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p Ft 1046 4634 a(H) p Fv 1122 4634 a(,) p 1172 4634 a(and) p 1334 4634 a(the) p 1477 4634 a(form) p 1673 4634 a(domains) p 2001 4634 a(are) p 2140 4634 a(giv) n(en) p 2356 4634 a(b) n(y) p Fm 1267 4816 a(Q) p Fv(\() p Ft(H) p Fp 1430 4782 a(D) 1423 4837 y(L) p Fv 1490 4816 a(\)) p 1546 4816 a(=) p Fm 1633 4816 a(D) p Fv(\() p Fs(j) p Ft(H) p Fp 1833 4782 a(D) 1826 4837 y(L) p Fs 1894 4816 a(j) p Fr 1917 4782 a(1) p Fp(=) p Fr(2) p Fv 2021 4816 a(\)) p 2076 4816 a(=) p Ft 2164 4816 a(H) p Fr 2240 4782 a(1) 2233 4837 y(0) p Fv 2277 4816 a(\(\012) p Fp 2369 4828 a(L) p Fv 2419 4816 a(\)) p Ft(;) p Fm 1264 4957 a(Q) p Fv(\() p Ft(H) p Fp 1427 4923 a(N) 1420 4978 y(L) p Fv 1490 4957 a(\)) p 1546 4957 a(=) p Fm 1633 4957 a(D) p Fv(\() p Fs(j) p Ft(H) p Fp 1833 4923 a(D) 1826 4978 y(L) p Fs 1894 4957 a(j) p Fr 1917 4923 a(1) p Fp(=) p Fr(2) p Fv 2021 4957 a(\)) p 2076 4957 a(=) p Ft 2164 4957 a(H) p Fr 2240 4923 a(1) p Fv 2277 4957 a(\(\012) p Fp 2369 4969 a(L) p Fv 2419 4957 a(\)) p Ft(:) p Fv 609 5140 a(In) p 713 5140 a(this) p 875 5140 a(pap) r(er,) p 1131 5140 a(w) n(e) p 1253 5140 a(supp) r(ose) p Fw 609 5289 a(Assumption) p 1139 5289 a(A.) p Ft 1279 5289 a(V) p Fv 1346 5289 a(\() p Ft(x) p Fv(\)) p 1485 5289 a(and) p Fs 1647 5289 a(f) p Ft(B) p Fp 1752 5301 a(ij) p Fv 1810 5289 a(\() p Ft(x) p Fv(\)) p Fs(g) p Fv 1991 5289 a(are) p 2130 5289 a(uniformly) p 2508 5289 a(b) r(ounded.) p Fy 1854 5652 a(1) p 90 rotate dyy eop %%Page: 2 2 2 1 bop Fw 609 548 a(Theorem) p 1015 548 a(1.) p Fl 1130 548 a(Supp) l(ose) p 1447 548 a(Assumption) p 1904 548 a(A.) p 2021 548 a(Then) p 2238 548 a(for) p 2371 548 a(any) p Ft 2531 548 a(f) p Fs 2604 548 a(2) p Ft 2683 548 a(C) p Fr 2748 518 a(1) 2742 569 y(0) p Fv 2786 548 a(\() p Fq(R) p Fv(\)) p Fl 2905 548 a(,) p 2966 548 a(ther) l(e) 609 648 y(exists) p Ft 837 648 a(C) p 925 648 a(>) p Fv 1012 648 a(0) p Fl 1084 648 a(such) p 1273 648 a(that) p Fn 1079 747 a(\014) 1079 797 y(\014) p Fv 1106 817 a(T) p 1159 817 a(r[) p Ft(f) p Fv 1265 817 a(\() p Ft(H) p Fp 1373 783 a(N) 1366 838 y(L) p Fv 1436 817 a(\)) p Fs 1486 817 a(\000) p Ft 1569 817 a(f) p Fv 1619 817 a(\() p Ft(H) p Fp 1727 783 a(D) 1720 