(e) 34 b(call) d(suc) m(h) p Fl 35 w(V) p Fo 0 3022 a(a) p Fy 32 w(subsp) -5 b(ac) g(e) 34 b(p) -5 b(otential) p Fo(.) 146 3143 y(In) 33 b(this) f(pap) s(er) h(w) m(e) h (study) g(scattering) e(prop) s(erties) g(of) g(the) h(op) s(erators) p Fl 1615 3353 a(H) p Fo 35 w(=) p Fl 28 w(H) p Ft 1916 3368 a(0) p Fo 1977 3353 a(+) p Fl 22 w(V) 5 b(:) p Fo 1415 w(\(1.1\)) 0 3562 y(The) 36 b(abstract) f(mo) s(del) e(\(1.1\)) i (is) f(a) h(natural) f(and) h(tec) m(hnically) f(con) m(v) m(enien) m (t) i(generalization) d(of) h(man) m(y) 0 3683 y(di\013eren) m(t) 39 b(sp) s(eci\014c) g(mo) s(dels) e(discussed) j(in) e(recen) m(t) i (literature) d([BBP,) i(CS,) g(JL1,) f(JL3,) g(JM3) q(,) g(MV1,) 0 3803 y(MV2].) 146 3924 y(Let) 33 b(us) g(recall) d(some) i(w) m (ell-kno) m(wn) h(facts.) 43 b(If) p Fl 33 w(A) p Fo 32 w(is) 32 b(a) g(self-adjoin) m(t) e(op) s(erator) i(on) g(a) g(Hilb) s(ert) e(space) p Fj 0 4044 a(H) p Fo 32 w(and) p Fl 33 w(\036;) 17 b( ) p Fm 31 w(2) p Fj 28 w(H) p Fo(,) 33 b(then) g(for) f(Leb) s(esgue) i(a.e.) p Fl 43 w(E) p Fm 34 w(2) p Fi 28 w(R) p Fo 5 w(,) 39 b(the) 33 b(limits) 879 4254 y(\() p Fl(\036) p Fm(j) p Fo(\() p Fl(A) p Fm 21 w(\000) p Fl 23 w(E) p Fm 28 w(\000) p Fo 22 w(i0\)) p Fq 1549 4213 a(\000) p Ft(1) p Fl 1643 4254 a( ) p Fo 4 w(\)) 27 b(:=) h(lim) p Fr 1924 4317 a(\017) p Fq(#) p Ft(0) p Fo 2041 4254 a(\() p Fl(\036) p Fm(j) p Fo(\() p Fl(A) p Fm 22 w(\000) p Fl 23 w(E) p Fm 28 w(\000) p Fo 22 w(i) p Fl(\017) p Fo(\)) p Fq 2702 4213 a(\000) p Ft(1) p Fl 2796 4254 a( ) p Fo 4 w(\)) 0 4512 y(exist) 33 b(and) g(are) h(\014nite) e(and) h(non-zero.) 45 b(W) -8 b(e) 34 b(denote) g(b) m(y) p Fh 34 w(1) p Ft 2135 4527 a(\002) p Fo 2194 4512 a(\() p Fl(A) p Fo(\)) f(the) g(sp) s(ectral) g (pro) 5 b(jection) 33 b(of) p Fl 32 w(A) p Fo 34 w(on) m(to) 0 4632 y(a) j(Borel) g(set) h(\002.) 54 b(A) 37 b(b) s(ounded) g(op) s (erator) p Fl 35 w(B) p Fo 42 w(is) f(called) p Fl 35 w(A) p Fo(-smo) s(oth) f(on) h(\002) g(if) f(there) i(is) f(a) g (constan) m(t) p Fl 38 w(C) p Fo 0 4753 a(suc) m(h) e(for) e(all) p Fl 31 w(\036) p Fm 27 w(2) p Fo 28 w(Ran) p Fh 16 w(1) p Ft 931 4768 a(\002) p Fo 990 4753 a(\() p Fl(A) p Fo(\),) p Fk 1328 4875 a(Z) p Fg 1383 5101 a(R) p Fm 1452 5011 a(k) p Fl(B) p Fo 5 w(e) p Fq 1624 4970 a(\000) p Ft(i) p Fr(tA) p Fl 1781 5011 a(\036) p Fm(k) p Ft 1889 4970 a(2) p Fo 1928 5011 a(d) p Fl(t) p Fm 28 w(\024) p Fl 29 w(C) p Fm 7 w(k) p Fl(\036) p Fm(k) p Ft 2386 4970 a(2) p Fl 2424 5011 a(:) p Fo 0 5268 a(If) h(\002) 27 b(=) p Fi 28 w(R) p Fo 5 w(,) 38 b(w) m(e) c(simply) d(sa) m(y) i(that) p Fl 33 w(B) p Fo 38 w(is) p Fl 32 w(A) p Fo(-smo) s(oth.) p 90 rotate dyy eop %%Page: 3 3 3 2 bop Fo 3731 100 a(3) 146 407 y(Let) p Fl 33 w(A) p Fo 33 w(and) p Fl 32 w(B) p Fo 38 w(b) s(e) 33 b(self-adjoin) m(t) e (op) s(erators) h(and) h(assume) g(that) f(the) h(w) m(a) m(v) m(e) i (op) s(erators) p Fl 1223 604 a(U) p Fq 1299 563 a(\006) p Fo 1387 604 a(:=) 27 b(s) p Fm 23 w(\000) p Fo 66 w(lim) p Fr 1678 664 a(t) p Fq(!\0061) p Fo 1916 604 a(e) p Ft 1959 563 a(i) p Fr(tB) p Fo 2065 604 a(e) p Fq 2108 563 a(\000) p Ft(i) p Fr(tA) p Fh 2265 604 a(1) p Ft 2321 619 a(\002) p Fo 2380 604 a(\() p Fl(A) p Fo(\)) p Fl(;) p Fo 0 842 a(exist.) 40 b(One) 23 b(easily) f(sho) m(ws) i(that) f(Ran) p Fl 16 w(U) p Fq 1459 806 a(\006) p Fm 1546 842 a(\032) p Fh 29 w(1) p Ft 1708 857 a(\002) p Fo 1767 842 a(\() p Fl(A) p Fo(\).) 40 b(The) 23 b(w) m(a) m(v) m(e) i(op) s(erators) p Fl 22 w(U) p Fq 2899 806 a(\006) p Fo 2981 842 a(are) e(called) e (complete) 0 963 y(on) i(\002) h(if) e(Ran) p Fl 16 w(U) p Fq 573 927 a(\006) p Fo 661 963 a(=) p Fh 27 w(1) p Ft 820 978 a(\002) p Fo 879 963 a(\() p Fl(B) p Fo 5 w(\).) 41 b(The) 24 b(w) m(a) m(v) m(e) h(op) s(erators) p Fl 23 w(U) p Fq 2020 927 a(\006) p Fo 2103 963 a(are) f(complete) f(on) g (\002) g(i\013) g(the) g(w) m(a) m(v) m(e) j(op) s(erators) 1384 1160 y(s) p Fm 22 w(\000) p Fo 66 w(lim) p Fr 1544 1220 a(t) p Fq(!\0061) p Fo 1782 1160 a(e) p Ft 1825 1119 a(i) p Fr(tA) p Fo 1928 1160 a(e) p Fq 1971 1119 a(\000) p Ft(i) p Fr(tB) p Fh 2132 1160 a(1) p Ft 2188 1175 a(\002) p Fo 2247 1160 a(\() p Fl(A) p Fo(\)) 0 1388 y(exist.) 146 1509 y(Let) p Fm 35 w(H) p Fr 407 1524 a(n) p Fo 488 1509 a(b) s(e) 34 b(the) h(cyclic) f(space) h(spanned) h(b) m(y) p Fl 35 w(H) p Fo 41 w(and) p Fl 35 w(\016) p Fr 2200 1524 a(n) p Fo 2247 1509 a(,) p Fl 34 w(n) p Fm 31 w(2) p Fo 31 w(\000,) f(and) f(let) 2980 1483 y(~) p Fm 2950 1509 a(H) p Fo 36 w(b) s(e) g(the) h(closure) f(of) 0 1629 y(the) e(linear) e(span) i(of) f(the) g(subspaces) p Fm 34 w(H) p Fr 1469 1644 a(n) p Fo 1516 1629 a(.) 43 b(If) 1712 1604 y(~) p Fm 1683 1629 a(H) p Fo 28 w(=) p Fm 28 w(H) p Fo 1 w(,) 32 b(w) m(e) g(sa) m(y) h(that) p Fm 31 w(f) p Fl(\016) p Fr 2656 1644 a(n) p Fm 2703 1629 a(g) p Fr 2753 1644 a(n) p Fq(2) p Ft(\000) p Fo 2922 1629 a(is) e(a) g(cyclic) g(family) e(for) p Fl 0 1749 a(H) p Fo 8 w(.) 47 b(It) 34 b(is) f(not) h(di\016cult) f(to) g(sho) m (w) i(\(see) g(the) f(pro) s(of) f(of) h(Prop) s(osition) e(3.1) h(in) g ([JL1]\)) h(that) 3328 1724 y(~) p Fm 3298 1749 a(H) p Fo 35 w(do) s(es) h(not) 0 1870 y(dep) s(end) 40 b(on) e(the) h(c) m (hoice) g(of) p Fl 38 w(V) p Fo 21 w(.) 61 b(Th) m(us,) 42 b(assuming) 37 b(that) p Fm 38 w(f) p Fl(\016) p Fr 2260 1885 a(n) p Fm 2307 1870 a(g) p Fr 2357 1885 a(n) p Fq(2) p Ft(\000) p Fo 2534 1870 a(is) h(a) g(cyclic) g(family) e(for) p Fl 38 w(H) p Ft 3539 1885 a(0) p Fo 3616 1870 a(also) 0 1990 y(implies) 31 b(that) h(it) g(is) h(a) f(cyclic) h(family) d(for) p Fl 33 w(H) p Fo 8 w(.) 44 b(F) -8 b(rom) 31 b(hereon) j(w) m(e) g (indeed) f(assume) h(that) p Fm 33 w(f) p Fl(\016) p Fr 3365 2005 a(n) p Fm 3411 1990 a(g) p Fr 3461 2005 a(n) p Fq(2) p Ft(\000) p Fo 3632 1990 a(is) f(a) 0 2111 y(cyclic) f(family) e(for) p Fl 32 w(H) p Ft 796 2126 a(0) p Fo 868 2111 a(and) j(th) m(us) g(for) p Fl 32 w(H) p Fo 8 w(.) 146 2231 y(Let) p Fl 33 w(R) p Fm 29 w(\025) p Fo 28 w(0) f(b) s(e) h(a) f(p) s(ositiv) m(e) g(in) m(teger) h (and) 1289 2428 y(\000) p Fr 1350 2443 a(R) p Fo 1436 2428 a(=) p Fm 27 w(f) p Fl(n) p Fm 28 w(2) 28 b(G) p Fo 34 w(:) p Fl 27 w(\032) p Fo(\() p Fl(n;) p Fo 17 w(\000\)) p Fm 28 w(\024) p Fl 28 w(R) p Fm 1 w(g) p Fl(:) p Fo 0 2626 a(\(Note) 33 b(that) f(\000) p Ft 546 2641 a(0) p Fo 613 2626 a(=) c(\000.\)) 43 b(W) -8 b(e) 33 b(denote) g(b) m(y) p Fh 34 w(1) p Fr 1560 2641 a(R) p Fo 1650 2626 a(the) g(orthogonal) e(pro) 5 b(jection) 32 b(on) p Fl 33 w(l) p Ft 2936 2590 a(2) p Fo 2975 2626 a(\(\000) p Fr 3074 2641 a(R) p Fo 3132 2626 a(\).) 146 2746 y(Our) h(main) e(result) h(is:) p Fh 0 2930 a(Theorem) 74 b(1.1) p Fy 49 w(L) -5 b(et) p Fo 36 w(\002) p Fm 27 w(\032) p Fi 28 w(R) p Fy 46 w(b) g(e) 35 b(an) f(op) -5 b(en) 34 b(set.) 45 b(Consider) 34 b(the) h(fol) 5 b(lowing) 33 b(assumptions:) p Fo 0 3050 a(\(a\)) p Fy 35 w(The) h(op) -5 b(er) g(ator) p Fl 34 w(H) p Fh 8 w(1) p Ft 887 3065 a(\002) p Fo 946 3050 a(\() p Fl(H) p Fo 8 w(\)) p Fy 34 w(has) 35 b(pur) -5 b(ely) 35 b(absolutely) g(c) -5 b(ontinuous) 34 b(sp) -5 b(e) g(ctrum.) p Fo 0 3171 a(\(b\)) p Fh 35 w(1) p Ft 221 3186 a(1) p Fy 295 3171 a(is) p Fl 35 w(H) p Ft 481 3186 a(0) p Fy 520 3171 a(-smo) g(oth) 34 b(on) p Fo 35 w(\002) p Fy(.) p Fo 0 3291 a(\(c\)) p Fy 35 w(The) g(wave) g(op) -5 b(er) g(ators) p Fl 1176 3489 a(W) p Fq 1282 3447 a(\006) p Fo 1368 3489 a(=) 28 b(s) p Fm 22 w(\000) p Fo 66 w(lim) p Fr 1632 3548 a(t) p Fq(!\0061) p Fo 1870 3489 a(e) p Ft 1913 3447 a(i) p Fr(tH) p Fo 2026 3489 a(e) p Fq 2069 3447 a(\000) p Ft(i) p Fr(tH) p Ff 2227 3456 a(0) p Fh 2266 3489 a(1) p Ft 2322 3504 a(\002) p Fo 2381 3489 a(\() p Fl(H) p Ft 2500 3504 a(0) p Fo 2539 3489 a(\)) p Fl(;) p Fy 0 3715 a(exist.) p Fo 0 3836 a(\(d\)) p Fy 35 w(Ther) -5 b(e) 34 b(is) h(a) f(set) p Fm 35 w(D) p Fy 38 w(dense) g(in) p Fo 34 w(Ran) p Fh 17 w(1) p Ft 1537 3851 a(\002) p Fo 1596 3836 a(\() p Fl(H) p Ft 1715 3851 a(0) p Fo 1754 3836 a(\)) p Fy 35 w(such) h(that) g(for) p Fl 34 w(\036) p Fm 28 w(2) 28 b(D) p Fy 3 w(,) p Fm 34 w(k) p Fh(1) p Ft 2833 3851 a(1) p Fo 2873 3836 a(e) p Fq 2916 3799 a(\000) p Ft(i) p Fr(tH) p Ff 3074 3808 a(0) p Fl 3113 3836 a(\036) p Fm(k) p Fo 27 w(=) p Fl 27 w(O) p Fo 3 w(\() p Fm(j) p Fl(t) p Fm(j) p Fq 3558 3799 a(\000) p Ft(2) p Fo 3652 3836 a(\)) p Fy(.) p Fo 0 3956 a(\(e\)) p Fy 35 w(F) -7 b(or) 34 b(L) -5 b(eb) g(esgue) 34 b(a.e.) p Fl 45 w(E) p Fm 33 w(2) p Fo 28 w(\002) p Fy 35 w(and) g(al) 5 b(l) p Fl 35 w(n) p Fm 28 w(2) p Fo 28 w(\000) p Fy(,) p Fo 1250 4153 a(Im) 16 b(\() p Fl(\016) p Fr 1464 4168 a(n) p Fm 1511 4153 a(j) p Fo(\() p Fl(H) p Fm 29 w(\000) p Fl 23 w(E) p Fm 28 w(\000) p Fo 23 w(i0\)) p Fq 2102 4112 a(\000) p Ft(1) p Fl 2195 4153 a(\016) p Fr 2238 4168 a(n) p Fo 2285 4153 a(\)) p Fl 27 w(>) p Fo 28 w(0) p Fl(:) p Fy 0 4351 a(Consider) 34 b(the) h(fol) 5 b(lowing) 33 b(statements:) p Fo 0 4471 a(\(1\)) p Fy 35 w(F) -7 b(or) 33 b(L) -5 b(eb) g(esgue) 35 b(a.e.) p Fl 44 w(E) p Fm 34 w(2) p Fo 28 w(\002) p Fy 35 w(and) f(al) 5 b(l) p Fl 34 w(n) p Fm 28 w(2) p Fo 28 w(\000) p Fy(,) p Fk 1087 4591 a(X) p Fr 1065 4803 a(m) p Fq(2) p Ft(\000) p Ff 1218 4812 a(1) p Fm 1269 4686 a(j) p Fo(Im) 16 b(\() p Fl(\016) p Fr 1511 4701 a(n) p Fm 1558 4686 a(j) p Fo(\() p Fl(H) p Fm 30 w(\000) p Fl 22 w(E) p Fm 28 w(\000) p Fo 23 w(i0\)) p Fq 2149 4645 a(\000) p Ft(1) p Fl 2242 4686 a(\016) p Fr 2285 4701 a(m) p Fo 2352 4686 a(\)) p Fm(j) p Ft 2418 4645 a(2) p Fl 2485 4686 a(<) p Fm 27 w(1) p Fl(:) p Fo 0 4982 a(\(2\)) p Fy 35 w(F) -7 b(or) 33 b(a) i(dense) f(set) h(of) p Fl 35 w(\036) p Fm 27 w(2) p Fo 28 w(Ran) p Fh 16 w(1) p Ft 1388 4997 a(\002) p Fo 1448 4982 a(\() p Fl(H) p Fo 8 w(\)) p Fy(,) p Fk 1403 5092 a(Z) p Fg 1458 5318 a(R) p Fm 1527 5228 a(k) p Fh(1) p Ft 1633 5243 a(1) p Fo 1672 5228 a(e) p Fq 1715 5187 a(\000) p Ft(i) p Fr(tH) p Fl 1883 5228 a(\036) p Fm(k) p Ft 1991 5187 a(2) p Fo 2030 5228 a(d) p Fl(t) 28 b(<) p Fm 27 w(1) p Fl(:) p 90 rotate dyy eop %%Page: 4 4 4 3 bop Fo 3731 100 a(4) 0 407 y(\(3\)) p Fy 35 w(The) 34 b(wave) g(op) -5 b(er) g(ators) p Fl 34 w(W) p Fq 1127 371 a(\006) p Fy 1221 407 a(ar) g(e) 34 b(c) -5 b(omplete) 34 b(on) p Fo 35 w(\002) p Fy(.) 146 648 y(If) p Fo 31 w(\(a\)) p Fy 30 w(holds,) d(then) p Fo 30 w(\(1\)) p Fm 31 w(\)) p Fo 30 w(\(2\)) p Fy(.) 43 b(If) p Fo 30 w(\(e\)) p Fy 31 w(holds,) 31 b(then) p Fo 30 w(\(2\)) p Fm 31 w(\)) p Fo 30 w(\(1\)) p Fy(.) 43 b(If) p Fo 30 w(\(b\)) p Fy 31 w(holds,) 31 b(then) p Fo 30 w(\(2\)) p Fm 31 w(\)) p Fo 30 w(\(3\)) p Fy(.) 0 768 y(If) p Fo 34 w(\(b\)) p Fy(,) p Fo 35 w(\(c\)) p Fy 35 w(and) p Fo 35 w(\(d\)) p Fy 34 w(hold,) k(then) p Fo 34 w(\(3\)) p Fm 28 w(\)) p Fo 27 w(\(2\)) p Fy(.) 44 b(Henc) -5 b(e,) 35 b(if) p Fo 34 w(\(a\)) p Fy({) p Fo(\(e\)) p Fy 35 w(hold,) f(then) p Fo 35 w(\(1\)) p Fm 27 w(,) p Fo 27 w(\(2\)) p Fm 28 w(,) p Fo 27 w(\(3\)) p Fy(.) p Fh 0 957 a(Remark) j(1.) p Fo 43 w(The) d(same) e(result) h(holds) f(if) p Fl 32 w(H) p Ft 1698 972 a(0) p Fo 1769 957 a(is) g(replaced) h(b) m(y) p Fl 34 w(H) p Ft 2469 972 a(0) p Fo 2530 957 a(+) p Fl 22 w(U) p Ft 2694 972 a(0) p Fo 2734 957 a(,) f(where) p Fl 1415 1172 a(U) p Ft 1481 1187 a(0) p Fo 1549 1172 a(=) p Fk 1652 1077 a(X) p Fr 1656 1289 a(n) p Fq(2G) p Fl 1813 1172 a(U) p Ft 1879 1187 a(0) p Fo 1918 1172 a(\() p Fl(n) p Fo(\)\() p Fl(\016) p Fr 2133 1187 a(n) p Fm 2181 1172 a(j\001) p Fo(\)) p Fl(\016) p Fr 2318 1187 a(n) p Fo 0 1466 a(is) g(an) h(arbitary) e(bac) m(kground) j(p) s(oten) m(tial.) p Fh 0 1586 a(Remark) 39 b(2.) p Fo 48 w(Since) p Fm 34 w(f) p Fl(\016) p Fr 911 1601 a(n) p Fm 958 1586 a(g) p Fr 1008 1601 a(n) p Fq(2) p Ft(\000) p Fo 1180 1586 a(is) 34 b(a) f(cyclic) h(family) e(for) p Fl 33 w(H) p Fo 8 w(,) i(\(2\)) p Fm 34 w(\)) p Fo 34 w(\(a\).) 