838 y(L) p Fv 1787 817 a(\)]) p Fn 1842 747 a(\014) 1842 797 y(\014) p Fs 1893 817 a(\024) p Ft 1981 817 a(C) p 2046 817 a(L) p Fp 2103 783 a(d) p Fk(\000) p Fr(1) p Ft 2226 817 a(;) p 2433 817 a(L) p Fs 2513 817 a(2) p Fq 2591 817 a(N) p Ft 2645 817 a(:) p Fl 609 987 a(In) p 718 987 a(p) l(articular,) p Fv 1184 1202 a(lim) p Fp 1153 1255 a(L) p Fk(!1) p Fn 1331 1085 a(\024) p Fv 1411 1146 a(1) p 1385 1183 96 4 v Ft 1385 1259 a(L) p Fp 1442 1235 a(d) p Fv 1490 1202 a(T) p 1543 1202 a(r) p Ft 1575 1202 a(f) p Fv 1625 1202 a(\() p Ft(H) p Fp 1733 1168 a(N) 1726 1222 y(L) p Fv 1796 1202 a(\)) p Fs 1847 1202 a(\000) p Fv 1967 1146 a(1) p 1940 1183 V Ft 1940 1259 a(L) p Fp 1997 1235 a(d) p Fv 2045 1202 a(T) p 2098 1202 a(r) p Ft 2130 1202 a(f) p Fv 2180 1202 a(\() p Ft(H) p Fp 2288 1168 a(D) 2281 1222 y(L) p Fv 2348 1202 a(\)) p Fn 2380 1085 a(\025) p Fv 2447 1202 a(=) p 2535 1202 a(0) p Ft(:) p Fl 609 1417 a(R) l(emark) p 917 1417 a(.) p Fv 984 1417 a(Ev) n(en) p 1189 1417 a(though) p 1466 1417 a(w) n(e) p 1584 1417 a(supp) r(ose) p 1893 1417 a(\012) p Fp 1953 1429 a(L) p Fv 2026 1417 a(is) p 2105 1417 a(a) p 2171 1417 a(b) r(o) n(x,) p 2350 1417 a(w) n(e) p 2468 1417 a(ma) n(y) p 2644 1417 a(consider) p 2963 1417 a(more) 609 1516 y(general) p 901 1516 a(domains.) p 1276 1516 a(F) p 1323 1516 a(or) p 1431 1516 a(example,) p 1785 1516 a(it) p 1874 1516 a(is) p 1962 1516 a(easy) p 2150 1516 a(to) p 2257 1516 a(extend) p 2532 1516 a(our) p 2685 1516 a(result) p 2921 1516 a(to) p 3028 1516 a(the) 609 1616 y(case) p 786 1616 a(\012) p Fp 846 1628 a(L) p Fv 924 1616 a(is) p 1008 1616 a(a) p 1078 1616 a(ball.) p 1274 1616 a(In) p 1378 1616 a(fact,) p 1566 1616 a(the) p 1710 1616 a(shap) r(e) p 1944 1616 a(ma) n(y) p 2124 1616 a(b) r(e) p 2238 1616 a(di\013eren) n(t) p 2566 1616 a(for) p 2694 1616 a(eac) n(h) p Ft 2882 1616 a(L) p Fv(.) p 3000 1616 a(The) 609 1716 y(constan) n(t) p Ft 950 1716 a(C) p Fv 1048 1716 a(in) p 1151 1716 a(the) p 1300 1716 a(theorem) p 1628 1716 a(dep) r(ends) p 1955 1716 a(only) p 2143 1716 a(on) p 2264 1716 a(the) p 2412 1716 a(constan) n(ts) p 2786 1716 a(app) r(earing) 609 1815 y(in) p 716 1815 a(Lemmas) p 1054 1815 a(7) p 1133 1815 a(and) p 1304 1815 a(10,) p 1450 1815 a(and) p 1621 1815 a(they) p 1817 1815 a(are) p 1965 1815 a(easily) p 2204 1815 a(extended) p 2566 1815 a(to) p 2677 1815 a(domain) p 2982 1815 a(with) 609 1915 y(piecewise) p Ft 982 1915 a(C) p Fr 1047 1885 a(1) p Fv 1085 1915 a(-b) r(oundary) p 1453 1915 a(.) p 1550 1915 a(Roughly) p 1894 1915 a(sp) r(eaking,) p 2275 1915 a(T) p 2328 1915 a(r\() p Ft(f) p Fv 2443 1915 a(\() p Ft(H) p Fp 2551 1885 a(N) 2544 1938 y(L) p Fv 2614 1915 a(\)) p Fs 2673 1915 a(\000) p Ft 2765 1915 a(f) p Fv 2815 1915 a(\() p Ft(H) p Fp 2923 1885 a(D) 2916 1938 y(L) p Fv 2982 1915 a(\)\)) p 3088 1915 a(is) 609 2014 y(b) r(ounded) p 947 2014 a(b) n(y) p 1062 2014 a(the) p 1204 2014 a(area) p 1384 2014 a(of) p Ft 1478 2014 a(@) p Fv 1527 2014 a(\012) p Fp 1587 2026 a(L) p Fv 1663 2014 a(in) p 1759 2014 a(general,) p 2068 2014 a(pro) n(vided) p Ft 2409 2014 a(@) p Fv 2458 2014 a(\012) p Fp 2518 2026 a(L) p Fv 2594 2014 a(is) p 2677 2014 a(not) p 2824 2014 a(to) r(o) p 2968 2014 a(wild.) 734 2136 y(No) n(w) p 927 2136 a(w) n(e) p 1054 2136 a(apply) p 1287 2136 a(this) p 1453 2136 a(result) p 1689 2136 a(to) p 1795 2136 a(the) p 1942 2136 a(in) n(tegrated) p 2341 2136 a(densit) n(y) p 2632 2136 a(of) p 2731 2136 a(states) p 2972 2136 a(\(IDS) 609 2236 y(for) p 736 2236 a(short\).) p 1014 2236 a(W) p 1092 2236 a(e) p 1157 2236 a(supp) r(ose) p Fw 609 2380 a(Assumption) p 1139 2380 a(B.) p Fv 1275 2380 a(F) p 1322 2380 a(or) p 1424 2380 a(almost) p 1691 2380 a(all) p Ft 1807 2380 a(\025) p Fs 1878 2380 a(2) p Fq 1957 2380 a(R) p Fv(,) p Ft 1089 2584 a(k) p Fp 1135 2550 a([) p Fv 1166 2584 a(\() p Ft(\025) p Fv(\)) p 1302 2584 a(=) p 1421 2584 a(lim) p Fp 1389 2638 a(L) p Fk(!1) p Fv 1618 2528 a(1) p 1591 2565 V Ft 1591 2641 a(L) p Fp 1648 2617 a(d) p Fv 1696 2584 a(#) p Fn 1765 2517 a(\010) p Fv 1814 2584 a(eigen) n(v) p 2036 2584 a(alues) p 2243 2584 a(of) p Ft 2338 2584 a(H) p Fp 2414 2550 a([) 2407 2605 y(L) p Fs 2479 2584 a(\024) p Ft 2567 2584 a(\025) p Fn 2615 2517 a(\011) p Fv 609 2787 a(exists,) p 861 2787 a(where) p Ft 1102 2787 a([) p Fv 1157 2787 a(=) p Ft 1245 2787 a(D) p Fv 1343 2787 a(or) p Ft 1445 2787 a(N) p Fv 1521 2787 a(.) 