47 b(Similarly) -8 b(,) 31 b(if) i(either) h(\(b\)) g(or) 0 1706 y(\(d\)) f(holds,) f(then) p Fl 33 w(H) p Ft 748 1721 a(0) p Fh 787 1706 a(1) p Ft 843 1721 a(\002) p Fo 902 1706 a(\() p Fl(H) p Ft 1021 1721 a(0) p Fo 1061 1706 a(\)) g(has) h(purely) g(absolutely) f(con) m(tin) m(uous) h(sp) s (ectrum.) p Fh 0 1827 a(Remark) d(3.) p Fo 42 w(The) e(assumption) e (that) h(\002) g(is) f(an) h(op) s(en) g(set) h(is) e(used) i(only) f (in) f(the) h(pro) s(of) f(of) h(implication) 0 1947 y(\(2\)) p Fm 32 w(\)) p Fo 31 w(\(3\),) 32 b(all) e(the) j(other) f (results) g(hold) f(for) h(an) m(y) h(Borel) e(set) i(\002) f(of) f(p) s (ositiv) m(e) h(Leb) s(esgue) h(measure.) 44 b(If) 0 2068 y(in) 30 b(\(b\)) h(w) m(e) h(assume) f(that) p Fh 31 w(1) p Ft 1016 2083 a(1) p Fo 1086 2068 a(is) p Fl 30 w(H) p Ft 1263 2083 a(0) p Fo 1302 2068 a(-smo) s(oth,) f(then) i (\(b\)) e(and) h(\(2\)) g(imply) e(\(3\)) h(for) g(an) m(y) i(Borel) e (set) h(\002) g(of) 0 2188 y(p) s(ositiv) m(e) h(Leb) s(esgue) i (measure.) 146 2308 y(Theorem) 29 b(1.1) f(is) g(based) i(on) e(a) g (simple) f(ph) m(ysical) i(principle) e(already) h(used) h(in) f(some) h (sp) s(ecial) e(cases) 0 2429 y(in) 37 b([JL3].) 58 b(If) 37 b(the) h(sp) s(ectrum) g(of) p Fl 37 w(H) p Fo 45 w(in) f(\002) g(is) g (purely) h(absolutely) e(con) m(tin) m(uous,) k(then) e(w) m(a) m(v) m (e) h(pac) m(k) m(ets) 0 2549 y(with) 31 b(energies) h(in) f(\002) h(m) m(ust) f(propagate.) 43 b(If) 32 b(propagation) e(along) g(the) i (subspace) p Fl 34 w(l) p Ft 3045 2513 a(2) p Fo 3084 2549 a(\(\000\)) g(is) f(supressed,) 0 2669 y(then) j(the) g(w) m(a) m (v) m(e) h(pac) m(k) m(ets) h(m) m(ust) d(propagate) g(in) m(to) p Fl 33 w(l) p Ft 1905 2633 a(2) p Fo 1945 2669 a(\(\000\)) p Fq 2082 2633 a(?) p Fo 2141 2669 a(.) 46 b(Theorem) 33 b(2.1) g(quan) m(ti\014es) i(this) e(heuristic) 0 2790 y(principle) 23 b(and) h(further) g(asserts) i(that) e(under) g(fairly) f(general) g(assumptions) h(the) h(\\lo) s(calization) 20 b(within) 0 2910 y(the) 39 b(subspace") h(is) e(the) g(only) g(ph) m (ysical) g(mec) m(hanism) g(relev) m(en) m(t) h(to) f(the) h (completeness) g(of) f(the) h(w) m(a) m(v) m(e) 0 3031 y(op) s(erators.) 146 3151 y(The) e(assumptions) e(\(a\)) g(and) g (\(e\)) h(of) f(Theorem) h(1.1) f(concern) h(the) g(in) m(teracting) e (Hamiltonian) p Fl 32 w(H) p Fo 0 3271 a(and) h(could) g(b) s(e) g (di\016cult) f(to) g(c) m(hec) m(k) k(in) c(practice.) 51 b(Ho) m(w) m(ev) m(er,) 37 b(due) f(to) f(results) g(in) f([JL2],) i (for) e(random) 0 3392 y(subspace) e(p) s(oten) m(tials) c(\(a\)) i (and) g(\(e\)) g(can) g(b) s(e) g(reduced) i(to) d(assumptions) h(on) p Fl 30 w(H) p Ft 2892 3407 a(0) p Fo 2961 3392 a(whic) m(h) g(can) g(b) s (e) g(easily) 0 3512 y(v) m(eri\014ed) j(in) f(concrete) i(mo) s(dels.) 42 b(Let) 33 b(us) g(describ) s(e) h(the) f(random) e(mo) s(del) g(and) i(this) f(result) h(in) e(detail.) 146 3633 y(Let) i(\012) g(b) s(e) g (the) g(set) g(of) f(all) f(b) s(oundary) i(p) s(oten) m(tials,) 1538 3836 y(\012) c(=) p Fi 27 w(R) p Ft 1806 3795 a(\000) p Fo 1888 3836 a(=) p Fe 28 w(\002) p Ft 2036 3912 a(\000) p Fi 2142 3836 a(R) p Fl 5 w(;) p Fo 0 4082 a(and) 36 b(let) p Fj 36 w(B) p Fo 36 w(the) g(Borel) p Fl 35 w(\033) p Fo 4 w(-algebra) f(in) g(\012.) 55 b(The) 37 b(mo) s(del) d(is) i(sp) s (eci\014ed) h(b) m(y) g(a) f(c) m(hoice) g(of) g(a) f(probabilit) m(y) 0 4203 y(measure) p Fl 33 w(P) p Fo 46 w(on) d(\(\012) p Fl(;) p Fm 17 w(B) p Fo 3 w(\).) 44 b(F) -8 b(or) 32 b(simplicit) m(y) -8 b(,) 30 b(w) m(e) k(will) c(consider) j(only) f (the) h(pro) s(duct) g(measures) p Fl 1644 4419 a(P) p Fo 41 w(=) p Fe 28 w(\002) p Ft 1897 4496 a(\000) p Fl 2002 4419 a(\026) p Fr 2061 4434 a(n) p Fl 2108 4419 a(;) p Fo 0 4666 a(where) e(eac) m(h) p Fl 30 w(\026) p Fr 554 4681 a(n) p Fo 630 4666 a(is) e(a) g(probabilit) m(y) f(measure) h(on) p Fi 30 w(R) p Fo 5 w(.) 48 b(Note) 30 b(that) p Fl 29 w(\026) p Fr 2449 4681 a(n) p Fo 2525 4666 a(is) f(the) h (probabilit) m(y) d(distribution) 0 4786 y(of) d(the) i(random) e(v) -5 b(ariable) 23 b(\012) p Fm 28 w(3) p Fl 28 w(V) p Fm 49 w(!) p Fl 28 w(V) p Fo 21 w(\() p Fl(n) p Fo(\).) 41 b(W) -8 b(e) 26 b(sa) m(y) f(that) g(the) h(random) e(v) -5 b(ariable) p Fl 23 w(V) p Fo 21 w(\() p Fl(n) p Fo(\)) 25 b(has) g(densit) m(y) 0 4907 y(if) j(the) i(measure) p Fl 30 w(\026) p Fr 687 4922 a(n) p Fo 764 4907 a(is) f(absolutely) g (con) m(tin) m(uous) h(w.r.t.) 43 b(Leb) s(esgue) 31 b(measure.) 43 b(By) 30 b(construction,) g(the) 0 5027 y(random) i(v) -5 b(ariables) p Fm 31 w(f) p Fl(V) p Fo 21 w(\() p Fl(n) p Fo(\)) p Fm(g) p Fr 1074 5042 a(n) p Fq(2) p Ft(\000) p Fo 1245 5027 a(are) 32 b(indep) s(enden) m(t.) 146 5147 y(The) 45 b(follo) m(wing) c(result) i(is) g(an) h(easy) h (consequence) h(of) d(the) h(main) e(theorem) i(in) f([JL2]) g(\(w) m (e) i(will) 0 5268 y(outline) 31 b(its) h(pro) s(of) g(in) g(Section) g (3.1\).) p 90 rotate dyy eop %%Page: 5 5 5 4 bop Fo 3731 100 a(5) p Fh 0 407 a(Prop) s(osition) 36 b(1.2) p Fy 49 w(Assume) 45 b(that) h(the) f(r) -5 b(andom) 45 b(variables) p Fm 44 w(f) p Fl(V) p Fo 22 w(\() p Fl(n) p Fo(\)) p Fm(g) p Fr 2648 422 a(n) p Fq(2) p Ft(\000) p Fy 2831 407 a(have) g(densities) f(and) h(let) p Fo 0 527 a(\002) p Fm 28 w(\032) p Fi 28 w(R) p Fy 46 w(b) -5 b(e) 34 b(a) h(Bor) -5 b(el) 34 b(set) h(of) f(p) -5 b(ositive) 35 b(L) -5 b(eb) g(esgue) 34 b(me) -5 b(asur) g(e.) 44 b(Consider) 34 b(the) h(assumption:) p Fo 0 648 a(\(g\)) p Fy 39 w(The) j(op) -5 b(er) g(ator) p Fl 39 w(H) p Ft 836 663 a(0) p Fh 876 648 a(1) p Ft 932 663 a(\002) p Fo 991 648 a(\() p Fl(H) p Ft 1110 663 a(0) p Fo 1149 648 a(\)) p Fy 39 w(has) 39 b(pur) -5 b(ely) 39 b(absolutely) g(c) -5 b(ontinuous) 39 b(sp) -5 b(e) g(ctrum) 39 b(and) g(for) g(L) -5 b(eb) g(esgue) 0 768 y(a.e.) p Fl 44 w(E) p Fm 34 w(2) p Fo 28 w(\002) p Fy(,) p Fk 1154 806 a(X) p Fr 1159 1017 a(n) p Fq(2) p Ft(\000) p Fo 1314 900 a(Im) 16 b(\() p Fl(\016) p Fr 1528 915 a(n) p Fm 1575 900 a(j) p Fo(\() p Fl(H) p Ft 1722 915 a(0) p Fm 1783 900 a(\000) p Fl 23 w(E) p Fm 28 w(\000) p Fo 23 w(i0\)) p Fq 2198 859 a(\000) p Ft(1) p Fl 2291 900 a(\016) p Fr 2334 915 a(n) p Fo 2381 900 a(\)) p Fl 28 w(>) p Fo 27 w(0) p Fl(:) p Fo 953 w(\(1.2\)) p Fy 0 1150 a(If) p Fo 39 w(\(g\)) p Fy 40 w(holds) 39 b(and) h(the) g(assumption) p Fo 39 w(\(c\)) p Fy 40 w(of) g(The) -5 b(or) g(em) 39 b(1.1) g(holds) h(with) f(pr) -5 b(ob) g(ability) 40 b(one,) h(then) f(the) 0 1270 y(assumptions) p Fo 34 w(\(a\)) p Fy 34 w(and) p Fo 35 w(\(e\)) p Fy 35 w(hold) 34 b(with) h(pr) -5 b(ob) g(ability) 34 b(one.) p Fo 146 1438 a(On) 23 b(a) g(tec) m(hnical) f(lev) m(el,) i(Theorem) f (1.1) g(is) f(a) h(v) -5 b(arian) m(t) 21 b(of) i(Kato's) f(theory) i (of) e(smo) s(oth) g(p) s(erturbations.) 0 1558 y(Its) k(main) e(in) m (terest) i(lies) e(in) h(applications) e(to) i(random) g(subspace) i(p) s(oten) m(tials.) 40 b(The) 27 b(scattering) e(theory) 0 1679 y(of) 32 b(random) f(Sc) m(hr\177) -49 b(odinger) 33 b(op) s(erators) f(has) h(receiv) m(ed) h(considerable) e(atten) m (tion) f(in) h(recen) m(t) i(literature.) 0 1799 y(Mo) s(dels) 40 b(that) g(ha) m(v) m(e) i(b) s(een) f(studied) g(include) f(slo) m(wly) g(deca) m(ying) h(random) e(p) s(oten) m(tials) g([B) q(,) h(CK,) h(Kr) o(,) 0 1919 y(RoSh],) 35 b(sparse) g(random) e(p) s(oten) m(tials) g ([HK,) h(MV1,) g(MV2) q(],) g(and) h(surface) f(random) g(p) s(oten) m (tials) e([JL1,) 0 2040 y(JL3].) 42 b(Theorem) 30 b(1.1) f(can) g(b) s (e) h(e\013ectiv) m(ely) g(applied) e(to) g(Anderson) j(mo) s(dels) d (with) h(surface) h(and) f(sparse) 0 2160 y(random) f(p) s(oten) m (tials.) 41 b(W) -8 b(e) 30 b(will) d(discuss) j(the) g(surface) g(mo) s (del) e(in) g(the) i(next) g(section.) 42 b(The) 30 b(analysis) f(of) 0 2280 y(sparse) 35 b(random) e(p) s(oten) m(tial) f(mo) s(dels) h(is) h (more) f(tec) m(hnical) h(and) g(is) f(based) i(on) f(a) g(fusion) f (of) h(tec) m(hniques) 0 2401 y(dev) m(elop) s(ed) 28 b(in) f(this) g(pap) s(er) h(and) g(in) e([JL1,) i(MV1,) g(MV2].) 42 b(The) 28 b(scattering) g(theory) g(of) f(sparse) h(random) 0 2521 y(p) s(oten) m(tials) j(will) g(b) s(e) h(discussed) j(in) d(a) g (con) m(tin) m(uation) f(of) i(this) f(pap) s(er.) p Fh 0 2717 a(Ac) m(kno) m(wledgmen) m(ts.) p Fo 86 w(VJ's) 47 b(w) m(ork) h(w) m(as) g(partially) d(supp) s(orted) j(b) m(y) g(NSER) m (C.) g(YL's) g(w) m(ork) g(w) m(as) 0 2838 y(partially) 21 b(supp) s(orted) j(b) m(y) h(THE) f(ISRAEL) h(SCIENCE) g(F) m(OUND) m (A) -8 b(TION) 24 b(\(gran) m(t) g(no.) 40 b(447/99\).) f(P) m(art) 0 2958 y(of) 28 b(this) h(w) m(ork) h(has) f(b) s(een) g(done) h(during) e (a) g(visit) g(of) g(the) i(\014rst) f(author) g(to) f(the) h(Hebrew) h (Univ) m(ersit) m(y) -8 b(.) 43 b(VJ) 0 3078 y(is) 32 b(grateful) f(to) i(Y) -8 b(oram) 31 b(Last) i(for) f(his) g (hospitalit) m(y) -8 b(.) p Fn 0 3405 a(2) 161 b(Surface) 53 b(random) g(p) t(oten) l(tials) p Fo 0 3624 a(W) -8 b(e) 35 b(consider) h(the) f(same) g(mo) s(del) e(as) i(in) f([JL1,) h(JL3]:) 48 b(Let) p Fl 35 w(d) p Fm 32 w(\025) p Fo 32 w(1) 34 b(b) s(e) h(giv) m (en) h(and) f(let) p Fm 34 w(G) p Fo 38 w(=) p Fi 31 w(Z) p Fr 3464 3588 a(d) p Fm 3526 3624 a(\002) p Fi 24 w(Z) p Ft 3696 3639 a(+) p Fo 3752 3624 a(,) 0 3744 y(where) p Fi 47 w(Z) p Ft 364 3759 a(+) p Fo 471 3744 a(=) p Fm 50 w(f) p Fo(0) p Fl(;) p Fo 17 w(1) p Fl(;) p Fm 17 w(\001) 17 b(\001) g(\001) c(g) p Fo(.) 83 b(W) -8 b(e) 46 b(denote) h(p) s(oin) m(ts) e(in) p Fm 45 w(G) p Fo 52 w(b) m(y) p Fl 47 w(n) p Fo 50 w(=) 50 b(\() p Fl(n) p Ft 2656 3759 a(1) p Fl 2696 3744 a(;) 17 b(:) g(:) g(:) e(;) i (n) p Fr 2972 3759 a(d) p Ft(+1) p Fo 3103 3744 a(\).) 83 b(W) -8 b(e) 46 b(consider) 0 3865 y(the) 41 b(usual) f(metric) f(on) p Fm 41 w(G) p Fo 6 w(,) p Fl 42 w(\032) p Fo(\() p Fl(n;) 17 b(m) p Fo(\)) 42 b(=) p Fm 41 w(j) p Fl(n) p Fm 27 w(\000) p Fl 28 w(m) p Fm(j) p Ft 1831 3880 a(+) p Fo 1890 3865 a(,) h(where) p Fm 41 w(j) p Fl(n) p Fm(j) p Ft 2363 3880 a(+) p Fo 2463 3865 a(=) e(max) p Fr 2761 3880 a(j) p Fm 2815 3865 a(j) p Fl(n) p Fr 2901 3880 a(j) p Fm 2937 3865 a(j) p Fo(.) 67 b(The) 41 b(sp) s(ectrum) g(of) 0 3985 y(the) j(corresp) s(onding) g(discrete) g(Laplacian) p Fl 41 w(H) p Ft 1723 4000 a(0) p Fo 1806 3985 a(is) f(purely) h (absolutely) f(con) m(tin) m(uous) h(and) p Fl 44 w(\033) p Fo 4 w(\() p Fl(H) p Ft 3580 4000 a(0) p Fo 3619 3985 a(\)) j(=) 0 4105 y([) p Fm(\000) p Fo(2\() p Fl(d) p Fo 22 w(+) 22 b(1\)) p Fl(;) p Fo 17 w(2\() p Fl(d) p Fo 21 w(+) g(1\)].) 146 4226 y(Let) 33 b(\000) 28 b(=) p Fm 27 w(f) p Fl(n) p Fm 28 w(2) g(G) p Fo 34 w(:) p Fl 28 w(n) p Fr 949 4241 a(d) p Ft(+1) p Fo 1107 4226 a(=) g(0) p Fm(g) p Fo 27 w(=) p Fl 28 w(@) p Fm 5 w(G) p Fo 6 w(,) 33 b(let) p Fl 32 w(V) p Fo 54 w(b) s(e) g(a) f(p) s(oten) m(tial) f(supp) s(orted) j(on) e(\000,) h(and) p Fl 1615 4404 a(H) p Fo 35 w(=) p Fl 28 w(H) p Ft 1916 4419 a(0) p Fo 1977 4404 a(+) p Fl 22 w(V) 5 b(:) p Fo 1415 w(\(2.3\)) 0 4582 y(This) 29 b(particular) e(mo) s(del) f(is) j(motiv) -5 b(ated) 26 b(b) m(y) k(the) f(ph) m(ysics) h(of) e(disordered) h (surfaces) h(\(see) g([JMP) q(,) e(KP]\).) 0 4702 y(It) 22 b(is) g(ob) m(viously) g(an) g(example) g(of) g(the) h(abstract) f (subspace) j(mo) s(del) 20 b(discussed) k(in) e(the) h(previous) f (section.) 146 4822 y(W) -8 b(e) 49 b(brie\015y) g(review) g(what) g (is) f(kno) m(wn) i(ab) s(out) e(the) h(scattering) f(theory) h(of) f (the) h(mo) s(del) e(\(2.3\),) 0 4943 y(refering) 41 b(the) h(reader) g(for) f(details) g(and) g(additional) e(information) f (to) k(the) g(original) c(literature.) 