734 2931 y(If) p Ft 818 2931 a(H) p Fv 922 2931 a(is) p 1006 2931 a(an) p 1122 2931 a(ergo) r(dic) p 1412 2931 a(\(random\)) p 1782 2931 a(op) r(erator,) p 2141 2931 a(Assumption) p 2601 2931 a(B) p 2689 2931 a(is) p 2773 2931 a(pro) n(v) n(ed) p 3044 2931 a(for) 609 3031 y(almost) p 887 3031 a(all) p 1012 3031 a(samples) p 1332 3031 a(b) n(y) p 1457 3031 a(the) p 1610 3031 a(subadditiv) n(e) p 2063 3031 a(ergo) r(dic) p 2361 3031 a(theorem) p 2694 3031 a(\(see) p 2871 3031 a([4) o(]) p 2996 3031 a(Sec-) 609 3130 y(tion) p 785 3130 a(7.3\).) p 1001 3130 a(Since) p Ft 1223 3130 a(L) p Fk 1280 3100 a(\000) p Fp(d) p Fv 1370 3130 a(#) p Fs(f) p Fv(e.v.'s) p 1691 3130 a(of) p Ft 1786 3130 a(H) p Fp 1862 3100 a([) 1855 3153 y(L) p Fs 1937 3130 a(\024) p Ft 2034 3130 a(\025) p Fs(g) p Fv 2157 3130 a(is) p 2246 3130 a(uniformly) p 2630 3130 a(b) r(ounded,) p 2999 3130 a(it) p 3088 3130 a(is) 609 3230 y(easy) p 792 3230 a(to) p 893 3230 a(see) p 1028 3230 a(that) 905 3438 y(lim) p Fp 874 3491 a(L) p Fk(!1) p Fv 1102 3381 a(1) p 1076 3419 V Ft 1076 3495 a(L) p Fp 1133 3471 a(d) p Fv 1181 3438 a(T) p 1234 3438 a(r) p Ft 1266 3438 a(f) p Fv 1316 3438 a(\() p Ft(H) p Fp 1424 3403 a([) 1417 3458 y(L) p Fv 1467 3438 a(\)) p 1522 3438 a(=) p Fs 1610 3438 a(\000) p Fn 1689 3325 a(Z) p Ft 1785 3438 a(f) p Fk 1835 3403 a(0) p Fv 1858 3438 a(\() p Ft(\025) p Fv(\)) p Ft(k) p Fp 2016 3403 a([) p Fv 2048 3438 a(\() p Ft(\025) p Fv(\)) p Ft(d\025) p Fv 2276 3438 a(=) p Fn 2363 3325 a(Z) p Ft 2460 3438 a(f) p Fv 2510 3438 a(\() p Ft(\025) p Fv(\)) p Ft(dk) p Fp 2711 3403 a([) p Fv 2743 3438 a(\() p Ft(\025) p Fv(\)) p Ft(;) p Fv 609 3650 a(where) p Ft 853 3650 a([) p Fv 915 3650 a(=) p Ft 1009 3650 a(D) p Fv 1112 3650 a(or) p Ft 1217 3650 a(N) p Fv 1293 3650 a(.) p 1365 3650 a(Then) p 1585 3650 a(the) p 1732 3650 a(next) p 1923 3650 a(corollary) p 2271 3650 a(follo) n(ws) p 2547 3650 a(from) p 2747 3650 a(Theorem) p 3102 3650 a(1) 609 3749 y(and) p 771 3749 a(the) p 914 3749 a(fundamen) n(tal) p 1393 3749 a(lemma) p 1661 3749 a(of) p 1755 3749 a(v) p 1794 3749 a(ariations.) p Fw 609 3894 a(Corollary) p 1034 3894 a(2.) p Fl 1150 3894 a(Supp) l(ose) p 1471 3894 a(Assumptions) p 1967 3894 a(A) p 2064 3894 a(and) p 2231 3894 a(B.) p 2350 3894 a(Then) p Ft 2572 3894 