69 b(In) 0 5063 y([CS) q(,) 32 b(JL1]) h(it) e(w) m(as) j(pro) m(v) m(en) g(that) e (for) g(all) p Fl 31 w(V) p Fo 54 w(the) h(w) m(a) m(v) m(e) h(op) s (erators) p Fl 1345 5241 a(W) p Fq 1451 5200 a(\006) p Fo 1537 5241 a(=) 28 b(s) p Fm 22 w(\000) p Fo 66 w(lim) p Fr 1801 5301 a(t) p Fq(!\0061) p Fo 2039 5241 a(e) p Ft 2082 5200 a(i) p Fr(tH) p Fo 2195 5241 a(e) p Fq 2238 5200 a(\000) p Ft(i) p Fr(tH) p Ff 2396 5209 a(0) p 90 rotate dyy eop %%Page: 6 6 6 5 bop Fo 3731 100 a(6) 0 407 y(exist.) 81 b(This) 45 b(implies) d(that) p Fl 44 w(\033) p Fo 4 w(\() p Fl(H) p Ft 1285 422 a(0) p Fo 1325 407 a(\)) p Fm 48 w(\032) p Fl 49 w(\033) p Ft 1592 422 a(ac) p Fo 1663 407 a(\() p Fl(H) p Fo 8 w(\).) 80 b(The) 46 b(question) f(of) f(completeness) i (of) e(the) i(w) m(a) m(v) m(e) 0 527 y(op) s(erators) p Fl 38 w(W) p Fq 543 491 a(\006) p Fo 639 527 a(has) 39 b(b) s(een) f(studied) h(in) e([JL1].) 60 b(In) 38 b(this) g(w) m(ork) g (the) h(notion) e(of) g(resonan) m(t) i(sp) s(ectrum) p Fj 0 648 a(R) p Fo(\() p Fl(H) p Fo 8 w(\)) 27 b(has) g(b) s(een) h(in) m(tro) s(duced,) g(and) f(it) f(w) m(as) i(sho) m(wn) g(that) e(the) i (w) m(a) m(v) m(e) g(op) s(erators) f(are) g(complete) f(on) h(the) 0 768 y(set) p Fl 36 w(\033) p Fo 4 w(\() p Fl(H) p Ft 333 783 a(0) p Fo 372 768 a(\)) p Fm 24 w(n) p Fj 24 w(R) p Fo(\() p Fl(H) p Fo 8 w(\).) 52 b(In) 35 b([JL1]) h(one) f(can) h (also) e(\014nd) i(v) -5 b(arious) 35 b(estimates) g(on) g(the) h(lo) s (cation) d(of) p Fj 35 w(R) p Fo(\() p Fl(H) p Fo 8 w(\)) 0 888 y(\(for) f(example,) g(if) p Fm 32 w(k) p Fl(V) p Fm 21 w(k) p Fl 27 w(<) p Fo 28 w(1,) g(then) p Fj 34 w(R) p Fo(\() p Fl(H) p Fo 8 w(\)) 27 b(=) p Fm 28 w(;) p Fo(\).) 146 1009 y(The) 35 b(resonan) m(t) f(sp) s(ectrum) g(is) f(c) m (haracterized) h(b) m(y) h(the) f(prop) s(ert) m(y) g(that) f(the) h (pro) 5 b(jection) p Fh 33 w(1) p Fr 3468 1024 a(R) p Fo 3559 1009 a(is) p Fl 33 w(H) p Fo 8 w(-) 0 1129 y(smo) s(oth) 39 b(on) h(an) m(y) h(compact) f(subset) h(of) p Fl 40 w(\033) p Fo 4 w(\() p Fl(H) p Fo 8 w(\)) p Fm 27 w(n) p Fj 27 w(R) p Fo(\() p Fl(H) p Fo 8 w(\).) 65 b(This) 41 b(is) e(a) h (restrictiv) m(e) g(condition) f(and) h(in) 0 1249 y(man) m(y) h(in) m (teresting) f(situations) p Fl 39 w(\033) p Fo 4 w(\() p Fl(H) p Ft 1393 1264 a(0) p Fo 1432 1249 a(\)) p Fm 42 w(\032) p Fj 42 w(R) p Fo(\() p Fl(H) p Fo 8 w(\).) 67 b(It) 41 b(is) f(also) f(kno) m(wn) j(that) f(in) f(general) g(the) h (w) m(a) m(v) m(e) 0 1370 y(op) s(erators) f(ma) m(y) g(not) g(b) s(e) g (complete) f(on) p Fl 40 w(\033) p Fo 4 w(\() p Fl(H) p Ft 1719 1385 a(0) p Fo 1758 1370 a(\)) p Fm 27 w(\\) p Fj 28 w(R) p Fo(\() p Fl(H) p Fo 8 w(\)) h([JL1,) g(MV1].) 66 b(The) 41 b(curren) m(t) g(pap) s(er) g(w) m(as) 0 1490 y(partly) 26 b(motiv) -5 b(ated) 26 b(b) m(y) h(the) h(question) f (under) h(what) f(conditions) f(one) h(ma) m(y) g(exp) s(ect) h(the) g (completeness) 0 1611 y(of) k(the) h(w) m(a) m(v) m(e) i(op) s(erators) d(on) p Fl 32 w(\033) p Fo 4 w(\() p Fl(H) p Ft 1262 1626 a(0) p Fo 1302 1611 a(\)) p Fm 22 w(\\) p Fj 22 w(R) p Fo(\() p Fl(H) p Fo 8 w(\).) 146 1731 y(F) -8 b(or) 45 b(the) h(mo) s(del) e(\(2.3\),) 49 b(it) 44 b(w) m(as) j(sho) m(wn) g(in) e([JL1]) g(that) p Fm 46 w(f) p Fl(\016) p Fr 2419 1746 a(n) p Fm 2466 1731 a(g) p Fr 2516 1746 a(n) p Fq(2) p Ft(\000) p Fo 2699 1731 a(is) g(a) h (cyclic) f(family) e(for) p Fl 45 w(H) p Ft 3740 1746 a(0) p Fo 0 1851 a(\(and) 34 b(hence) h(for) p Fl 34 w(H) p Fo 8 w(\)) e(and) h(that) g(the) g(conditions) f(\(b\),) h (\(c\)) g(and) g(\(d\)) g(of) g(Theorem) g(1.1) f(hold.) 47 b(Hence,) 0 1972 y(Theorem) 33 b(1.1) f(and) h(Remark) f(3) g(after) g (it) g(yield:) p Fh 0 2200 a(Theorem) 74 b(2.1) p Fy 49 w(L) -5 b(et) p Fo 32 w(\002) p Fm 28 w(\032) p Fl 28 w(\033) p Fo 4 w(\() p Fl(H) p Ft 1256 2215 a(0) p Fo 1295 2200 a(\)) p Fy 32 w(b) g(e) 31 b(a) g(Bor) -5 b(el) 31 b(set) h(of) f(p) -5 b(ositive) 31 b(L) -5 b(eb) g(esgue) 31 b(me) -5 b(asur) g(e.) 43 b(Consider) 31 b(the) 0 2320 y(assumptions:) p Fo 0 2441 a(\(a\)) p Fy 35 w(The) j(op) -5 b(er) g(ator) p Fl 34 w(H) p Fh 8 w(1) p Ft 887 2456 a(\002) p Fo 946 2441 a(\() p Fl(H) p Fo 8 w(\)) p Fy 34 w(has) 35 b(pur) -5 b(ely) 35 b(absolutely) g(c) -5 b(ontinuous) 34 b(sp) -5 b(e) g(ctrum.) p Fo 0 2561 a(\(b\)) p Fy 35 w(F) e(or) 34 b(L) -5 b(eb) g(esgue) 34 b(a.e.) p Fl 44 w(E) p Fm 34 w(2) p Fo 28 w(\002) p Fy 35 w(and) g(al) 5 b(l) p Fl 35 w(n) p Fm 28 w(2) p Fo 28 w(\000) p Fy(,) p Fo 34 w(Im) 17 b(\() p Fl(\016) p Fr 2108 2576 a(n) p Fm 2155 2561 a(j) p Fo(\() p Fl(H) p Fm 29 w(\000) p Fl 23 w(E) p Fm 28 w(\000) p Fo 22 w(i0\)) p Fq 2745 2525 a(\000) p Ft(1) p Fl 2839 2561 a(\016) p Fr 2882 2576 a(n) p Fo 2929 2561 a(\)) p Fl 27 w(>) p Fo 28 w(0) p Fy(.) 0 2682 y(Consider) 34 b(the) h(statements:) p Fo 0 2802 a(\(1\)) p Fy 35 w(F) -7 b(or) 33 b(L) -5 b(eb) g(esgue) 35 b(a.e.) p Fl 44 w(E) p Fm 34 w(2) p Fo 28 w(\002) p Fy 35 w(and) f(al) 5 b(l) p Fl 34 w(n) p Fm 28 w(2) p Fo 28 w(\000) p Fk 1106 2945 a(X) p Fr 1084 3156 a(m) p Fq(2) p Ft(\000) p Ff 1237 3165 a(1) p Fm 1288 3039 a(j) p Fo(Im) 16 b(\() p Fl(\016) p Fr 1530 3054 a(n) p Fm 1577 3039 a(j) p Fo(\() p Fl(H) p Fm 30 w(\000) p Fl 22 w(E) p Fm 28 w(\000) p Fo 23 w(i0\)) p Fq 2168 2998 a(\000) p Ft(1) p Fl 2261 3039 a(\016) p Fr 2304 3054 a(m) p Fm 2371 3039 a(j) p Ft 2399 2998 a(2) p Fl 2466 3039 a(<) p Fm 27 w(1) p Fl(:) p Fo 0 3358 a(\(2\)) p Fy 35 w(F) -7 b(or) 33 b(a) i(dense) f(set) h(of) p Fl 35 w(\036) p Fm 27 w(2) p Fo 28 w(Ran) p Fh 16 w(1) p Ft 1388 3373 a(\002) p Fo 1448 3358 a(\() p Fl(H) p Fo 8 w(\)) p Fy(,) p Fk 1403 3491 a(Z) p Fg 1458 3716 a(R) p Fm 1527 3626 a(k) p Fh(1) p Ft 1633 3641 a(1) p Fo 1672 3626 a(e) p Fq 1715 3585 a(\000) p Ft(i) p Fr(tH) p Fl 1883 3626 a(\036) p Fm(k) p Ft 1991 3585 a(2) p Fo 2030 3626 a(d) p Fl(t) 28 b(<) p Fm 27 w(1) p Fl(:) p Fo 0 3901 a(\(3\)) p Fy 35 w(The) 34 b(wave) g(op) -5 b(er) g(ators) p Fl 34 w(W) p Fq 1127 3864 a(\006) p Fy 1221 3901 a(ar) g(e) 34 b(c) -5 b(omplete) 34 b(on) p Fo 35 w(\002) p Fy(.) 146 4141 y(Then) p Fo 36 w(\(2\)) p Fm 31 w(\)) p Fo 31 w(\(3\)) p Fy(.) 49 b(If) p Fo 36 w(\(b\)) p Fy 37 w(holds,) p Fo 36 w(\(2\)) p Fm 31 w(\)) p Fo 31 w(\(1\)) p Fy(.) h(If) p Fo 36 w(\(a\)) p Fy 36 w(holds,) 36 b(then) p Fo 37 w(\(1\)) p Fm 30 w(\)) p Fo 31 w(\(2\)) p Fy(.) 50 b(If) p Fo 36 w(\(a\)) p Fy 36 w(and) p Fo 36 w(\(b\)) p Fy 0 4262 a(hold,) 34 b(then) p Fo 35 w(\(3\)) p Fm 27 w(\)) p Fo 27 w(\(2\)) p Fy(.) 45 b(Henc) -5 b(e,) 34 b(if) p Fo 35 w(\(a\)) p Fy 34 w(and) p Fo 35 w(\(b\)) p Fy 35 w(hold,) g(then) p Fo 34 w(\(1\)) p Fm 28 w(,) p Fo 27 w(\(2\)) p Fm 27 w(,) p Fo 28 w(\(3\)) p Fy -1 w(.) p Fo 146 4490 a(W) -8 b(e) 31 b(no) m(w) g(assume) g(that) p Fl 30 w(V) p Fo 52 w(is) e(a) h(random) g(subspace) i(p) s(oten) m (tial.) 41 b(An) 31 b(explicit) e(computation) f(\(see) 0 4610 y([JL2]\)) k(sho) m(ws) j(that) d(for) g(all) p Fl 31 w(n) p Fm 27 w(2) p Fo 29 w(\000) g(and) p Fl 33 w(E) p Fm 33 w(2) p Fo 29 w(in) m(t) p Fl 16 w(\033) p Fo 4 w(\() p Fl(H) p Ft 2035 4625 a(0) p Fo 2074 4610 a(\),) 1234 4830 y(Im) 16 b(\() p Fl(\016) p Fr 1448 4845 a(n) p Fm 1495 4830 a(j) p Fo(\() p Fl(H) p Ft 1642 4845 a(0) p Fm 1703 4830 a(\000) p Fl 23 w(E) p Fm 28 w(\000) p Fo 22 w(i0\)) p Fq 2117 4789 a(\000) p Ft(1) p Fl 2211 4830 a(\016) p Fr 2254 4845 a(n) p Fo 2301 4830 a(\)) p Fl 27 w(>) p Fo 28 w(0) p Fl(:) p Fo 0 5051 a(Hence,) 34 b(Prop) s(osition) d(1.2) h(and) h(Theorem) g(1.1) f(yield:) p 90 rotate dyy eop %%Page: 7 7 7 6 bop Fo 3731 100 a(7) p Fh 0 407 a(Theorem) 74 b(2.2) p Fy 49 w(Assume) 33 b(that) f(the) g(r) -5 b(andom) 31 b(variables) p Fm 31 w(f) p Fl(V) p Fo 22 w(\() p Fl(n) p Fo(\)) p Fm(g) p Fr 2488 422 a(n) p Fq(2) p Ft(\000) p Fy 2658 407 a(have) g(densities) h(and) f(let) p Fo 32 w(\002) p Fm 28 w(\032) p Fl 0 527 a(\033) p Fo 4 w(\() p Fl(H) p Ft 178 542 a(0) p Fo 217 527 a(\)) p Fy 45 w(b) -5 b(e) 44 b(a) g(Bor) -5 b(el) 44 b(set) g(of) g(p) -5 b(ositive) 44 b(L) -5 b(eb) g(esgue) 44 b(me) -5 b(asur) g(e.) 73 b(Then) 43 b(the) i(fol) 5 b(lowing) 43 b(statements) h(ar) -5 b(e) 0 648 y(e) g(quivalent:) p Fo 0 768 a(\(1\)) p Fy 35 w(F) e(or) p Fo 33 w(d) p Fl(P) p Fm 36 w(\012) p Fo 23 w(d) p Fl(E) p Fy 6 w(-a.e.) p Fo 44 w(\() p Fl(V) 5 b(;) 17 b(E) p Fo 6 w(\)) p Fm 28 w(2) p Fo 28 w(\012) p Fm 22 w(\002) p Fo 23 w(\002) p Fy 35 w(and) 34 b(for) h(al) 5 b(l) p Fl 34 w(n) p Fm 28 w(2) p Fo 28 w(\000) p Fy(,) p Fk 1087 911 a(X) p Fr 1065 1122 a(m) p Fq(2) p Ft(\000) p Ff 1218 1131 a(1) p Fm 1269 1005 a(j) p Fo(Im) 16 b(\() p Fl(\016) p Fr 1511 1020 a(n) p Fm 1558 1005 a(j) p Fo(\() p Fl(H) p Fm 30 w(\000) p Fl 22 w(E) p Fm 28 w(\000) p Fo 23 w(i0\)) p Fq 2149 964 a(\000) p Ft(1) p Fl 2242 1005 a(\016) p Fr 2285 1020 a(m) p Fo 2352 1005 a(\)) p Fm(j) p Ft 2418 964 a(2) p Fl 2485 1005 a(<) p Fm 27 w(1) p Fl(:) p Fo 0 1324 a(\(2\)) p Fy 35 w(F) -7 b(or) p Fl 33 w(P) p Fy 14 w(-a.e.) p Fl 44 w(V) p Fy 56 w(ther) i(e) 35 b(is) g(a) g(dense) f(set) g(of) p Fl 35 w(\036) p Fm 27 w(2) p Fo 29 w(Ran) p Fh 16 w(1) p Ft 2160 1339 a(\002) p Fo 2219 1324 a(\() p Fl(H) p Fo 8 w(\)) p Fy 34 w(such) h(that) p Fk 1403 1457 a(Z) p Fg 1458 1682 a(R) p Fm 1527 1592 a(k) p Fh(1) p Ft 1633 1607 a(1) p Fo 1672 1592 a(e) p Fq 1715 1551 a(\000) p Ft(i) p Fr(tH) p Fl 1883 1592 a(\036) p Fm(k) p Ft 1991 1551 a(2) p Fo 2030 1592 a(d) p Fl(t) 28 b(<) p Fm 27 w(1) p Fl(:) p Fo 0 1867 a(\(3\)) p Fy 35 w(The) 34 b(wave) g(op) -5 b(er) g(ators) p Fl 34 w(W) p Fq 1127 1830 a(\006) p Fy 1221 1867 a(ar) g(e) p Fl 34 w(P) p Fy 14 w(-a.s.) 44 b(c) -5 b(omplete) 34 b(on) p Fo 35 w(\002) p Fy(.) p Fo 146 2190 a(The) g(follo) m(wing) c (corollary) g(follo) m(ws) i(easily) g(from) f(Theorem) i(2.2) f(and) h (Prop) s(osition) d(3.1) j(in) e([JL1].) p Fh 0 2419 a(Corollary) 36 b(2.3) p Fy 49 w(Assume) d(that) h(the) g(r) -5 b(andom) 33 b(variables) p Fm 32 w(f) p Fl(V) p Fo 22 w(\() p Fl(n) p Fo(\)) p Fm(g) p Fr 2481 2434 a(n) p Fq(2) p Ft(\000) p Fy 2652 2419 a(have) g(densities) g(and) g(let) p Fo 33 w(\002) p Fm 28 w(\032) p Fl 0 2539 a(\033) p Fo 4 w(\() p Fl(H) p Ft 178 2554 a(0) p Fo 217 2539 a(\)) p Fy 29 w(b) -5 b(e) 28 b(a) g(Bor) -5 b(el) 28 b(set) g(of) h(p) -5 b(ositive) 28 b(L) -5 b(eb) g(esgue) 27 b(me) -5 b(asur) g(e.) 43 b(Assume) 28 b(that) h(for) p Fo 28 w(d) p Fl(P) p Fm 22 w(\012) p Fo 8 w(d) p Fl(E) p Fy 6 w(-a.e.) p Fo 43 w(\() p Fl(V) 5 b(;) 17 b(E) p Fo 6 w(\)) p Fm 27 w(2) p Fo 0 2659 a(\012) p Fm 23 w(\002) p Fo 22 w(\002) p Fy 35 w(and) 34 b(for) h(al) 5 b(l) p Fl 34 w(n) p Fm 28 w(2) p Fo 28 w(\000) p Fy(,) p Fo 1010 2891 a(lim) 17 b(inf) p Fr 1095 2954 a(\017) p Fq(#) p Ft(0) p Fk 1301 2797 a(X) p Fr 1297 3008 a(m) p Fq(2) p Ft(\000) p Fm 1467 2891 a(j) p Fo(\() p Fl(\016) p Fr 1576 2906 a(n) p Fm 1622 2891 a(j) p Fo(\() p Fl(H) p Fm 30 w(\000) p Fl 22 w(E) p Fm 29 w(\000) p Fo 22 w(i) p Fl(\017) p Fo(\)) p Fq 2203 2850 a(\000) p Ft(1) p Fl 2297 2891 a(\016) p Fr 2340 2906 a(m) p Fo 2407 2891 a(\)) p Fm(j) p Ft 2473 2850 a(2) p Fl 2539 2891 a(<) p Fm 28 w(1) p Fl(:) p Fo 809 w(\(2.4\)) p Fy 0 3207 a(Then) 34 b(the) h(wave) f(op) -5 b(er) g(ators) p Fl 34 w(W) p Fq 1184 3170 a(\006) p Fy 1278 3207 a(ar) g(e) p Fl 35 w(P) p Fy 14 w(-a.s.) 43 b(c) -5 b(omplete) 34 b(on) p Fo 35 w(\002) p Fy(.) p Fo 146 3435 a(The) 29 b(condition) c(\(2.4\)) i(should) g(b) s(e) h (compared) f(with) g(the) h(w) m(ell-kno) m(wn) f(Simon-W) -8 b(ol\013) 24 b(lo) s(calization) 0 3555 y(criterion) 31 b([SW].) 44 b(F) -8 b(or) 31 b(comparison,) g(w) m(e) i(also) e(recall) g(the) h(follo) m(wing) e(result) i(pro) m(v) m(en) h(in) f([JM1,) g (JM2]:) 0 3676 y(if) p Fm 29 w(f) p Fl(V) p Fo 21 w(\() p Fl(n) p Fo(\)) p Fm(g) p Fr 399 3691 a(n) p Fq(2) p Ft(\000) p Fo 567 3676 a(ha) m(v) m(e) g(densities,) f(\002) p Fm 27 w(\032) p Fi 28 w(R) p Fm 29 w(n) p Fl 17 w(\033) p Fo 4 w(\() p Fl(H) p Ft 1755 3691 a(0) p Fo 1794 3676 a(\),) f(and) h(for) e(d) p Fl(P) p Fm 31 w(\012) p Fo 18 w(d) p Fl(E) p Fo 6 w(-a.e.) 42 b(\() p Fl(V) 5 b(;) 17 b(E) p Fo 6 w(\)) p Fm 28 w(2) p Fo 28 w(\012) p Fm 18 w(\002) p Fo 17 w(\002) 30 b(and) h(all) p Fl 0 3796 a(n) p Fm 28 w(2) p Fo 28 w(\000,) 1010 3928 y(lim) 17 b(inf) p Fr 1095 3991 a(\017) p Fq(#) p Ft(0) p Fk 1301 3834 a(X) p Fr 1297 4045 a(m) p Fq(2) p Ft(\000) p Fm 1467 3928 a(j) p Fo(\() p Fl(\016) p Fr 1576 3943 a(n) p Fm 1622 3928 a(j) p Fo(\() p Fl(H) p Fm 30 w(\000) p Fl 22 w(E) p Fm 29 w(\000) p Fo 22 w(i) p Fl(\017) p Fo(\)) p Fq 2203 3887 a(\000) p Ft(1) p Fl 2297 3928 a(\016) p Fr 2340 3943 a(m) p Fo 2407 3928 a(\)) p Fm(j) p Ft 2473 3887 a(2) p Fl 2539 3928 a(<) p Fm 28 w(1) p Fl(;) p Fo 809 w(\(2.5\)) 0 4191 y(then) 32 b(the) g(sp) s(ectrum) f (of) p Fl 31 w(H) p Fo 39 w(in) g(\002) g(is) p Fl 31 w(P) p Fo 14 w(-a.s.) 42 b(pure) 32 b(p) s(oin) m(t.) 42 b(If) 32 b(supp) p Fl 1 w(\026) p Fr 2522 4206 a(n) p Fo 2596 4191 a(=) p Fi 27 w(R) p Fo 43 w(for) e(at) h(least) g(one) p Fl 32 w(n) p Fo(,) h(then) 0 4311 y(the) h(condition) e(\(2.5\)) h(is) g (also) g(necessary) j(for) p Fl 32 w(H) p Fo 40 w(to) d(ha) m(v) m(e) p Fl 34 w(P) p Fo 14 w(-a.s.) 43 b(pure) 33 b(p) s(oin) m(t) f(sp) s (ectrum) h(in) e(\002.) 146 4431 y(W) -8 b(e) 31 b(no) m(w) h(discuss) g (an) e(application) f(of) h(Theorem) h(2.2.) 42 b(F) -8 b(or) 30 b(simplicit) m(y) -8 b(,) 29 b(w) m(e) j(assume) f(that) f (all) f(the) 0 4552 y(measures) p Fl 33 w(\026) p Fr 477 4567 a(n) p Fo 557 4552 a(are) j(the) h(same) g(and) f(equal) h(to) p Fl 32 w(\026) p Fo(,) f(and) h(that) f(d) p Fl(\026) p Fo 28 w(=) p Fl 27 w(p) p Fo(\() p Fl(x) p Fo(\)d) p Fl(x) p Fo(.) p Fh 0 4780 a(Theorem) 74 b(2.4) p Fy 49 w(Assume) 39 b(that) p Fl 38 w(d) p Fo 25 w(+) 24 b(1) 34 b(=) g(2) p Fy(.) 54 b(L) -5 b(et) p Fl 39 w(U) p Ft 2014 4795 a(p) r(er) p Fy 2157 4780 a(b) g(e) 37 b(a) i(p) -5 b(erio) g(dic) 37 b(p) -5 b(otential) 38 b(supp) -5 b(orte) g(d) 38 b(on) p Fo 38 w(\000) p Fy 0 4900 a(and) p Fl 39 w(H) p Fo 43 w(=) p Fl 36 w(H) p Ft 511 4915 a(0) p Fo 575 4900 a(+) p Fl 26 w(U) p Ft 743 4915 a(p) r(er) p Fo 873 4900 a(+) p Fl 25 w(\025V) p Fy 21 w(,) p Fl 41 w(V) p Fm 57 w(2) p Fo 36 w(\012) p Fy(,) j(wher) -5 b(e) p Fl 39 w(\025) p Fy 39 w(is) 39 b(a) g(r) -5 b(e) g(al) 39 b(c) -5 b(onstant.) 58 b(Assume) 39 b(that) p Fm 40 w(h) p Fl(x) p Fm(i) p Fr 3448 4864 a(\013) p Fl 3497 4900 a(p) p Fo(\() p Fl(x) p Fo(\)) p Fm 36 w(2) p Fl 0 5021 a(L) p Ft 66 4985 a(1) p Fo 106 5021 a(\() p Fi(R) p Fo 5 w(\)) p Fm 27 w(\\) p Fl 22 w(L) p Fq 429 4985 a(1) p Fo 505 5021 a(\() p Fi(R) p Fo 4 w(\)) p Fy 41 w(for) c(some) p Fl 33 w(\013) 29 b(>) p Fo 27 w(2) p Fl(=) p Fo(3) p Fy(.) 44 b(Then) 34 b(ther) -5 b(e) 35 b(is) f(a) h(c) -5 b(onstant) p Fo 34 w(\003) p Fl 27 w(>) p Fo 28 w(0) p Fy 34 w(such) 35 b(that) g(for) p Fm 34 w(j) p Fl(\025) p Fm(j) p Fl 27 w(>) p Fo 28 w(\003) p Fy(,) 0 5141 y(the) g(wave) f(op) -5 b(er) g(ators) p Fl 34 w(W) p Fq 930 5105 a(\006) p Fy 1024 5141 a(ar) g(e) p Fl 35 w(P) p Fy 14 w(-a.s.) 43 b(c) -5 b(omplete.) p 90 rotate dyy eop %%Page: 8 8 8 7 bop Fo 3731 100 a(8) p Fh 0 407 a(Theorem) 74 b(2.5) p Fy 49 w(Assume) 44 b(that) p Fl 44 w(d) p Fo 29 w(+) 28 b(1) 44 b(=) g(2) p Fy 44 w(and) f(let) p Fl 44 w(H) p Fo 52 w(=) p Fl 44 w(H) p Ft 2453 422 a(0) p Fo 2521 407 a(+) p Fl 29 w(\025V) p Fy 21 w(,) p Fl 46 w(V) p Fm 66 w(2) p Fo 44 w(\012) p Fy(.) 72 b(Assume) 44 b(that) p Fm 0 527 a(h) p Fl(x) p Fm(i) p Fr 133 491 a(\013) p Fl 182 527 a(p) p Fo(\() p Fl(x) p Fo(\)) p Fy 35 w(is) 33 b(in) p Fl 34 w(L) p Ft 685 491 a(1) p Fo 725 527 a(\() p Fi(R) p Fo 5 w(\)) p Fy 40 w(for) g(some) p Fl 33 w(\013) c(>) p Fo 27 w(2) p Fl(=) p Fo(3) p Fy 34 w(and) k(in) p Fl 34 w(L) p Fq 2057 491 a(1) p Fo 2132 527 a(\() p Fi(R) p Fo 5 w(\)) p Fy 39 w(for) h(some) p Fl 33 w(\013) 29 b(>) p Fo 27 w(5) p Fl(=) p Fo(3) p Fy(.) 44 b(Then) 33 b(ther) -5 b(e) 34 b(is) g(a) 0 648 y(c) -5 b(onstant) p Fo 35 w(\003) p Fl 27 w(>) p Fo 27 w(0) p Fy 35 w(such) 35 b(that) g(for) p Fm 35 w(j) p Fl(\025) p Fm(j) p Fl 27 w(<) p Fo 27 w(\003) p Fy 35 w(the) g(wave) f(op) -5 b(er) g(ators) p Fl 34 w(W) p Fq 2524 611 a(\006) p Fy 2618 648 a(ar) g(e) p Fl 34 w(P) p Fy 14 w(-a.s.) 44 b(c) -5 b(omplete.) p Fo 0 876 a(In) 40 b([JM1]) g(it) f(w) m(as) h (sho) m(wn) i(that) d(under) i(the) f(conditions) e(of) i(these) h (theorems) f(the) g(sp) s(ectrum) g(of) p Fl 39 w(H) p Fo 0 996 a(outside) p Fl 36 w(\033) p Fo 4 w(\() p Fl(H) p Ft 518 1011 a(0) p Fo 557 996 a(\)) c(is) p Fl 35 w(P) p Fo 14 w(-a.s.) 53 b(dense) 37 b(pure) f(p) s(oin) m(t) f(with) h(exp) s (onen) m(tially) f(deca) m(ying) h(eigenfunction) f(\(for) 0 1117 y(related) d(results) h(see) h([AM,) f(G,) f(JM2) q(]\).) 146 1237 y(Assume) i(no) m(w) g(that) p Fl 33 w(U) p Ft 990 1252 a(0) p Fo 1059 1237 a(=) 28 b(0) 33 b(and) h(let) e(supp) p Fl 1 w(\026) p Fo 33 w(b) s(e) h(the) h(supp) s(ort) g(of) e(the) i (probabilit) m(y) d(measure) p Fl 34 w(\026) p Fo(.) 0 1357 y(Then,) j(for) p Fl 32 w(P) p Fo 14 w(-a.e.) p Fl 42 w(V) p Fo 22 w(,) p Fl 651 1577 a(\033) p Fo 4 w(\() p Fl(H) p Fo 8 w(\)) 27 b(=) p Fl 27 w(\033) p Fo 4 w(\() p Fl(H) p Ft 1183 1592 a(0) p Fo 1223 1577 a(\)) p Fm 22 w([) 22 b(f) p Fo([2) p Fl(d;) p Fo 17 w(2) p Fl(d) p Fo(]) f(+) p Fl 22 w(x) p Fo 22 w(+) p Fl 23 w(x) p Fq 2069 1536 a(\000) p Ft(1) p Fo 2191 1577 a(:) p Fl 28 w(x) p Fm 28 w(2) p Fo 28 w(supp) p Fl 1 w(\026;) p Fm 50 w(j) p Fl(x) p Fm(j) 27 b(\025) p Fo 28 w(1) p Fm(g) p Fl(;) p Fo 0 1797 a(see) 49 b([JL1].) 89 b(F) -8 b(or) 46 b(example,) 51 b(if) p Fl 47 w(\026) p Fo 47 w(is) c(Gaussian,) k(then) p Fl 49 w(\033) p Fo 4 w(\() p Fl(H) p Fo 8 w(\)) i(=) p Fi 53 w(R) p Fl 58 w(P) p Fo 14 w(-a.s.) 88 b(\(In) 48 b(this) g(case,) k(the) 0 1918 y(resonan) m(t) 28 b(sp) s(ectrum) g(of) p Fl 27 w(H) p Fo 35 w(is) f(also) f(equal) h(to) p Fi 27 w(R) p Fl 39 w(P) p Fo 14 w(-a.s.\)) 41 b(Th) m(us,) 30 b(Theorems) e(2.4) f (and) g(2.5) g(pro) m(vide) h(\(to) 0 2038 y(the) 33 b(b) s(est) h(of) f(our) f(kno) m(wledge\)) i(the) g(\014rst) f (non-trivial) d(examples) j(of) g(Anderson) h(t) m(yp) s(e) g (Hamiltonians) p Fl 0 2159 a(H) p Fo 35 w(=) p Fl 28 w(H) p Ft 301 2174 a(0) p Fo 351 2159 a(+) p Fl 11 w(V) p Fo 49 w(whic) m(h) 28 b(ha) m(v) m(e) p Fl 29 w(P) p Fo 14 w(-a.s.) 41 b(dense) 29 b(p) s(oin) m(t) d(sp) s(ectrum) i (outside) p Fl 27 w(\033) p Fo 4 w(\() p Fl(H) p Ft 2770 2174 a(0) p Fo 2809 2159 a(\),) h(purely) e(a.c.) 42 b(sp) s(ectrum) 0 2279 y(in) p Fl 32 w(\033) p Fo 4 w(\() p Fl(H) p Ft 292 2294 a(0) p Fo 331 2279 a(\),) 33 b(and) f(the) h (scattering) g(b) s(et) m(w) m(een) p Fl 34 w(H) p Fo 40 w(and) p Fl 33 w(H) p Ft 2005 2294 a(0) p Fo 2077 2279 a(is) f(complete.) 146 2399 y(Theorems) 37 b(2.4) e(and) h(2.5) g (are) g(closely) f(related) h(to) f(the) h(results) h(of) e([JL3].) 53 b(There) 38 b(it) c(w) m(as) j(sho) m(wn) 0 2520 y(that) 32 b(under) i(the) f(same) f(conditions,) g(for) g(all) p Fl 30 w(\036) p Fm 28 w(2) c(H) p Fo 34 w(and) p Fl 32 w(R) p Fm 29 w(\025) p Fo 28 w(0,) 1015 2802 y(lim) p Fr 987 2864 a(T) p Fq 10 w(!1) p Fo 1241 2734 a(1) p 1206 2779 120 4 v 1206 2870 a(2) p Fl(T) p Fk 1352 2666 a(Z) p Fr 1451 2692 a(T) p Fq 1407 2892 a(\000) p Fr(T) p Fi 1534 2802 a(E) p Fk 1611 2721 a(\000) p Fm 1662 2802 a(k) p Fh(1) p Fr 1768 2817 a(R) p Fo 1826 2802 a(e) p Fq 1869 2761 a(\000) p Ft(i) p Fr(tH) p Fh 2036 2802 a(1) p Fr 2092 2817 a(\033) p Ft 2 w(\() p Fr(H) p Ff 2219 2826 a(0) p Ft 2255 2817 a(\)) p Fl 2287 2802 a(\036) p Fm(k) p Ft 2395 2761 a(2) p Fk 2434 2721 a(\001) p Fo 2496 2802 a(d) p Fl(t) p Fo 28 w(=) g(0) p Fl(:) p Fo 786 w(\(2.6\)) 0 3081 y(\() p Fi(E) p Fo 45 w(stands) f(for) f(the) h (exp) s(ectation\).) 42 b(A) 27 b(consequence) i(of) d(\(2.6\)) g(is) g (that) h(the) g(op) s(erators) p Fl 26 w(H) p Fo 34 w(ha) m(v) m(e) p Fl 28 w(P) p Fo 14 w(-a.s.) 0 3202 y(no) 32 b(surface) i(sp) s(ectrum) f (in) p Fl 32 w(\033) p Fo 4 w(\() p Fl(H) p Ft 1182 3217 a(0) p Fo 1221 3202 a(\),) f(see) i([KP,) f(JL3,) g(JMP].) 146 3322 y(The) 49 b(pro) s(of) e(of) h(\(2.6\)) f(is) h(based) h(on) f (the) g(follo) m(wing) d(estimate) i(pro) m(v) m(en) j(in) d([JL3]:) 74 b(under) 49 b(the) 0 3442 y(conditions) 32 b(of) g(either) g(Theorem) h (2.4) f(or) g(Theorem) h(2.5,) g(for) p Fl 32 w(n) p Fm 27 w(2) p Fo 29 w(\000,) p Fl 32 w(R) p Fm 29 w(\025) p Fo 28 w(0) f(and) h(2) p Fl(=) p Fo(3) p Fl 27 w(<) 28 b(s) f(<) p Fo 28 w(1,) 927 3750 y(sup) p Fr 856 3832 a(E) p Fq 4 w(2) p Fg(R) p Fr(;\017) p Fq(6) p Ft(=0) p Fi 1162 3750 a(E) p Fk 1239 3579 a( ) 1354 3655 y(X) p Fr 1324 3866 a(m) p Fq(2) p Ft(\000) p Fd 1477 3877 a(R) p Fm 1544 3750 a(j) p Fo(\() p Fl(\016) p Fr 1653 3765 a(m) p Fm 1719 3750 a(j) p Fo(\() p Fl(H) p Fm 30 w(\000) p Fl 22 w(E) p Fm 28 w(\000) p Fo 23 w(i) p Fl(\017) p Fo(\)) p Fq 2300 3708 a(\000) p Ft(1) p Fl 2394 3750 a(\016) p Fr 2437 3765 a(n) p Fo 2484 3750 a(\)) p Fm(j) p Fr 2550 3708 a(s) p Fk 2586 3579 a(!) p Fl 2693 3750 a(<) p Fm 27 w(1) p Fl(:) p Fo 0 4070 a(This) 33 b(estimate) e(and) i (F) -8 b(atou's) 32 b(lemma) f(yield) h(that) g(for) g(d) p Fl(P) p Fm 36 w(\012) p Fo 22 w(d) p Fl(E) p Fo 39 w(a.e.) 44 b(\() p Fl(V) 5 b(;) 17 b(E) p Fo 6 w(\)) p Fm 27 w(2) p Fo 28 w(\012) p Fm 23 w(\002) p Fi 23 w(R) p Fo 5 w(,) p Fk 1075 4230 a( ) 1184 4306 y(X) p Fr 1154 4518 a(m) p Fq(2) p Ft(\000) p Fd 1307 4529 a(R) p Fm 1374 4401 a(j) p Fo(\() p Fl(\016) p Fr 1483 4416 a(m) p Fm 1550 4401 a(j) p Fo(\() p Fl(H) p Fm 29 w(\000) p Fl 22 w(E) p Fm 29 w(\000) p Fo 22 w(i0\)) p Fq 2140 4360 a(\000) p Ft(1) p Fl 2233 4401 a(\016) p Fr 2276 4416 a(n) p Fo 2324 4401 a(\)) p Fm(j) p Ft 2390 4360 a(2) p Fk 2429 4230 a(!) p Fr 2507 4253 a(s=) p Ft(2) p Fm 1086 4788 a(\024) p Fk 1221 4693 a(X) p Fr 1191 4905 a(m) p Fq(2) p Ft(\000) p Fd 1344 4916 a(R) p Fm 1412 4788 a(j) p Fo(\() p Fl(\016) p Fr 1521 4803 a(m) p Fm 1587 4788 a(j) p Fo(\() p Fl(H) p Fm 29 w(\000) p Fl 23 w(E) p Fm 28 w(\000) p Fo 23 w(i0\)) p Fq 2178 4747 a(\000) p Ft(1) p Fl 2271 4788 a(\016) p Fr 2314 4803 a(n) p Fo 2361 4788 a(\)) p Fm(j) p Fr 2427 4747 a(s) p Fl 2491 4788 a(<) p Fm 28 w(1) p Fl(:) p Fo 0 5110 a(By) 36 b(Theorem) g(2.2,) g(the) g(last) e (estimate) h(implies) e(that) i(the) h(w) m(a) m(v) m(e) h(op) s (erators) p Fl 36 w(W) p Fq 3012 5074 a(\006) p Fo 3106 5110 a(are) p Fl 35 w(P) p Fo 14 w(-a.s.) 52 b(com-) 0 5231 y(plete,) 33 b(and) f(Theorems) i(2.4) e(and) g(2.5) h(follo) m (w.) p 90 rotate dyy eop %%Page: 9 9 9 8 bop Fo 3731 100 a(9) p Fn 0 407 a(3) 161 b(Pro) t(ofs) p Fc 0 655 a(3.1) 135 b(Preliminaries) p Fh 0 840 a(Pro) s(of) 36 b(of) h(Prop) s(osition) d(1.2.) p Fo 44 w(By) e(the) g(remark) f(b) s (efore) h(Theorem) g(1.1,) g(w) m(e) g(ma) m(y) g(assume) g(that) p Fm 31 w(H) p Fo 0 960 a(is) 38 b(spanned) i(b) m(y) p Fl 40 w(H) p Fo 46 w(and) p Fm 38 w(f) p Fl(\016) p Fr 1047 975 a(n) p Fm 1094 960 a(g) p Fr 1144 975 a(n) p Fq(2) p Ft(\000) p Fo 1321 960 a(for) e(all) p Fl 36 w(V) p Fo 22 w(.) 62 b(One) 39 b(also) e(easily) h(v) m(eri\014es) i (that) e(the) h(subspaces) p Fm 41 w(H) p Fr 3732 975 a(n) p Fo 0 1080 a(and) p Fm 36 w(H) p Fr 277 1095 a(m) p Fo 379 1080 a(are) d(not) f(orthogonal) f(for) h(all) p Fl 33 w(V) p Fo 57 w(and) p Fl 36 w(n;) 17 b(m) p Fm 33 w(2) 33 b(G) p Fo 6 w(.) 53 b(The) 37 b(condition) d(\(1.2\)) h (ensures) i(that) f(\002) 0 1201 y(is) k(con) m(tained) g(in) f(an) h (essen) m(tial) h(supp) s(ort) f(of) g(the) g(absolutely) g(con) m(tin) m(uous) h(sp) s(ectrum) f(of) p Fl 40 w(H) p Ft 3479 1216 a(0) p Fo 3518 1201 a(.) 