a(k) p Fp 2618 3863 a(D) p Fv 2678 3894 a(\() p Ft(\025) p Fv(\)) p 2824 3894 a(=) p Ft 2922 3894 a(k) p Fp 2968 3863 a(N) p Fv 3030 3894 a(\() p Ft(\025) p Fv(\)) p Fl 609 3993 a(for) p 740 3993 a(almost) p 1003 3993 a(al) t(l) p Ft 1119 3993 a(\025) p Fs 1191 3993 a(2) p Fq 1269 3993 a(R) p Fl(.) p 1392 3993 a(Mor) l(e) l(over,) p 1781 3993 a(if) p 1860 3993 a(either) p Ft 2094 3993 a(k) p Fp 2140 3963 a(D) p Fl 2228 3993 a(or) p Ft 2333 3993 a(k) p Fp 2379 3963 a(N) p Fl 2469 3993 a(is) p 2556 3993 a(c) l(ontinuous) p 2972 3993 a(at) p Ft 3070 3993 a(\025) p Fl(,) 609 4093 y(then) p Ft 794 4093 a(k) p Fp 840 4063 a(D) p Fv 900 4093 a(\() p Ft(\025) p Fv(\)) p 1036 4093 a(=) p Ft 1124 4093 a(k) p Fp 1170 4063 a(N) p Fv 1232 4093 a(\() p Ft(\025) p Fv(\)) p Fl(.) 609 4265 y(Pr) l(o) l(of.) p Fv 873 4265 a(It) p 970 4265 a(remains) p 1286 4265 a(only) p 1475 4265 a(to) p 1583 4265 a(sho) n(w) p 1795 4265 a(the) p 1945 4265 a(latter) p 2178 4265 a(statemen) n(t.) p 2615 4265 a(Supp) r(ose) p Ft 2947 4265 a(k) p Fp 2993 4234 a(D) p Fv 3088 4265 a(is) 609 4364 y(con) n(tin) n(uous) p 1025 4364 a(at) p Ft 1127 4364 a(\025) p Fr 1175 4376 a(0) p Fv 1240 4364 a(and) p Ft 1402 4364 a(k) p Fp 1448 4334 a(D) p Fv 1508 4364 a(\() p Ft(\025) p Fr 1588 4376 a(0) p Fv 1626 4364 a(\)) p Fs 1682 4364 a(6) p Fv(=) p Ft 1770 4364 a(k) p Fp 1816 4334 a(N) p Fv 1879 4364 a(\() p Ft(\025) p Fr 1959 4376 a(0) p Fv 1997 4364 a(\).) p 2090 4364 a(Since) p Ft 2307 4364 a(k) p Fp 2353 4334 a(N) p Fv 2415 4364 a(\() p Ft(\025) p Fv(\)) p Fs 2552 4364 a(\025) p Ft 2640 4364 a(k) p Fp 2686 4334 a(D) p Fv 2746 4364 a(\() p Ft(\025) p Fv(\)) p 2887 4364 a(for) p 3014 4364 a(an) n(y) p Ft 609 4464 a(\025) p Fv(,) p Ft 719 4464 a(k) p Fp 765 4434 a(N) p Fv 828 4464 a(\() p Ft(\025) p Fr 908 4476 a(0) p Fv 946 4464 a(\)) p Ft 1016 4464 a(<) p 1118 4464 a(k) p Fp 1164 4434 a(D) p Fv 1224 4464 a(\() p Ft(\025) p Fr 1304 4476 a(0) p Fv 1342 4464 a(\).) p 1461 4464 a(By) p 1600 4464 a(the) p 1752 4464 a(assumption,) p 2225 4464 a(there) p 2446 4464 a(is) p Ft 2538 4464 a(") p 2615 4464 a(>) p Fv 2717 4464 a(0) p 2795 4464 a(suc) n(h) p 2991 4464 a(that) p Ft 609 4564 a(k) p Fp 655 4533 a(D) p Fv 715 4564 a(\() p