67 b(The) 0 1321 y(existence) 30 b(of) e(w) m(a) m(v) m(e) i(op) s(erators) e(implies) e(that) j(the) f (op) s(erators) p Fl 29 w(H) p Fb 35 w(\026) p Fo 27 w(Ran) p Fl 16 w(W) p Fq 2787 1285 a(\006) p Fo 2874 1321 a(and) p Fl 29 w(H) p Ft 3141 1336 a(0) p Fb 3208 1321 a(\026) p Fo 27 w(Ran) p Fh 16 w(1) p Ft 3524 1336 a(\002) p Fo 3583 1321 a(\() p Fl(H) p Ft 3702 1336 a(0) p Fo 3742 1321 a(\)) 0 1442 y(are) 48 b(unitarily) d(equiv) -5 b(alen) m(t.) 89 b(Hence,) 53 b(with) 47 b(probabilit) m(y) f(one,) 52 b(\002) c(is) f(con) m(tained) h(in) f(an) g(essen) m(tial) 0 1562 y(supp) s(ort) e(of) e(the) i(absolutely) f(con) m(tin) m(uous) h (sp) s(ectrum) g(of) p Fl 43 w(H) p Fo 52 w(and) g(the) f(prop) s (osition) f(follo) m(ws) g(from) 0 1682 y(Corollaries) 30 b(1.1.1) i(and) h(1.1.3) f(in) g([JL2].) p Fa 43 w(2) p Fo 146 1848 a(The) i(next) f(lemma) e(holds) h(for) g(an) g(arbitrary) g (subspace) j(p) s(oten) m(tial) p Fl 30 w(V) p Fo 22 w(.) p Fh 0 2060 a(Lemma) i(3.1) p Fy 49 w(F) -7 b(or) 45 b(any) p Fl 46 w(m;) 17 b(n) p Fm 48 w(2) p Fo 48 w(\000) p Fy(,) 48 b(the) e(sp) -5 b(e) g(ctr) g(al) 46 b(me) -5 b(asur) g(e) p Fl 45 w(\027) p Fr 2436 2075 a(\016) p Fd 2467 2083 a(m) p Fr 2526 2075 a(;\016) p Fd 2577 2083 a(n) p Fy 2669 2060 a(for) p Fl 46 w(H) p Fy 53 w(and) p Fl 45 w(\016) p Fr 3213 2075 a(m) p Fy 3280 2060 a(,) p Fl(\016) p Fr 3353 2075 a(n) p Fy 3400 2060 a(,) 48 b(is) e(r) -5 b(e) g(al-) 0 2181 y(value) g(d.) p Fh 0 2393 a(Pro) s(of.) p Fo 64 w(Let) p Fm 33 w(C) p Fo 6 w(\() p Fi(R) p Fo 5 w(\)) 38 b(b) s(e) 32 b(the) h(set) g(of) f(all) e(real-v) -5 b(alued,) 31 b(b) s(ounded,) i(con) m(tin) m(uous) g(functions) f(on) p Fi 32 w(R) p Fo 5 w(.) 50 b(The) 0 2513 y(measure) p Fl 30 w(\027) p Fr 425 2528 a(\016) p Fd 456 2536 a(m) p Fr 515 2528 a(;\016) p Fd 566 2536 a(n) p Fo 643 2513 a(is) 29 b(real-v) -5 b(alued) 28 b(i\013) h(for) g(all) p Fl 28 w(f) p Fm 39 w(2) f(C) p Fo 6 w(\() p Fi(R) p Fo 5 w(\),) 36 b(\() p Fl(\016) p Fr 2149 2528 a(m) p Fm 2216 2513 a(j) p Fl(f) p Fo 11 w(\() p Fl(H) p Fo 8 w(\)) p Fl(\016) p Fr 2511 2528 a(n) p Fo 2557 2513 a(\)) 30 b(is) f(a) h(real) f(n) m(um) m(b) s(er.) 43 b(Note) 30 b(\014rst) 0 2634 y(that) g(for) g(an) m(y) i(p) s(ositiv) m(e) e(in) m (teger) p Fl 30 w(k) p Fo 3 w(,) h(\() p Fl(\016) p Fr 1411 2649 a(m) p Fm 1478 2634 a(j) p Fo(\() p Fl(H) p Ft 1625 2649 a(0) p Fo 1682 2634 a(+) p Fl 18 w(V) p Fo 22 w(\)) p Fr 1893 2597 a(k) p Fl 1935 2634 a(\016) p Fr 1978 2649 a(n) p Fo 2025 2634 a(\)) g(is) f(a) g(real) g(n) m(um) m (b) s(er.) 43 b(It) 31 b(follo) m(ws) e(that) i(for) f(an) m(y) 0 2754 y(p) s(olynomial) p Fl 33 w(p) p Fo 37 w(with) 37 b(real) f(co) s(e\016cen) m(ts,) k(\() p Fl(\016) p Fr 1595 2769 a(m) p Fm 1661 2754 a(j) p Fl(p) p Fo(\() p Fl(H) p Fo 8 w(\)) p Fl(\016) p Fr 1946 2769 a(n) p Fo 1993 2754 a(\)) c(is) h(a) f(real) g(n) m(um) m(b) s(er.) 57 b(Assume) 38 b(no) m(w) f(that) g(the) 0 2874 y(p) s(oten) m(tial) p Fl 25 w(V) p Fo 48 w(is) 26 b(b) s(ounded.) 42 b(Then) p Fl 28 w(\033) p Fo 4 w(\() p Fl(H) p Fo 8 w(\)) 26 b(is) g(a) g (compact) g(set) i(and,) f(b) m(y) h(an) e(appro) m(ximation) f (argumen) m(t,) 0 2995 y(for) 32 b(all) p Fl 31 w(f) p Fm 38 w(2) c(C) p Fo 6 w(\() p Fi(R) p Fo 5 w(\),) 39 b(\() p Fl(\016) p Fr 812 3010 a(m) p Fm 878 2995 a(j) p Fl(f) p Fo 11 w(\() p Fl(H) p Fo 8 w(\)) p Fl(\016) p Fr 1173 3010 a(n) p Fo 1219 2995 a(\)) 33 b(is) f(a) g(real) g(n) m(um) m(b) s(er.) 146 3115 y(If) p Fl 32 w(V) p Fo 53 w(is) f(un) m(b) s (ounded,) j(let) p Fl 31 w(V) p Fr 1178 3130 a(`) p Fo 1211 3115 a(\() p Fl(j) p Fo 6 w(\)) 27 b(=) p Fl 28 w(V) p Fo 21 w(\() p Fl(j) p Fo 6 w(\)) 32 b(if) p Fm 30 w(j) p Fl(j) p Fm 6 w(j) 27 b(\024) p Fl 29 w(`) p Fo(,) k(otherwise) p Fl 33 w(V) p Fr 2609 3130 a(`) p Fo 2642 3115 a(\() p Fl(j) p Fo 6 w(\)) c(=) h(0.) 42 b(Set) p Fl 33 w(H) p Fr 3262 3130 a(`) p Fo 3322 3115 a(=) p Fl 28 w(H) p Ft 3507 3130 a(0) p Fo 3566 3115 a(+) p Fl 21 w(V) p Fr 3720 3130 a(`) p Fo 3752 3115 a(.) 0 3236 y(Then) p Fl 45 w(H) p Fr 347 3251 a(`) p Fm 427 3236 a(!) p Fl 47 w(H) p Fo 51 w(in) h(the) h(strong) g(resolv) m (en) m(t) i(sense) f(and) f(this) g(implies) d(that) j(for) g(an) m(y) p Fl 44 w(f) p Fm 58 w(2) j(C) p Fo 6 w(\() p Fi(R) p Fo 5 w(\),) p Fl 0 3356 a(f) p Fo 11 w(\() p Fl(H) p Fr 178 3371 a(`) p Fo 210 3356 a(\)) p Fm 28 w(!) p Fl 27 w(f) p Fo 11 w(\() p Fl(H) p Fo 8 w(\)) 32 b(strongly) -8 b(.) 43 b(Hence,) 1200 3562 y(\() p Fl(\016) p Fr 1281 3577 a(m) p Fm 1348 3562 a(j) p Fl(f) p Fo 11 w(\() p Fl(H) p Fo 8 w(\)) p Fl(\016) p Fr 1643 3577 a(n) p Fo 1689 3562 a(\)) 27 b(=) 45 b(lim) p Fr 1858 3624 a(`) p Fq(!1) p Fo 2028 3562 a(\() p Fl(\016) p Fr 2109 3577 a(m) p Fm 2175 3562 a(j) p Fl(f) p Fo 11 w(\() p Fl(H) p Fr 2381 3577 a(`) p Fo 2414 3562 a(\)) p Fl(\016) p Fr 2495 3577 a(n) p Fo 2542 3562 a(\)) 0 3797 y(is) 32 b(a) g(real) g(n) m (um) m(b) s(er.) p Fa 44 w(2) p Fo 146 3963 a(One) h(consequence) j(of) c(this) g(lemma) f(is) h(the) h(iden) m(tit) m(y) 718 4169 y(Im) 16 b(\() p Fl(\016) p Fr 932 4184 a(m) p Fm 999 4169 a(j) p Fo(\() p Fl(H) p Fm 29 w(\000) p Fl 23 w(E) p Fm 28 w(\000) p Fo 22 w(i0\)) p Fq 1589 4128 a(\000) p Ft(1) p Fl 1682 4169 a(\016) p Fr 1725 4184 a(n) p Fo 1773 4169 a(\)) 27 b(=) h(Im) 16 b(\() p Fl(\016) p Fr 2156 4184 a(n) p Fm 2203 4169 a(j) p Fo(\() p Fl(H) p Fm 29 w(\000) p Fl 23 w(E) p Fm 28 w(\000) p Fo 23 w(i0\)) p Fq 2794 4128 a(\000) p Ft(1) p Fl 2887 4169 a(\016) p Fr 2930 4184 a(m) p Fo 2997 4169 a(\)) p Fl(;) p Fo 0 4375 a(whic) m(h) 33 b(w) m(e) h(will) c(often) j(use) g(in) f(the) h (sequel.) 146 4495 y(W) -8 b(e) 33 b(also) f(recall) f(the) i(follo) m (wing) d(w) m(ell-kno) m(wn) j(result) f(\(see,) i(e.g.,) f([S]\).) p Fh 0 4686 a(Lemma) k(3.2) p Fy 49 w(L) -5 b(et) p Fl 28 w(\026) p Fy 26 w(b) g(e) 27 b(a) g(\014nite) g(r) -5 b(e) g(gular) 27 b(c) -5 b(omplex) 25 b(me) -5 b(asur) g(e) 27 b(and) p Fo 26 w(d) p Fl(\026) p Fo 28 w(=) p Fl 27 w(f) p Fo 11 w(d) p Fl(E) p Fo 11 w(+) 5 b(d) p Fl(\026) p Ft 3129 4701 a(sing) p Fy 3283 4686 a(its) 27 b(L) -5 b(eb) g(esgue) 0 4807 y(de) g(c) g(omp) g(osition.) 43 b(Then) 34 b(for) h(L) -5 b(eb) g(esgue) 34 b(a.e.) p Fl 44 w(E) p Fm 34 w(2) p Fi 28 w(R) p Fy 5 w(,) p Fl 1187 5062 a(f) p Fo 11 w(\() p Fl(E) p Fo 6 w(\)) 27 b(=) g(lim) p Fr 1549 5125 a(\017) p Fq(#) p Ft(0) p Fl 1683 5062 a(\031) p Fq 1742 5021 a(\000) p Ft(1) p Fk 1852 4926 a(Z) p Fg 1908 5152 a(R) p Fl 2129 4995 a(\017) p Fo(d) p Fl(\026) p Fo(\() p Fl(x) p Fo(\)) p 1986 5039 570 4 v 1986 5130 a(\() p Fl(x) p Fm 23 w(\000) p Fl 23 w(E) p Fo 6 w(\)) p Ft 2318 5102 a(2) p Fo 2379 5130 a(+) p Fl 22 w(\017) p Ft 2516 5102 a(2) p Fl 2566 5062 a(:) p 90 rotate dyy eop %%Page: 10 10 10 9 bop Fo 3682 100 a(10) 0 407 y(Com) m(bining) 31 b(the) i(last) f(t) m(w) m(o) h(lemmas) e(w) m(e) i(deriv) m(e) p Fh 0 608 a(Lemma) k(3.3) p Fy 49 w(The) 43 b(absolutely) h(c) -5 b(ontinouous) 43 b(p) -5 b(art) 44 b(of) f(the) h(sp) -5 b(e) g(ctr) g(al) 43 b(me) -5 b(asur) g(e) p Fl 43 w(\027) p Fr 3097 623 a(\016) p Fd 3128 631 a(m) p Fr 3187 623 a(;\016) p Fd 3238 631 a(n) p Fy 3329 608 a(is) 43 b(e) -5 b(qual) 43 b(to) p Fl 0 728 a(\031) p Fq 59 692 a(\000) p Ft(1) p Fo 153 728 a(Im) 16 b(\() p Fl(\016) p Fr 367 743 a(m) p Fm 434 728 a(j) p Fo(\() p Fl(H) p Fm 29 w(\000) p Fl 23 w(E) p Fm 28 w(\000) p Fo 23 w(i0\)) p Fq 1025 692 a(\000) p Ft(1) p Fl 1118 728 a(\016) p Fr 1161 743 a(n) p Fo 1208 728 a(\)d) p Fl(E) p Fy 6 w(.) p Fc 0 1016 a(3.2) 135 b(Pro) t(of) 45 b(of) g(Theorem) h(1.1) p Fo 0 1201 a(Theorem) 33 b(1.1) f(follo) m(ws) f(from) h(Prop) s (ositions) f(3.4) h(and) h(3.5) f(b) s(elo) m(w.) 146 1321 y(In) g(Prop) s(osition) d(3.4) i(w) m(e) i(use) f(the) f(same) g (notation) f(as) i(in) e(Theorem) i(1.1.) 42 b(In) 32 b(particular,) e(\(b\)-\(d\)) 0 1442 y(refer) 25 b(to) g(the) g (assumptions) g(of) f(Theorem) h(1.1.) 41 b(W) -8 b(e) 25 b(assume) g(that) p Fm 25 w(H) p Fo 26 w(is) f(spanned) i(b) m(y) p Fl 26 w(H) p Fo 33 w(and) p Fm 25 w(f) p Fl(\016) p Fr 3518 1457 a(n) p Fm 3565 1442 a(g) p Fr 3615 1457 a(n) p Fq(2) p Ft(\000) p Fo 3752 1442 a(.) p Fh 0 1667 a(Prop) s(osition) 36 b(3.4) p Fy 49 w(L) -5 b(et) p Fo 35 w(\002) p Fm 27 w(\032) p Fi 28 w(R) p Fy 46 w(b) g(e) 35 b(an) f(op) -5 b(en) 34 b(set.) 45 b(Consider) 34 b(the) h(fol) 5 b(lowing) 33 b(statements:) p Fo 0 1787 a(\(1\)) p Fy 35 w(F) -7 b(or) 33 b(a) i(dense) f(set) h(of) p Fl 35 w(\036) p Fm 27 w(2) p Fo 28 w(Ran) p Fh 16 w(1) p Ft 1388 1802 a(\002) p Fo 1448 1787 a(\() p Fl(H) p Fo 8 w(\)) p Fy(,) p Fk 1403 1917 a(Z) p Fg 1458 2142 a(R) p Fm 1527 2052 a(k) p Fh(1) p Ft 1633 2067 a(1) p Fo 1672 2052 a(e) p Fq 1715 2011 a(\000) p Ft(i) p Fr(tH) p Fl 1883 2052 a(\036) p Fm(k) p Ft 1991 2011 a(2) p Fo 2030 2052 a(d) p Fl(t) 28 b(<) p Fm 27 w(1) p Fl(:) p Fo 1202 w(\(3.7\)) 0 2322 y(\(2\)) p Fy 35 w(The) 34 b(wave) g(op) -5 b(er) g(ators) p Fo 1233 2417 a(~) p Fl 1205 2442 a(W) p Fq 1311 2401 a(\006) p Fo 1397 2442 a(=) 28 b(s) p Fm 23 w(\000) p Fo 65 w(lim) p Fr 1661 2502 a(t) p Fq(!\0061) p Fo 1899 2442 a(e) p Ft 1942 2401 a(i) p Fr(tH) p Ff 2045 2410 a(0) p Fo 2084 2442 a(e) p Fq 2127 2401 a(\000) p Ft(i) p Fr(tH) p Fh 2295 2442 a(1) p Ft 2351 2457 a(\002) p Fo 2410 2442 a(\() p Fl(H) p Fo 8 w(\)) p Fy 0 2645 a(exist.) 0 2765 y(If) p Fo 34 w(\(b\)) p Fy 35 w(holds,) 34 b(then) p Fo 35 w(\(1\)) p Fm 34 w(\)) p Fo 35 w(\(2\)) p Fy(.) 44 b(If) p Fo 35 w(\(b\)) p Fy(,) p Fo 34 w(\(c\)) p Fy 35 w(and) p Fo 35 w(\(d\)) p Fy 35 w(hold,) 34 b(then) p Fo 35 w(\(2\)) p Fm 34 w(\)) p Fo 35 w(\(1\)) p Fy(.) p Fh 0 2990 a(Pro) s(of.) p Fo 72 w(The) j(pro) s(of) e(is) h(a) g(based) h(on) f(the) h(argumen) m(ts) f(used) h(in) f([JL1,) g(JL3,) g(JM3]) g (in) g(the) g(analysis) 0 3110 y(of) d(some) g(sp) s(eci\014c) i (examples) e(of) g(the) h(abstract) g(mo) s(del) e(\(1.1\).) 46 b(These) 35 b(argumen) m(ts) f(ha) m(v) m(e) h(their) e(ro) s(ots) 0 3230 y(in) f(Kato's) g(theory) h(of) g(smo) s(oth) e(p) s (erturbations.) 146 3351 y(Assume) 36 b(\014rst) f(that) f(\(b\)) h (holds.) 49 b(T) -8 b(o) 34 b(pro) m(v) m(e) i(that) e(\(1\)) p Fm 34 w(\)) p Fo 34 w(\(2\)) h(it) e(su\016ces) k(to) d(sho) m(w) h (that) g(for) f(an) m(y) p Fl 0 3471 a(\036) p Fm 27 w(2) p Fo 28 w(Ran) p Fh 17 w(1) p Ft 427 3486 a(\002) p Fo 486 3471 a(\() p Fl(H) p Fo 8 w(\)) e(for) g(whic) m(h) h(\(3.7\)) f (holds,) g(the) h(limits) 1587 3688 y(lim) p Fr 1544 3748 a(t) p Fq(!\0061) p Fo 1782 3688 a(e) p Ft 1825 3647 a(i) p Fr(tH) p Ff 1928 3656 a(0) p Fo 1967 3688 a(e) p Fq 2010 3647 a(\000) p Ft(i) p Fr(tH) p Fl 2178 3688 a(\036) p Fo 1343 w(\(3.8\)) 0 3939 y(exist.) 44 b(In) 33 b(what) g(follo) m(ws) e(w) m(e) i(\014x) p Fl 34 w(\036) p Fo(.) 146 4059 y(Note) g(\014rst) g(that) 1528 4180 y(lim) p Fq 1493 4246 a(j) p Fr(t) p Fq(j!1) p Fh 1716 4180 a(1) p Ft 1772 4195 a(0) p Fo 1811 4180 a(e) p Fq 1854 4139 a(\000) p Ft(i) p Fr(tH) p Fl 2022 4180 a(\036) p Fo 27 w(=) 28 b(0) p Fl(:) p Fo 1292 w(\(3.9\)) 0 4403 y(T) -8 b(o) 33 b(pro) m(v) m(e) g(this) g(relation,) d(let) p Fl 1446 4523 a(w) p Fo 3 w(\() p Fl(t) p Fo(\)) d(:=) g(e) p Ft 1830 4482 a(i) p Fr(tH) p Fh 1943 4523 a(1) p Ft 1999 4538 a(0) p Fo 2039 4523 a(e) p Fq 2082 4482 a(\000) p Ft(i) p Fr(tH) p Fl 2249 4523 a(\036:) p Fo 0 4696 a(Then,) 36 b(b) m(y) g(\(3.7\),) p Fk 684 4616 a(R) p Fm 767 4696 a(k) p Fl(w) p Fo 3 w(\() p Fl(t) p Fo(\)) p Fm(k) p Ft 1051 4660 a(2) p Fo 1090 4696 a(d) p Fl(t) 31 b(<) p Fm 31 w(1) p Fo(.) 49 b(Since) p Fm 35 w(k) p Fl(w) p Fq 1873 4660 a(0) p Fo 1895 4696 a(\() p Fl(t) p Fo(\)) p Fm(k) 31 b(\024) p Fo 32 w(2) p Fm(k) p Fl(H) p Ft 2376 4711 a(0) p Fm 2414 4696 a(k) p Fo(,) k(it) f(follo) m(ws) f(from) g (Exercise) j(62) e(in) 0 4817 y([RS]) f(that) f(lim) p Fq 560 4832 a(j) p Fr(t) p Fq(j!1) p Fl 786 4817 a(w) p Fo 3 w(\() p Fl(t) p Fo(\)) 27 b(=) h(0.) 146 4937 y(W) -8 b(e) 33 b(adopt) g(the) g(shorthand) p Fh 33 w(1) p 1276 4909 40 3 v Ft 1276 4960 a(0) p Fo 1343 4937 a(:=) p Fh 28 w(1) p Fm 22 w(\000) p Fh 22 w(1) p Ft 1707 4952 a(0) p Fo 1747 4937 a(.) 43 b(Let) p Fl 33 w(T) p Fo 46 w(b) s(e) 33 b(a) f(linear) f(op) s(erator) h(de\014ned) i(b) m(y) p Fl 1084 5170 a(T) 14 b(\016) p Fr 1198 5185 a(n) p Fo 1272 5170 a(=) p Fm 28 w(\000) p Fk 1637 5075 a(X) p Fr 1470 5291 a(m) p Fq(62) p Ft(\000) p Fr(;\032) p Ft(\() p Fr(m;n) p Ft(\)=1) p Fl 1966 5170 a(\016) p Fr 2009 5185 a(m) p Fl 2075 5170 a(;) p Fo 212 w(if) p Fl 56 w(n) p Fm 28 w(2) p Fo 28 w(\000) p Fl(;) p 90 rotate dyy eop %%Page: 11 11 11 10 bop Fo 3682 100 a(11) p Fl 1033 412 a(T) 14 b(\016) p Fr 1147 427 a(n) p Fo 1222 412 a(=) p Fk 1493 317 a(X) p Fr 1326 533 a(m) p Fq(2) p Ft(\000) p Fr(;\032) p Ft(\() p Fr(m;n) p Ft(\)=1) p Fl 1821 412 a(\016) p Fr 1864 427 a(m) p Fl 1931 412 a(;) p Fo 212 w(if) p Fl 56 w(n) p Fm 27 w(2) p Fo 29 w(\000) p Ft 2525 427 a(1) p Fm 2586 412 a(n) p Fo 22 w(\000) p Fl(;) p Fo 0 691 a(and) p Fl 33 w(T) g(\016) p Fr 304 706 a(n) p Fo 378 691 a(=) 28 b(0) k(if) p Fl 32 w(n) p Fm 27 w(62) p Fo 29 w(\000) p Ft 894 706 a(1) p Fo 933 691 a(.) 43 b(A) 33 b(simple) e(calculation) f (yields) p Fl 1295 911 a(H) p Ft 1376 926 a(0) p Fh 1415 911 a(1) p 1471 883 40 3 v Ft 1471 934 a(0) p Fm 1533 911 a(\000) p Fh 22 w(1) p 1688 883 V Ft 1688 934 a(0) p Fl 1728 911 a(H) p Fo 35 w(=) d([) p Fl(H) p Ft 2055 926 a(0) p Fl 2095 911 a(;) p Fh 17 w(1) p 2195 883 V Ft 2195 934 a(0) p Fo 2234 911 a(]) h(=) p Fl 27 w(T) 8 b(:) p Fo 1046 w(\(3.10\)) 0 1131 y(Ob) m(viously) -8 b(,) p Fm 32 w(k) p Fl(T) p Fm 14 w(k) 27 b(\024) p Fo 29 w(2) p Fm(k) p Fl(H) p Ft 961 1146 a(0) p Fm 999 1131 a(k) p Fo 33 w(and) p Fl 32 w(T) p Fh 14 w(1) p Ft 1398 1146 a(1) p Fo 1465 1131 a(=) p Fh 28 w(1) p Ft 1625 1146 a(1) p Fl 1664 1131 a(T) p Fo 14 w(.) 146 1251 y(Let) p Fl 45 w(I) p Fo 57 w(=) 48 b([) p Fl(a;) 17 b(b) p Fo(]) p Fm 50 w(\032) p Fo 49 w(\002) 45 b(and) g(let) p Fl 44 w(\015) p Fo 50 w(b) s(e) g(a) f(simple) g(closed) h(curv) m(e) h(in) e (the) i(complex) e(plane) h(that) 0 1372 y(separates) 34 b([) p Fl(a;) 17 b(b) p Fo(]) 33 b(and) p Fi 33 w(R) p Fm 33 w(n) p Fo 22 w(\002) f(and) h(encloses) p Fl 33 w(I) p Fo 8 w(.) 44 b(Then,) 33 b(for) f(an) m(y) p Fl 34 w( ) p Fm 31 w(2) c(H) p Fo 1 w(,) p Fh 233 1650 a(1) p Fg 289 1665 a(R) p Fq(n) p Ft(\002) p Fo 431 1650 a(\() p Fl(H) p Ft 550 1665 a(0) p Fo 589 1650 a(\)e) p Ft 670 1609 a(i) p Fr(tH) p Ff 773 1618 a(0) p Fh 812 1650 a(1) p 868 1622 V Ft 868 1673 a(0) p Fo 908 1650 a(e) p Fq 951 1609 a(\000) p Ft(i) p Fr(tH) p Fh 1118 1650 a(1) p Fr 1174 1665 a(I) p Fo 1214 1650 a(\() p Fl(H) p Fo 8 w(\)) p Fl( ) p Fo 31 w(=) p Fm 255 1962 a(\000) p Fo 22 w(\(2) p Fl(\031) p Fo 4 w(i\)) p Fq 566 1920 a(\000) p Ft(1) p Fk 676 1826 a(I) p Fr 731 2051 a(\015) p Fh 792 1962 a(1) p Fg 848 1977 a(R) p Fq(n) p Ft(\002) p Fo 991 1962 a(\() p Fl(H) p Ft 1110 1977 a(0) p Fo 1149 1962 a(\)e) p Ft 1230 1920 a(i) p Fr(tH) p Ff 1333 1929 a(0) p Fo 1372 1962 a(\() p Fl(H) p Ft 1491 1977 a(0) p Fm 1552 1962 a(\000) p Fl 23 w(z) p Fo 4 w(\)) p Fq 1739 1920 a(\000) p Ft(1) p Fo 1834 1962 a(\() p Fl(H) p Ft 1953 1977 a(0) p Fh 1992 1962 a(1) p 2048 1933 V Ft 2048 1985 a(0) p Fm 2110 1962 a(\000) p Fh 22 w(1) p 2265 1933 V Ft 2265 1985 a(0) p Fl 2305 1962 a(H) p Fo 8 w(\)\() p Fl(H) p Fm 29 w(\000) p Fl 22 w(z) p Fo 4 w(\)) p Fq 2766 1920 a(\000) p Ft(1) p Fo 2861 1962 a(e) p Fq 2904 1920 a(\000) p Ft(i) p Fr(tH) p Fh 3072 1962 a(1) p Fr 3128 1977 a(I) p Fo 3168 1962 a(\() p Fl(H) p Fo 8 w(\)) p Fl( ) p Fo 20 w(d) p Fl(z) t(;) p Fo 0 2247 a(see) 34 b(the) f(pro) s(of) e(of) i (Theorem) f(XI) s(I) s(I.31) h(in) e([RS) q(].) 43 b(Hence,) 34 b(for) e(some) h(constan) m(t) p Fl 33 w(C) p Fo 7 w(,) p Fm 407 2515 a(k) p Fh(1) p Fg 513 2531 a(R) p Fq(n) p Ft(\002) p Fo 655 2515 a(\() p Fl(H) p Ft 774 2530 a(0) p Fo 813 2515 a(\)e) p Ft 894 2474 a(i) p Fr(tH) p Ff 997 2483 a(0) p Fh 1036 2515 a(1) p 1092 2487 V Ft 1092 2539 a(0) p Fo 1132 2515 a(e) p Fq 1175 2474 a(\000) p Ft(i) p Fr(tH) p Fh 1342 2515 a(1) p Fr 1398 2530 a(I) p Fo 1438 2515 a(\() p Fl(H) p Fo 8 w(\)) p Fl( ) p Fm 4 w(k) 27 b(\024) p Fl 28 w(C) p Fk 1946 2380 a(I) p Fr 2001 2605 a(\015) p Fm 2062 2515 a(k) p Fh(1) p Ft 2168 2530 a(1) p Fo 2207 2515 a(\() p Fl(H) p Fm 30 w(\000) p Fl 23 w(z) p Fo 4 w(\)) p Fq 2543 2474 a(\000) p Ft(1) p Fo 2638 2515 a(e) p Fq 2681 2474 a(\000) p Ft(i) p Fr(tH) p Fh 2848 2515 a(1) p Fr 2904 2530 a(I) p Fo 2944 2515 a(\() p Fl(H) p Fo 8 w(\)) p Fl( ) p Fm 4 w(k) p Fo 17 w(d) p Fl(z) t(:) p Fo 0 2794 a(Set) p Fl 1142 2915 a(`) p Fo(\() p Fl(z) t(;) 17 b(t) p Fo(\)) 29 b(:=) p Fh 27 w(1) p Ft 1602 2930 a(1) p Fo 1642 2915 a(\() p Fl(H) p Fm 29 w(\000) p Fl 23 w(z) p Fo 4 w(\)) p Fq 1977 2874 a(\000) p Ft(1) p Fo 2072 2915 a(e) p Fq 2115 2874 a(\000) p Ft(i) p Fr(tH) p Fh 2283 2915 a(1) p Fr 2339 2930 a(I) p Fo 2379 2915 a(\() p Fl(H) p Fo 8 w(\)) p Fl( ) t(:) p Fo 0 3089 a(The) 41 b(v) m(ector-v) -5 b(alued) 41 b(function) p Fl 40 w(`) p Fo(\() p Fl(z) t(;) 17 b(t) p Fo(\)) 41 b(is) f(uniformly) e(b) s (ounded) j(on) p Fl 40 w(\015) p Fm 33 w(\002) p Fi 28 w(R) p Fo 51 w(and) g(has) g(a) f(uniformly) 0 3209 y(b) s(ounded) 28 b(deriv) -5 b(ativ) m(e) 28 b(in) p Fl 26 w(t) p Fo(.) 42 b(Moreo) m(v) m(er,) 31 b(for) c(all) p Fl 25 w(z) p Fm 33 w(2) p Fl 28 w(\015) p Fo 5 w(,) p Fl 29 w(`) p Fo(\() p Fl(z) t(;) 17 b(t) p Fo(\)) 28 b(is) f(square-in) m(tegrable) g (in) p Fl 27 w(t) p Fo(.) 42 b(It) 28 b(follo) m(ws) 0 3330 y(that) k(lim) p Fq 347 3345 a(j) p Fr(t) p Fq(j!1) p Fl 573 3330 a(`) p Fo(\() p Fl(z) t(;) 17 b(t) p Fo(\)) 29 b(=) e(0) 32 b(and) h(so) 1014 3550 y(s) p Fm 23 w(\000) p Fo 65 w(lim) p Fr 1174 3610 a(t) p Fq(!\0061) p Fh 1412 3550 a(1) p Fg 1468 3565 a(R) p Fq(n) p Ft(\002) p Fo 1611 3550 a(\() p Fl(H) p Ft 1730 3565 a(0) p Fo 1769 3550 a(\)e) p Ft 1850 3509 a(i) p Fr(tH) p Ff 1953 3518 a(0) p Fh 1992 3550 a(1) p 2048 3522 V Ft 2048 3573 a(0) p Fo 2087 3550 a(e) p Fq 2130 3509 a(\000) p Ft(i) p Fr(tH) p Fh 2298 3550 a(1) p Fr 2354 3565 a(I) p Fo 2394 3550 a(\() p Fl(H) p Fo 8 w(\)) 27 b(=) h(0) p Fl(:) p Fo 0 3804 a(Since) 33 b(\002) f(is) g(a) g(coun) m(table) h(union) f(of) g (closed) h(in) m(terv) -5 b(als,) 32 b(w) m(e) h(conclude) h(that) 1004 4024 y(s) p Fm 23 w(\000) p Fo 66 w(lim) p Fr 1165 4083 a(t) p Fq(!\0061) p Fh 1403 4024 a(1) p Fg 1459 4039 a(R) p Fq(n) p Ft(\002) p Fo 1601 4024 a(\() p Fl(H) p Ft 1720 4039 a(0) p Fo 1759 4024 a(\)e) p Ft 1840 3983 a(i) p Fr(tH) p Ff 1943 3992 a(0) p Fh 1982 4024 a(1) p 2038 3995 V Ft 2038 4047 a(0) p Fo 2078 4024 a(e) p Fq 2121 3983 a(\000) p Ft(i) p Fr(tH) p Fh 2288 4024 a(1) p Ft 2344 4039 a(\002) p Fo 2404 4024 a(\() p Fl(H) p Fo 8 w(\)) 27 b(=) g(0) p Fl(:) p Fo 755 w(\(3.11\)) 146 4283 y(W) -8 b(e) 33 b(are) g(no) m(w) g(ready) g(to) g(pro) m(v) m(e) g (that) g(the) g(limits) c(\(3.8\)) j(exist.) 44 b(Let) p Fl 1286 4503 a(\020) p Fo 8 w(\() p Fl(t) p Fo(\)) 27 b(:=) p Fh 28 w(1) p Ft 1662 4518 a(\002) p Fo 1721 4503 a(\() p Fl(H) p Ft 1840 4518 a(0) p Fo 1879 4503 a(\)e) p Ft 1960 4462 a(i) p Fr(tH) p Ff 2063 4471 a(0) p Fh 2102 4503 a(1) p 2158 4475 V Ft 2158 4526 a(0) p Fo 2198 4503 a(e) p Fq 2241 4462 a(\000) p Ft(i) p Fr(tH) p Fl 2408 4503 a(\036:) p Fo 0 4723 a(By) 35 b(\(3.9\)) f(and) h(\(3.11\),) f(it) f(su\016ces) k(to) d(sho) m(w) i(that) e(lim) p Fr 2040 4738 a(t) p Fq(!\0061) p Fl 2282 4723 a(\020) p Fo 8 w(\() p Fl(t) p Fo(\)) g(exist.) 49 b(Let) p Fl 35 w( ) p Fm 35 w(2) 31 b(H) p Fo 36 w(b) s(e) j(arbitrary) -8 b(.) 0 4843 y(It) 33 b(follo) m(ws) e(from) g(\(3.10\)) h(that) 1074 5045 y(d) p 1056 5090 90 4 v 1056 5181 a(d) p Fl(t) p Fo 1155 5113 a(\() p Fl( ) p Fm 4 w(j) p Fl(\020) p Fo 8 w(\() p Fl(t) p Fo(\)\)) 27 b(=) g(i\(e) p Fq 1727 5072 a(\000) p Ft(i) p Fr(tH) p Ff 1885 5081 a(0) p Fh 1924 5113 a(1) p Ft 1980 5128 a(\002) p Fo 2039 5113 a(\() p Fl(H) p Ft 2158 5128 a(0) p Fo 2197 5113 a(\)) p Fl( ) p Fm 4 w(j) p Fl(T) p Fo 14 w(e) p Fq 2444 5072 a(\000) p Ft(i) p Fr(tH) p Fl 2611 5113 a(\036) p Fo(\)) p Fl(;) p 90 rotate dyy eop %%Page: 12 12 12 11 bop Fo 3682 100 a(12) 0 407 y(and) 33 b(so,) g(for) p Fl 32 w(t) 27 b(>) h(s) p Fo(,) p Fm 139 708 a(j) p Fo(\() p Fl( ) p Fm 4 w(j) p Fo(\() p Fl(\020) p Fo 8 w(\() p Fl(t) p Fo(\)) p Fm 20 w(\000) p Fl 23 w(\020) p Fo 8 w(\() p Fl(s) p Fo(\)\)) p Fm(j) e(\024) i(k) p Fl(T) p Fm 14 w(k) p Fk 1178 567 a(\022) 1251 572 y(Z) p Fr 1350 598 a(t) 1306 798 y(s) p Fm 1397 708 a(k) p Fh(1) p Ft 1503 723 a(1) p Fo 1542 708 a(e) p Fq 1585 667 a(\000) p Ft(i) p Fr(\034) 8 b(H) p Ff 1757 676 a(0) p Fh 1795 708 a(1) p Ft 1851 723 a(\002) p Fo 1910 708 a(\() p Fl(H) p Ft 2029 723 a(0) p Fo 2069 708 a(\)) p Fl( ) p Fm 4 w(k) p Ft 2224 667 a(2) p Fo 2263 708 a(d) p Fl(\034) p Fk 2370 567 a(\023) p Ft 2444 584 a(1) p Fr(=) p Ft(2) p Fk 2570 567 a(\022) 2644 572 y(Z) p Fr 2743 598 a(t) 2699 798 y(s) p Fm 2790 708 a(k) p Fh(1) p Ft 2896 723 a(1) p Fo 2935 708 a(e) p Fq 2978 667 a(\000) p Ft(i) p Fr(\034) g(H) p Fl 3159 708 a(\036) p Fm(k) p Ft 3267 667 a(2) p Fo 3306 708 a(d) p Fl(\034) p Fk 3413 567 a(\023) p Ft 3487 584 a(1) p Fr(=) p Ft(2) p Fl 3614 708 a(:) p Fo 0 980 a(Since) p Fh 33 w(1) p Ft 311 995 a(1) p Fo 383 980 a(is) p Fl 32 w(H) p Ft 562 995 a(0) p Fo 601 980 a(-smo) s(oth) 31 b(on) i(\002,) f(there) i(is) e(a) g(constan) m(t) p Fl 33 w(C) p Fo 40 w(suc) m(h) i(that) e(for) g(all) p Fl 31 w( ) p Fm 31 w(2) c(H) p Fo 1 w(,) p Fk 1128 1107 a(Z) p Fg 1184 1332 a(R) p Fm 1252 1242 a(k) p Fh(1) p Ft 1358 1257 a(1) p Fo 1398 1242 a(e) p Fq 1441 1201 a(\000) p Ft(i) p Fr(\034) 8 b(H) p Ff 1613 1210 a(0) p Fh 1651 1242 a(1) p Ft 1707 1257 a(\002) p Fo 1766 1242 a(\() p Fl(H) p Ft 1885 1257 a(0) p Fo 1925 1242 a(\)) p Fl( ) p Fm 4 w(k) p Ft 2080 1201 a(2) p Fo 2119 1242 a(d) p Fl(t) p Fm 28 w(\024) p Fl 28 w(C) p Fm 7 w(k) p Fl( ) p Fm 4 w(k) p Ft 2585 1201 a(2) p Fl 2624 1242 a(:) p Fo 0 1510 a(Hence,) 34 b(for) e(some) g(constan) m(t) p Fl 34 w(C) p Fo 7 w(,) p Fm 964 1811 a(k) p Fl(\020) p Fo 8 w(\() p Fl(t) p Fo(\)) p Fm 21 w(\000) p Fl 23 w(\020) p Fo 8 w(\() p Fl(s) p Fo(\)) p Fm(k) 26 b(\024) p Fl 29 w(C) p Fk 1745 1670 a(\022) 1818 1675 y(Z) p Fr 1918 1701 a(t) 1874 1900 y(s) p Fm 1964 1811 a(k) p Fh(1) p Ft 2070 1826 a(1) p Fo 2110 1811 a(e) p Fq 2153 1769 a(\000) p Ft(i) p Fr(\034) 8 b(H) p Fl 2334 1811 a(\036) p Fm(k) p Ft 2442 1769 a(2) p Fo 2481 1811 a(d) p Fl(\034) p Fk 2588 1670 a(\023) p Ft 2662 1687 a(1) p Fr(=) p Ft(2) p Fl 2788 1811 a(:) p Fo 0 2088 a(By) 29 b(\(3.7\),) g(the) g(sequence) p Fl 31 w(\020) p Fo 8 w(\() p Fl(t) p Fo(\)) e(is) h(Cauc) m(h) m(y) j (as) p Fl 28 w(t) p Fm 28 w(!) c(\0061) p Fo(,) j(and) e(the) h(limits) d(lim) p Fr 2890 2103 a(t) p Fq(!\0061) p Fl 3132 2088 a(\020) p Fo 8 w(\() p Fl(t) p Fo(\)) i(exist.) 42 b(This) 0 2208 y(\014nishes) 34 b(the) f(pro) s(of) e(that) i(if) e(\(b\)) i (holds,) f(then) h(\(1\)) p Fm 32 w(\)) p Fo 32 w(\(2\).) 146 2329 y(Assume) 23 b(no) m(w) f(that) g(in) f(addition,) h(\(c\)) g(and) g(\(d\)) g(also) e(hold.) 39 b(If) 22 b(\(2\)) f(holds,) j(then) p Fl 22 w(W) p Fq 3127 2292 a(\006) p Fo 3214 2329 a(:) k(Ran) p Fh 16 w(1) p Ft 3516 2344 a(\002) p Fo 3575 2329 a(\() p Fl(H) p Ft 3694 2344 a(0) p Fo 3733 2329 a(\)) p Fm 28 w(7!) p Fo 0 2449 a(Ran) p Fh 16 w(1) p Ft 247 2464 a(\002) p Fo 306 2449 a(\() p Fl(H) p Fo 8 w(\)) k(are) h(norm-preserving) f (bijections.) 