Ft(\025) p Fv(\)) p Ft 852 4564 a(<) p 940 4564 a(k) p Fp 986 4533 a(N) p Fv 1049 4564 a(\() p Ft(\025) p Fr 1129 4576 a(0) p Fv 1167 4564 a(\)) p 1227 4564 a(for) p Ft 1355 4564 a(\025) p Fs 1427 4564 a(2) p Fv 1506 4564 a([) p Ft(\025) p Fr 1577 4576 a(0) p Fs 1633 4564 a(\000) p Ft 1717 4564 a(";) p 1793 4564 a(\025) p Fr 1841 4576 a(0) p Fv 1897 4564 a(+) p Ft 1980 4564 a(") p Fv(].) p 2103 4564 a(On) p 2242 4564 a(the) p 2385 4564 a(other) p 2602 4564 a(hand,) p Ft 2833 4564 a(k) p Fp 2879 4533 a(N) p Fv 2942 4564 a(\() p Ft(\025) p Fv(\)) p Fs 3079 4564 a(\025) p Ft 609 4663 a(k) p Fp 655 4633 a(N) p Fv 718 4663 a(\() p Ft(\025) p Fr 798 4675 a(0) p Fv 836 4663 a(\)) p 904 4663 a(for) p Ft 1040 4663 a(\025) p Fs 1125 4663 a(2) p Fv 1218 4663 a([) p Ft(\025) p Fr 1289 4675 a(0) p Ft 1326 4663 a(;) p 1363 4663 a(\025) p Fr 1411 4675 a(0) p Fv 1473 4663 a(+) p Ft 1562 4663 a(") p Fv(]) p 1660 4663 a(since) p Ft 1871 4663 a(k) p Fp 1917 4633 a(N) p Fv 1980 4663 a(\() p Ft(\025) p Fv(\)) p 2129 4663 a(is) p 2221 4663 a(monotone) p 2612 4663 a(nondecreasing.) 609 4763 y(Th) n(us) p 827 4763 a(w) n(e) p 957 4763 a(ha) n(v) n(e) p Ft 1156 4763 a(k) p Fp 1202 4733 a(N) p Fv 1265 4763 a(\() p Ft(\025) p Fv(\)) p Ft 1413 4763 a(>) p 1514 4763 a(k) p Fp 1560 4733 a(D) p Fv 1620 4763 a(\() p Ft(\025) p Fv(\)) p 1768 4763 a(on) p Ft 1891 4763 a(\025) p Fs 1975 4763 a(2) p Fv 2066 4763 a([) p Ft(\025) p Fr 2137 4775 a(0) p Ft 2175 4763 a(;) p 2212 4763 a(\025) p Fr 2260 4775 a(0) p Fv 2321 4763 a(+) p Ft 2409 4763 a(") p Fv(]) p 2506 4763 a(,) p 2566 4763 a(a) p 2643 4763 a(con) n(tradiction.) 609 4862 y(The) p 780 4862 a(other) p 997 4862 a(case) p 1173 4862 a(is) p 1256 4862 a(pro) n(v) n(ed) p 1526 4862 a(similarly) p 1832 4862 a(.) p 3083 4862 4 57 v 3087 4810 50 4 v 3087 4862 V 3136 4862 4 57 v 734 5034 a(By) p 864 5034 a(the) p 1006 5034 a(same) p 1214 5034 a(argumen) n(t,) p 1608 5034 a(w) n(e) p 1729 5034 a(can) p 1881 5034 a(also) p 2047 5034 a(pro) n(v) n(e) p 2271 5034 a(that) p 2450 5034 a(the) p 2593 5034 a(in) n(tegrated) p 2987 5034 a(den-) 609 5134 y(sit) n(y) p 765 5134 a(of) p 859 5134 a(states) p 1094 5134 a(is) p 1176 5134 a(indep) r(enden) n(t) p 1646 5134 a(of) p 1739 5