43 b(Hence,) 34 b(it) d(su\016ces) k(to) d(sho) m(w) i (that) e(for) g(all) p Fl 31 w( ) p Fm 31 w(2) d(D) p Fo 3 w(,) p Fk 1316 2582 a(Z) p Fg 1371 2807 a(R) p Fm 1440 2717 a(k) p Fh(1) p Ft 1546 2732 a(1) p Fo 1585 2717 a(e) p Fq 1628 2676 a(\000) p Ft(i) p Fr(tH) p Fl 1796 2717 a(W) p Ft 1902 2676 a(+) p Fl 1960 2717 a( ) p Fm 4 w(k) p Ft 2077 2676 a(2) p Fo 2117 2717 a(d) p Fl(t) f(<) p Fm 27 w(1) p Fl(:) p Fo 1066 w(\(3.12\)) 0 2990 y(By) 33 b(\(3.8\),) p Fl 1391 3110 a(W) p Ft 1497 3069 a(+) p Fo 1583 3110 a(=) 44 b(lim) p Fr 1687 3170 a(t) p Fq(!1) p Fo 1870 3110 a(e) p Ft 1913 3069 a(i) p Fr(tH) p Fh 2026 3110 a(1) p 2082 3082 40 3 v Ft 2082 3134 a(0) p Fo 2121 3110 a(e) p Fq 2164 3069 a(\000) p Ft(i) p Fr(tH) p Ff 2322 3078 a(0) p Fl 2361 3110 a(;) p Fo 0 3312 a(and) 33 b(so) p Fl 1152 3565 a(W) p Ft 1258 3524 a(+) p Fl 1316 3565 a( ) p Fm 27 w(\000) p Fl 22 w( ) p Fo 32 w(=) p Fk 1703 3429 a(Z) p Fq 1803 3455 a(1) p Ft 1759 3655 a(0) p Fo 1931 3497 a(d) p 1904 3542 108 4 v 1904 3633 a(d) p Fl(\034) p Fo 2022 3565 a(e) p Ft 2065 3524 a(i) p Fr(\034) 8 b(H) p Fh 2191 3565 a(1) p 2247 3536 40 3 v Ft 2247 3588 a(0) p Fo 2287 3565 a(e) p Fq 2330 3524 a(\000) p Ft(i) p Fr(\034) g(H) p Ff 2502 3533 a(0) p Fl 2540 3565 a( ) p Fo 1600 3904 a(=) 27 b(i) p Fk 1748 3768 a(Z) p Fq 1847 3795 a(1) p Ft 1802 3994 a(0) p Fo 1938 3904 a(e) p Ft 1981 3863 a(i) p Fr(\034) 8 b(H) p Fl 2107 3904 a(T) p Fo 14 w(e) p Fq 2221 3863 a(\000) p Ft(i) p Fr(\034) g(H) p Ff 2393 3872 a(0) p Fl 2431 3904 a( ) p Fo 4 w(d) p Fl(\034) e(:) p Fo 0 4170 a(Hence,) p Fm 1027 4443 a(k) p Fh(1) p Ft 1133 4458 a(1) p Fo 1173 4443 a(e) p Fq 1216 4401 a(\000) p Ft(i) p Fr(tH) p Fl 1383 4443 a(W) p Ft 1489 4401 a(+) p Fl 1548 4443 a( ) p Fm 4 w(k) p Ft 1665 4401 a(2) p Fo 1732 4443 a(=) p Fm 27 w(k) p Fh(1) p Ft 1941 4458 a(1) p Fl 1981 4443 a(W) p Ft 2087 4401 a(+) p Fo 2145 4443 a(e) p Fq 2188 4401 a(\000) p Ft(i) p Fr(tH) p Ff 2346 4410 a(0) p Fl 2385 4443 a( ) p Fm 4 w(k) p Ft 2502 4401 a(2) p Fm 1732 4754 a(\024) p Fl 28 w(L) p Fo(\() p Fl(t) p Fo(\)) 23 b(+) f(2) p Fm(k) p Fh(1) p Ft 2290 4769 a(1) p Fo 2329 4754 a(e) p Fq 2372 4713 a(\000) p Ft(i) p Fr(tH) p Ff 2530 4722 a(0) p Fl 2569 4754 a( ) p Fm 4 w(k) p Ft 2686 4713 a(2) p Fl 2725 4754 a(;) p Fo 0 4994 a(where) p Fl 1098 5163 a(L) p Fo(\() p Fl(t) p Fo(\)) p Fm 28 w(\024) p Fl 28 w(C) p Fk 1502 5022 a(\022) 1575 5027 y(Z) p Fq 1675 5054 a(1) p Ft 1631 5253 a(0) p Fm 1766 5163 a(k) p Fh(1) p Ft 1872 5178 a(1) p Fo 1912 5163 a(e) p Fq 1955 5122 a(\000) p Ft(i\() p Fr(t) p Ft(+) p Fr(\034) p Ft 8 w(\)) p Fr(H) p Ff 2261 5131 a(0) p Fl 2300 5163 a( ) p Fm 4 w(k) p Fo(d) p Fl(\034) p Fk 2524 5022 a(\023) p Ft 2598 5045 a(2) p Fl 2654 5163 a(:) p 90 rotate dyy eop %%Page: 13 13 13 12 bop Fo 3682 100 a(13) 0 407 y(By) 33 b(the) g(de\014nition) f(of) p Fm 32 w(D) p Fo 3 w(,) p Fl 32 w(L) p Fo(\() p Fl(t) p Fo(\)) c(=) p Fl 28 w(O) p Fo 3 w(\() p Fm(j) p Fl(t) p Fm(j) p Fq 1521 371 a(\000) p Ft(2) p Fo 1614 407 a(\),) k(and) h (\(3.12\)) f(follo) m(ws.) p Fa 42 w(2) p Fo 146 577 a(W) -8 b(e) 34 b(ha) m(v) m(e) g(used) g(the) f(assumption) f(that) h (\002) f(is) h(an) g(op) s(en) g(set) g(only) f(in) g(the) i(pro) s(of) e(of) g(the) h(estimate) 0 697 y(\(3.11\).) 49 b(If) p Fh 34 w(1) p Ft 481 712 a(1) p Fo 555 697 a(is) p Fl 35 w(H) p Ft 737 712 a(0) p Fo 776 697 a(-smo) s(oth) 33 b(this) h(estimate) g(is) g(not) h(needed) h(and) f(the) g(prop) s (osition) d(holds) j(for) f(an) m(y) 0 818 y(Borel) e(set) h(\002.) 146 938 y(The) i(next) g(prop) s(osition) c(is) j(of) f(an) g(indep) s (enden) m(t) i(in) m(terest) g(and) e(w) m(e) i(pro) m(v) m(e) g(it) e (in) g(a) g(more) g(general) 0 1059 y(setting.) p Fh 0 1287 a(Prop) s(osition) j(3.5) p Fy 49 w(L) -5 b(et) p Fl 42 w(A) p Fy 42 w(b) g(e) 42 b(a) f(self-adjoint) g(op) -5 b(e) g(artor) 42 b(on) f(a) h(Hilb) -5 b(ert) 43 b(sp) -5 b(ac) g(e) p Fj 41 w(H) p Fy 41 w(and) p Fm 42 w(f) p Fl(\016) p Fr 3440 1302 a(n) p Fm 3487 1287 a(g) p Fr 3537 1302 a(n) p Fq(2F) p Fy 3730 1287 a(a) 0 1407 y(c) g(ountable) 29 b(orthonormal) g(set) g(in) p Fj 29 w(H) p Fy(.) 43 b(Assume) 30 b(that) p Fm 29 w(f) p Fl(\016) p Fr 2026 1422 a(n) p Fm 2073 1407 a(g) p Fr 2123 1422 a(n) p Fq(2F) p Fy 2304 1407 a(is) f(a) g(cyclic) h(family) f(for) p Fl 29 w(A) p Fy(.) 43 b(L) -5 b(et) p Fo 30 w(\002) p Fm 27 w(\032) p Fi 29 w(R) p Fy 0 1528 a(b) g(e) 35 b(a) f(Bor) -5 b(el) 34 b(set) h(of) g(p) -5 b(ositive) 34 b(L) -5 b(eb) g(esgue) 34 b(me) -5 b(asur) g(e.) 45 b(Consider) 33 b(the) i(fol) 5 b(lowing) 34 b(assumptions:) p Fo 0 1648 a(\(a\)) p Fy 35 w(The) g(op) -5 b(er) g(ator) p Fl 34 w(A) p Fh(1) p Ft 871 1663 a(\002) p Fo 931 1648 a(\() p Fl(A) p Fo(\)) p Fy 34 w(has) 35 b(pur) -5 b(ely) 35 b(absolutely) g(c) -5 b(ontinuous) 34 b(sp) -5 b(e) g(ctrum.) p Fo 0 1768 a(\(b\)) p Fy 35 w(The) 34 b(sp) -5 b(e) g(ctr) g(al) 35 b(me) -5 b(asur) g(e) 34 b(for) p Fl 35 w(A) p Fy 35 w(and) p Fl 34 w(\016) p Fr 1585 1783 a(n) p Fl 1632 1768 a(;) 17 b(\016) p Fr 1719 1783 a(m) p Fy 1820 1768 a(is) 35 b(r) -5 b(e) g(al-value) g(d) 34 b(for) h(al) 5 b(l) p Fl 34 w(n;) 17 b(m) p Fm 28 w(2) 28 b(F) p Fy 10 w(.) p Fo 0 1889 a(\(c\)) p Fy 35 w(F) -7 b(or) 34 b(L) -5 b(eb) g(esgue) 34 b(a.e.) p Fl 45 w(E) p Fm 33 w(2) p Fo 28 w(\002) p Fy 35 w(and) g(al) 5 b(l) p Fl 35 w(n) p Fm 28 w(2) 28 b(F) p Fy 10 w(,) p Fo 1257 2109 a(Im) 16 b(\() p Fl(\016) p Fr 1471 2124 a(n) p Fm 1519 2109 a(j) p Fo(\() p Fl(A) p Fm 21 w(\000) p Fl 23 w(E) p Fm 28 w(\000) p Fo 23 w(i0\)) p Fq 2094 2068 a(\000) p Ft(1) p Fl 2187 2109 a(\016) p Fr 2230 2124 a(n) p Fo 2277 2109 a(\)) p Fl 28 w(>) p Fo 27 w(0) p Fl(:) p Fy 0 2329 a(Consider) 34 b(the) h(fol) 5 b(lowing) 33 b(statements:) p Fo 0 2449 a(\(1\)) p Fy 35 w(F) -7 b(or) 33 b(L) -5 b(eb) g(esgue) 35 b(a.e.) p Fl 44 w(E) p Fm 34 w(2) p Fo 28 w(\002) p Fy 35 w(and) f(for) h(al) 5 b(l) p Fl 34 w(n) p Fm 28 w(2) 28 b(F) p Fy 10 w(,) p Fk 1094 2592 a(X) p Fr 1083 2803 a(m) p Fq(2F) p Fm 1266 2687 a(j) p Fo(Im) 16 b(\() p Fl(\016) p Fr 1508 2702 a(n) p Fm 1555 2687 a(j) p Fo(\() p Fl(A) p Fm 22 w(\000) p Fl 23 w(E) p Fm 28 w(\000) p Fo 22 w(i0\)) p Fq 2130 2645 a(\000) p Ft(1) p Fl 2224 2687 a(\016) p Fr 2267 2702 a(m) p Fo 2333 2687 a(\)) p Fm(j) p Ft 2399 2645 a(2) p Fl 2466 2687 a(<) p Fm 28 w(1) p Fl(:) p Fo 0 3000 a(\(2\)) p Fy 35 w(F) -7 b(or) 33 b(a) i(dense) f(set) h(of) p Fl 35 w(\036) p Fm 27 w(2) p Fh 28 w(1) p Ft 1197 3015 a(\002) p Fo 1256 3000 a(\() p Fl(A) p Fo(\)) p Fj(H) p Fy(,) p Fk 1291 3174 a(X) p Fr 1279 3385 a(m) p Fq(2F) p Fk 1463 3133 a(Z) p Fg 1518 3359 a(R) p Fm 1587 3269 a(j) p Fo(\() p Fl(\016) p Fr 1696 3284 a(m) p Fm 1762 3269 a(j) p Fo(e) p Fq 1833 3227 a(\000) p Ft(i) p Fr(tA) p Fl 1990 3269 a(\036) p Fo(\)) p Fm(j) p Ft 2114 3227 a(2) p Fo 2153 3269 a(d) p Fl(t) 28 b(<) p Fm 27 w(1) p Fl(:) p Fo 1030 w(\(3.13\)) p Fy 0 3582 a(If) p Fo 34 w(\(a\)) p Fy 35 w(and) p Fo 34 w(\(b\)) p Fy 35 w(hold,) 34 b(then) p Fo 35 w(\(1\)) p Fm 34 w(\)) p Fo 35 w(\(2\)) p Fy(.) 44 b(If) p Fo 35 w(\(b\)) p Fy 34 w(and) p Fo 35 w(\(c\)) p Fy 35 w(hold,) 34 b(then) p Fo 35 w(\(2\)) p Fm 34 w(\)) p Fo 35 w(\(1\)) p Fy(.) p Fh 0 3811 a(Pro) s(of.) p Fo 70 w(W) -8 b(e) 36 b(\014rst) g(assume) g(that) f(\(a\)) g(and) g (\(b\)) g(hold) g(and) g(sho) m(w) i(that) e(\(1\)) p Fm 35 w(\)) p Fo 34 w(\(2\).) 52 b(F) -8 b(or) p Fl 34 w(n) p Fm 33 w(2) 32 b(F) p Fo 45 w(let) p Fj 0 3931 a(H) p Fr 72 3946 a(n) p Fo 153 3931 a(b) s(e) j(the) g(cyclic) g (space) g(spanned) h(b) m(y) p Fl 36 w(A) p Fo 35 w(and) p Fl 35 w(\016) p Fr 1854 3946 a(n) p Fo 1901 3931 a(.) 49 b(It) 35 b(su\016ces) i(to) d(sho) m(w) i(that) f(for) f(all) p Fl 32 w(n) p Fo 35 w(there) i(is) e(a) 0 4051 y(dense) g(set) f(of) p Fl 32 w(\036) p Fm 28 w(2) p Fh 28 w(1) p Ft 765 4066 a(\002) p Fo 824 4051 a(\() p Fl(A) p Fo(\)) p Fj(H) p Fr 1045 4066 a(n) p Fo 1124 4051 a(for) f(whic) m(h) h(\(3.13\)) f (holds.) 43 b(In) 33 b(what) g(follo) m(ws) f(w) m(e) h(\014x) p Fl 33 w(n) p Fo(.) 146 4172 y(Let) 721 4354 y(\002) p Fr 797 4369 a(j) p Fo 861 4354 a(=) p Fk 964 4184 a(\() p Fl 1044 4354 a(E) p Fm 34 w(2) p Fo 28 w(\002) 28 b(:) p Fk 1414 4260 a(X) p Fr 1403 4471 a(m) p Fq(2F) p Fm 1586 4354 a(j) p Fo(Im) 15 b(\() p Fl(\016) p Fr 1827 4369 a(n) p Fm 1875 4354 a(j) p Fo(\() p Fl(A) p Fm 21 w(\000) p Fl 23 w(E) p Fm 28 w(\000) p Fo 23 w(i0\)) p Fq 2450 4313 a(\000) p Ft(1) p Fl 2543 4354 a(\016) p Fr 2586 4369 a(m) p Fo 2653 4354 a(\)) p Fm(j) p Ft 2719 4313 a(2) p Fl 2785 4354 a(<) 28 b(j) p Fk 2935 4184 a(\)) p Fl 3032 4354 a(:) p Fo 0 4622 a(The) i(set) p Fm 30 w(f) p Fl(\037) p Fo(\() p Fl(A) p Fo(\)) p Fh(1) p Ft 662 4637 a(\002) p Fd 717 4647 a(j) p Fo 753 4622 a(\() p Fl(A) p Fo(\)) p Fl(\016) p Fr 945 4637 a(n) p Fo 1020 4622 a(:) p Fl 28 w(\037) p Fm 28 w(2) p Fl 28 w(L) p Fq 1324 4586 a(1) p Fo 1399 4622 a(\() p Fi(R) p Fo 5 w(\)) p Fl(;) 17 b(j) 39 b(>) p Fo 27 w(0) p Fm(g) p Fo 29 w(is) 28 b(dense) j(in) p Fh 28 w(1) p Ft 2418 4637 a(\002) p Fo 2477 4622 a(\() p Fl(A) p Fo(\)) p Fj(H) p Fr 2698 4637 a(n) p Fo 2774 4622 a(and) e(so) g(it) f(su\016es) j(to) d(sho) m(w) 0 4743 y(that) k(\(3.13\)) g(holds) g(for) p Fl 32 w(\036) p Fo('s) h(in) f(this) g(set.) 45 b(In) 32 b(what) h(follo) m(ws) f(w) m (e) h(\014x) p Fl 33 w(\036) p Fo 28 w(=) p Fl 27 w(\037) p Fo(\() p Fl(H) p Fo 8 w(\)) p Fh(1) p Ft 2986 4758 a(\002) p Fd 3041 4768 a(j) p Fo 3077 4743 a(\() p Fl(H) p Fo 8 w(\)) p Fl(\016) p Fr 3285 4758 a(n) p Fo 3332 4743 a(.) 146 4863 y(The) h(sp) s(ectral) e(theorem) h(and) f(the) h(assumption) f (\(b\)) h(yield,) 603 5131 y(\() p Fl(\016) p Fr 684 5146 a(m) p Fm 750 5131 a(j) p Fo(e) p Fq 821 5090 a(\000) p Ft(i) p Fr(tA) p Fl 978 5131 a(\036) p Fo(\)) 28 b(=) p Fk 1205 4996 a(Z) p Ft 1261 5221 a(\002) p Fd 1316 5231 a(j) p Fo 1369 5131 a(e) p Fq 1412 5090 a(\000) p Ft(i) p Fr(tE) p Fl 1572 5131 a(\037) p Fo(\() p Fl(E) p Fo 6 w(\)) p Fl(\031) p Fq 1846 5090 a(\000) p Ft(1) p Fo 1940 5131 a(Im) 16 b(\() p Fl(\016) p Fr 2154 5146 a(m) p Fm 2221 5131 a(j) p Fo(\() p Fl(A) p Fm 22 w(\000) p Fl 22 w(E) p Fm 29 w(\000) p Fo 22 w(i0\)) p Fq 2796 5090 a(\000) p Ft(1) p Fl 2889 5131 a(\016) p Fr 2932 5146 a(n) p Fo 2979 5131 a(\)d) p Fl(E) 6 b(:) p 90 rotate dyy eop %%Page: 14 14 14 13 bop Fo 3682 100 a(14) 0 407 y(Hence) p Fk 431 515 a(Z) p Fg 487 740 a(R) p Fm 555 650 a(j) p Fo(\() p Fl(\016) p Fr 664 665 a(m) p Fm 731 650 a(j) p Fo(e) p Fq 802 609 a(\000) p Ft(i) p Fr(tH) p Fl 969 650 a(\036) p Fo(\)) p Fm(j) p Ft 1093 609 a(2) p Fo 1132 650 a(d) p Fl(t) p Fo 28 w(=) 27 b(2) p Fl(\031) p Fq 1460 609 a(\000) p Ft(1) p Fk 1571 515 a(Z) p Ft 1626 740 a(\002) p Fd 1681 750 a(j) p Fm 1735 650 a(j) p Fl(\037) p Fo(\() p Fl(E) p Fo 6 w(\)) p Fm(j) p Ft 2006 609 a(2) p Fm 2044 650 a(j) p Fo(Im) 16 b(\() p Fl(\016) p Fr 2286 665 a(n) p Fo 2333 650 a(\() p Fl(A) p Fm 22 w(\000) p Fl 23 w(E) p Fm 28 w(\000) p Fo 23 w(i0\)) p Fq 2881 609 a(\000) p Ft(1) p Fl 2974 650 a(\016) p Fr 3017 665 a(m) p Fo 3084 650 a(\)) p Fm(j) p Ft 3150 609 a(2) p Fo 3189 650 a(d) p Fl(E) 6 b(;) p Fo 0 937 a(and) p Fk 242 1096 a(X) p Fr 231 1307 a(m) p Fq(2F) p Fk 414 1055 a(Z) p Fg 469 1280 a(R) p Fm 538 1190 a(j) p Fo(\() p Fl(\016) p Fr 647 1205 a(m) p Fm 713 1190 a(j) p Fl(e) p Fq 786 1149 a(\000) p Ft(i) p Fr(tA) p Fl 943 1190 a(\036) p Fo(\)) p Fm(j) p Ft 1067 1149 a(2) p Fo 1106 1190 a(d) p Fl(t) p Fo 28 w(=) p Fk 1338 1096 a(X) p Fr 1327 1307 a(m) p Fq(2F) p Fo 1510 1190 a(2) p Fl(\031) p Fq 1618 1149 a(\000) p Ft(1) p Fk 1728 1055 a(Z) p Ft 1784 1280 a(\002) p Fd 1839 1290 a(j) p Fm 1892 1190 a(j) p Fl(\037) p Fo(\() p Fl(E) p Fo 6 w(\)) p Fm(j) p Ft 2163 1149 a(2) p Fm 2202 1190 a(j) p Fo(Im) 16 b(\() p Fl(\016) p Fr 2444 1205 a(n) p Fm 2491 1190 a(j) p Fo(\() p Fl(H) p Fm 29 w(\000) p Fl 23 w(E) p Fm 28 w(\000) p Fo 22 w(i0\)) p Fq 3081 1149 a(\000) p Ft(1) p Fl 3174 1190 a(\016) p Fr 3217 1205 a(m) p Fo 3284 1190 a(\)) p Fm(j) p Ft 3350 1149 a(2) p Fo 3389 1190 a(d) p Fl(E) p Fm 1223 1571 a(\024) p Fo 28 w(4\() p Fl(d) p Fo 22 w(+) 22 b(1\)) p Fl(j) 6 b(\031) p Fq 1778 1530 a(\000) p Ft(1) p Fm 1872 1571 a(k) p Fl(\037) p Fm(k) p Fq 2033 1586 a(1) p Fl 2107 1571 a(:) 3522 1364 y(:) p Fo 146 1816 a(Assume) 31 b(no) m(w) f(that) f (\(b\)) g(and) g(\(c\)) h(hold.) 42 b(Assume) 30 b(that) f(\(2\)) g (holds) g(but) g(\(1\)) g(do) s(es) h(not) f(\(note) h(that) 0 1936 y(\(2\)) d(implies) d(\(a\)\).) 42 b(Then) 28 b(there) g(is) p Fl 26 w(n) p Fm 28 w(2) g(F) p Fo 37 w(and) f(a) g(Borel) f(set) 2257 1911 y(~) 2243 1936 y(\002) p Fm 28 w(\032) p Fo 28 w(\002) h(of) f(p) s (ositiv) m(e) h(Leb) s(esgue) h(measure) 0 2056 y(suc) m(h) 34 b(that) e(for) p Fl 32 w(E) p Fm 34 w(2) p Fo 794 2031 a(~) 780 2056 y(\002,) p Fk 1094 2094 a(X) p Fr 1083 2305 a(m) p Fq(2F) p Fm 1266 2189 a(j) p Fo(Im) 16 b(\() p Fl(\016) p Fr 1508 2204 a(n) p Fm 1555 2189 a(j) p Fo(\() p Fl(A) p Fm 22 w(\000) p Fl 23 w(E) p Fm 28 w(\000) p Fo 22 w(i0\)) p Fq 2130 2148 a(\000) p Ft(1) p Fl 2224 2189 a(\016) p Fr 2267 2204 a(m) p Fo 2333 2189 a(\)) p Fm(j) p Ft 2399 2148 a(2) p Fo 2466 2189 a(=) p Fm 28 w(1) p Fl(:) p Fo 146 2457 a(By) 39 b(assumption) f(\(c\),) p Fh 41 w(1) p Ft 1080 2467 a(~) 1070 2484 y(\002) p Fo 1129 2457 a(\() p Fl(H) p Fo 8 w(\)) p Fj(H) p Fr 1366 2472 a(n) p Fo 1450 2457 a(is) h(a) f(non-trivial) d(subspace) 41 b(of) p Fj 38 w(H) p Fo(.) 61 b(Let) p Fl 39 w(\027) p Fr 3047 2472 a(n) p Fo 3133 2457 a(b) s(e) 39 b(the) g(sp) s(ectral) 0 2577 y(measure) c(for) p Fl 34 w(A) p Fo 35 w(and) p Fl 35 w(\016) p Fr 876 2592 a(n) p Fo 923 2577 a(.) 50 b(By) 35 b(the) g(sp) s(ectral) g(theorem,) g(for) f(ev) m(ery) p Fl 37 w(\036) p Fm 31 w(2) p Fh 32 w(1) p Ft 2767 2587 a(~) 2757 2604 y(\002) p Fo 2816 2577 a(\() p Fl(A) p Fo(\)) p Fj(H) p Fr 3037 2592 a(n) p Fo 3118 2577 a(there) i(is) e(a) g (Borel) 0 2697 y(function) p Fl 32 w(\037) p Fr 443 2712 a(\036) p Fm 517 2697 a(2) p Fl 28 w(L) p Ft 677 2661 a(2) p Fo 717 2697 a(\() p Fi(R) p Fl 5 w(;) p Fo 17 w(d) p Fl(\027) p Fr 967 2712 a(\016) p Fd 998 2720 a(n) p Fo 1051 2697 a(\)) e(suc) m(h) i(that) 257 2969 y(\() p Fl(\036) p Fm(j) p Fl(\036) p Fo(\)) 27 b(=) p Fk 608 2833 a(Z) p Fg 663 3059 a(R) p Fm 732 2969 a(j) p Fl(\037) p Fr 821 2984 a(\036) p Fo 867 2969 a(\() p Fl(E) p Fo 6 w(\)) p Fm(j) p Ft 1049 2928 a(2) p Fo 1088 2969 a(d) p Fl(\027) p Fr 1190 2984 a(\016) p Fd 1221 2992 a(n) p Fo 1295 2969 a(=) p Fl 28 w(\031) p Fq 1458 2928 a(\000) p Ft(1) p Fk 1569 2833 a(Z) p Ft 1634 3042 a(~) 1624 3059 y(\002) p Fm 1700 2969 a(j) p Fl(\037) p Fr 1789 2984 a(\036) p Fo 1835 2969 a(\() p Fl(E) p Fo 6 w(\)) p Fm(j) p Ft 2017 2928 a(2) p Fo 2056 2969 a(Im) 16 b(\() p Fl(\016) p Fr 2270 2984 a(n) p Fm 2317 2969 a(j) p Fo(\() p Fl(A) p Fm 22 w(\000) p Fl 22 w(E) p Fm 29 w(\000) p Fo 22 w(i0\)) p Fq 2892 2928 a(\000) p Ft(1) p Fl 2985 2969 a(\016) p Fr 3028 2984 a(n) p Fo 3076 2969 a(\)d) p Fl(E) 6 b(:) p Fo 257 w(\(3.14\)) 0 3258 y(Ob) m(viously) -8 b(,) 36 b(if) p Fl 34 w(\036) p Fm 32 w(6) p Fo(=) c(0,) j(then) p Fl 36 w(\037) p Fr 1168 3273 a(\036) p Fo 1215 3258 a(\() p Fl(E) p Fo 6 w(\)) p Fm 32 w(6) p Fo(=) d(0) j(for) f(a) h(set) h(of) p Fl 35 w(E) p Fo 6 w('s) g(in) 2406 3233 y(~) 2392 3258 y(\002) f(of) g(p) s(ositiv) m(e) g(Leb) s(esgue) h(measure.) 0 3378 y(Moreo) m(v) m(er,) 611 3641 y(\() p Fl(\016) p Fr 692 3656 a(m) p Fm 758 3641 a(j) p Fo(e) p Fq 829 3600 a(\000) p Ft(i) p Fr(tA) p Fl 986 3641 a(\036) p Fo(\)) 28 b(=) p Fl 27 w(\031) p Fq 1272 3600 a(\000) p Ft(1) p Fk 1383 3506 a(Z) p Ft 1448 3714 a(~) 1439 3731 y(\002) p Fo 1514 3641 a(e) p Fq 1557 3600 a(\000) p Ft(i) p Fr(tE) p Fl 1717 3641 a(\037) p Fo(\() p Fl(E) p Fo 6 w(\)Im) 16 b(\() p Fl(\016) p Fr 2146 3656 a(m) p Fm 2213 3641 a(j) p Fo(\() p Fl(A) p Fm 22 w(\000) p Fl 22 w(E) p Fm 29 w(\000) p Fo 22 w(i0\)) p Fq 2788 3600 a(\000) p Ft(1) p Fl 2881 3641 a(\016) p Fr 2924 3656 a(n) p Fo 2972 3641 a(\)d) p Fl(E) 6 b(;) p Fo 0 3914 a(and) 33 b(so) f(for) g(all) f(non-zero) p Fl 33 w(\036) p Fm 27 w(2) p Fh 28 w(1) p Ft 1235 3924 a(~) 1225 3941 y(\002) p Fo 1284 3914 a(\() p Fl(A) p Fo(\)) p Fj(H) p Fr 1505 3929 a(n) p Fo 1552 3914 a(,) p Fk 258 4099 a(X) p Fr 246 4311 a(m) p Fq(2F) p Fk 429 4058 a(Z) p Fg 485 4284 a(R) p Fm 553 4194 a(j) p Fo(\() p Fl(\016) p Fr 662 4209 a(m) p Fm 729 4194 a(j) p Fo(e) p Fq 800 4153 a(\000) p Ft(i) p Fr(tA) p Fl 957 4194 a(\036) p Fo(\)) p Fm(j) p Ft 1081 4153 a(2) p Fo 1120 4194 a(d) p Fl(t) p Fo 28 w(=) c(2) p Fl(\031) p Fq 1448 4153 a(\000) p Ft(1) p Fk 1570 4099 a(X) p Fr 1559 4311 a(m) p Fq(2F) p Fk 1742 4058 a(Z) p Ft 1807 4267 a(~) 1797 4284 y(\002) p Fm 1873 4194 a(j) p Fl(\037) p Fr 1962 4209 a(\036) p Fo 2008 4194 a(\() p Fl(E) p Fo 6 w(\)) p Fm(j) p Ft 2190 4153 a(2) p Fm 2229 4194 a(j) p Fo(Im) 16 b(\() p Fl(\016) p Fr 2471 4209 a(n) p Fm 2518 4194 a(j) p Fo(\() p Fl(A) p Fm 22 w(\000) p Fl 22 w(E) p Fm 28 w(\000) p Fo 23 w(i0\)) p Fq 3093 4153 a(\000) p Ft(1) p Fl 3186 4194 a(\016) p Fr 3229 4209 a(m) p Fo 3296 4194 a(\)) p Fm(j) p Ft 3362 4153 a(2) p Fo 3401 4194 a(d) p Fl(E) p Fo 1237 4574 a(=) p Fm 27 w(1) p Fl(:) p Fo 0 4819 a(This) 33 b(con) m(tradicts) g(\(2\).) p Fa 43 w(2) p 90 rotate dyy eop %%Page: 15 15 15 14 bop Fo 3682 100 a(15) p Fn 0 407 a(References) p Fo 0 626 a([AM]) 131 b(Aizenman,) 50 b(M.,) h(Molc) m(hano) m(v,) g (S.:) 72 b(Lo) s(calization) 44 b(at) j(large) e(disorder) i(and) g(at) g(extreme) 347 746 y(energies:) d(An) 33 b(elemen) m(tary) f(deriv) -5 b(ation.) 31 b(Comm) m(un.) h(Math.) h(Ph) m(ys.) p Fh 35 w(157) p Fo(,) f(245) g(\(1993\)) 0 950 y([B]) 224 b(Bourgain,) 31 b(J.:) 44 b(On) 32 b(random) g(Sc) m(hr\177) -49 b(odinger) 33 b(op) s(erators) f(on) p Fi 32 w(Z) p Ft 2632 913 a(2) p Fo 2669 950 a(.) h(Discrete) f(and) h(Con) m(tin) m (uous) 347 1070 y(Dynamical) d(Systems) p Fh 34 w(8) p Fo(,) i(1) h(\(2002\)) 0 1273 y([BBP]) 89 b(Ben) m(tosela,) 38 b(F.,) g(Briet,) g(Ph.,) h(P) m(astur,) g(L.:) 52 b(Sp) s(ectral) 37 b(and) g(w) m(a) m(v) m(e) i(propagation) d(prop) s(erties) 347 1394 y(of) c(the) h(surface) g(Maryland) g(mo) s(del.) e(Preprin) m(t.) 0 1597 y([CK]) 147 b(Christ,) 60 b(M.,) h(Kiselev,) e(A.:) 88 b(Scattering) 54 b(and) g(w) m(a) m(v) m(e) j(op) s(erators) d(for) g (one-dimensional) 347 1718 y(Sc) m(hr\177) -49 b(odinger) 33 b(op) s(erators) f(with) g(slo) m(wly) g(deca) m(ying) h(nonsmo) s(oth) f(p) s(oten) m(tials.) g(Preprin) m(t.) 0 1921 y([CS]) 169 b(Chahrour,) 69 b(A.,) h(Sah) m(bani,) e(J.:) 102 b(On) 62 b(the) g(sp) s(ectral) f(and) h(scattering) g(theory) g(of) f(the) 347 2041 y(Sc) m(hr\177) -49 b(odinger) 33 b(op) s(erator) f(with) g (surface) h(p) s(oten) m(tial.) e(Rev.) i(Math.) g(Ph) m(ys.) p Fh 35 w(12) p Fo(,) f(561) g(\(2000\)) 0 2245 y([G]) 216 b(Grinshpun,) 26 b(V.:) 40 b(Lo) s(calization) 21 b(for) j(random) g(p) s(oten) m(tials) f(supp) s(orted) j(on) e(a) h(subspace.) h(Lett.) 347 2365 y(Math.) 33 b(Ph) m(ys.) p Fh 34 w(34) p Fo(,) g(103) f(\(1995\)) 0 2569 y([HK]) 144 b(Hundertmark,) 32 b(D.,) f(Kirsc) m(h,) h(W.:) 43 b(Sp) s(ectral) 31 b(theory) h(of) f(sparse) i(p) s(oten) m(tials.) d (In) p Fy 32 w(Sto) -5 b(chastic) 347 2689 y(pr) g(o) g(c) g(esses,) 42 b(physics) f(and) g(ge) -5 b(ometry:) 58 b(new) 41 b(interplays,) h(I) f (\(L) -5 b(eipzig,) 43 b(1999\)) p Fo(,) d(CMS) h(Conf.) 347 2809 y(Pro) s(c.) p Fh 33 w(28) p Fo(,) 32 b(213) g(\(2000\)) 0 3013 y([JL1]) 133 b(Jak) -5 b(\024) -44 b(s) q(i) m(\023) e(c,) 40 b(V.,) h(Last,) g(Y.:) 57 b(Corrugated) 39 b(surfaces) h(and) f(a.c.) h (sp) s(ectrum.) f(Rev.) h(Math.) g(Ph) m(ys.) p Fh 347 3133 a(12) p Fo(,) 33 b(1465) e(\(2000\)) 0 3337 y([JL2]) 133 b(Jak) -5 b(\024) -44 b(s) q(i) m(\023) e(c,) 37 b(V.,) h(Last,) g(Y.:) 52 b(Sp) s(ectral) 36 b(structure) i(of) e(Anderson) i(t) m(yp) s(e) g (Hamiltonians.) 33 b(In) m(v) m(en) m(t.) 347 3457 y(Math.) p Fh 33 w(141) p Fo(,) f(561) g(\(2000\)) 0 3660 y([JL3]) 133 b(Jak) -5 b(\024) -44 b(s) q(i) m(\023) e(c,) 30 b(V.,) i(Last,) f(Y.:) 43 b(Surface) 31 b(states) h(and) f(sp) s(ectra.) h(Comm) m(un.) e (Math.) h(Ph) m(ys.) p Fh 33 w(218) p Fo(,) g(459) 347 3781 y(\(2001\)) 0 3984 y([JM1]) 105 b(Jak) -5 b(\024) -44 b(s) q(i) m(\023) e(c,) 28 b(V.,) g(Molc) m(hano) m(v,) h(S.:) 41 b(Lo) s(calization) 24 b(of) j(surface) h(sp) s(ectra.) g(Comm) m(un.) f (Math.) h(Ph) m(ys.) p Fh 347 4104 a(208) p Fo(,) 33 b(153) e(\(1999\)) 0 4308 y([JM2]) 105 b(Jak) -5 b(\024) -44 b(s) q(i) m(\023) e(c,) 26 b(V.,) h(Molc) m(hano) m(v,) g(S.:) 41 b(On) 25 b(the) h(surface) g(sp) s(ectrum) g(in) e(dimension) g(t) m(w) m(o.) i(Helv.) g(Ph) m(ys.) 347 4428 y(Acta) p Fh 33 w(71) p Fo(,) 32 b(629) g(\(1999\)) 0 4632 y([JM3]) 105 b(Jak) -5 b(\024) -44 b(s) q(i) m(\023) e(c,) 42 b(V.,) h(Molc) m(hano) m(v,) g(S.:) 60 b(W) -8 b(a) m(v) m(e) 42 b(op) s(erators) f(for) f (the) h(surface) h(Maryland) e(mo) s(del.) f(J.) 347 4752 y(Math.) 33 b(Ph) m(ys.) p Fh 34 w(41) p Fo(,) g(4452) e(\(2000\)) 0 4955 y([JMP]) 88 b(Jak) -5 b(\024) -44 b(s) q(i) m(\023) e(c,) 37 b(V.,) h(Molc) m(hano) m(v,) h(S.,) f(P) m(astur,) h(L.:) 52 b(On) 36 b(the) h(propagation) f(prop) s(erties) g(of) h(surface) 347 5076 y(w) m(a) m(v) m(es.) i(In) p Fy 37 w(Wave) g(Pr) -5 b(op) g(agation) 38 b(in) g(Complex) f(Me) -5 b(dia) p Fo(,) 38 b(IMA) f(V) -8 b(ol.) 36 b(Math.) h(Appl.) p Fh 36 w(96) p Fo(,) h(143) 347 5196 y(\(1998\)) p 90 rotate dyy eop %%Page: 16 16 16 15 bop Fo 3682 100 a(16) 0 407 y([Kr]) 179 b(Krishna,) 35 b(M.:) 49 b(Anderson) 36 b(mo) s(del) d(with) i(deca) m(ying) g (randomness-extended) i(states.) f(Pro) s(c.) 347 527 y(Indian) c(Acad.) h(Sci.) f(\(MathSci.\)) p Fh 33 w(100) p Fo(,) g(285) g(\(1990\)) 0 731 y([KP]) 151 b(Khoruzenk) m(o,) 54 b(B.A.,) f(P) m(astur) c(L.:) 76 b(The) 49 b(lo) s(calization) c(of) j (surface) h(states:) 76 b(An) 49 b(exactly) 347 851 y(solv) -5 b(able) 31 b(mo) s(del.) g(Ph) m(ysics) k(Rep) s(orts) p Fh 32 w(288) p Fo(,) e(109) f(\(1997\)) 0 1054 y([MV1]) 82 b(Molc) m(hano) m(v,) 52 b(S.,) f(V) -8 b(ain) m(b) s(erg,) 51 b(B.:) 74 b(Scattering) 47 b(on) g(the) h(system) g(of) f(the) h (sparse) h(bumps:) 347 1175 y(m) m(ultidimensional) 28 b(case.) 33 b(Appl.) g(Anal.) p Fh 31 w(71) p Fo(,) g(167) f(\(1999\)) 0 1378 y([MV2]) 82 b(Molc) m(hano) m(v,) 39 b(S.,) h(V) -8 b(ain) m(b) s(erg,) 38 b(B.:) 54 b(Sp) s(ectrum) 37 b(of) g(m) m (ultidimensional) c(Sc) m(hr\177) -49 b(odinger) 38 b(op) s(era-) 347 1499 y(tors) c(with) f(sparse) j(p) s(oten) m(tials.) c(Analytical) g (and) i(computational) e(metho) s(ds) h(in) h(scattering) 347 1619 y(and) 28 b(applied) g(mathematics) f(Chapman) h(Hall/CR) m(C) f (Res.) i(Notes) h(Math.) p Fh 28 w(417) p Fo(,) g(231) e(\(2000\)) 0 1822 y([RS]) 167 b(Reed,) 37 b(M.,) f(Simon,) f(B.:) p Fy 49 w(Metho) -5 b(ds) 37 b(of) g(Mo) -5 b(dern) 37 b(Mathematic) -5 b(al) 37 b(Physics) g(IV.) g(A) n(nalysis) g(of) 347 1943 y(Op) -5 b(er) g(ators.) p Fo 32 w(Academic) 32 b(Press) i(Inc,) f(Boston) g(1978) 0 2146 y([RoSh]) 64 b(Ro) s(dnianski,) 35 b(I.,) h(Sc) m(hlag,) g(W.:) 49 b(Classical) 34 b(and) i(quan) m(tum) f(scattering) g(for) g(a) g (class) h(of) f(long) 347 2267 y(range) d(random) g(p) s(oten) m (tials.) g(Preprin) m(t.) 0 2470 y([S]) 239 b(Simon,) 39 b(B.:) 58 b(Sp) s(ectral) 38 b(analysis) h(of) g(rank) h(one) f(p) s (erturbations) g(and) h(applications.) d(CRM) 347 2590 y(Pro) s(c.) c(Lecture) g(Notes,) g(Pro) m(vidence,) i(RI,) p Fh 32 w(8) p Fo(,) e(1995) 0 2794 y([SW]) 139 b(Simon,) 26 b(B.,) h(W) -8 b(ol\013,) 27 b(T.:) 41 b(Singular) 24 b(con) m(tin) m(uous) j(sp) s(ectrum) f(under) h(rank) f(one) h(p) s (erturbations) 347 2914 y(and) 41 b(lo) s(calization) 36 b(for) 41 b(random) e(Hamiltonians.) f(Comm) m(un.) i(Pure) i(Appl.) e (Math.) p Fh 41 w(39) p Fo(,) j(75) 347 3034 y(\(1986\)) p 90 rotate dyy eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF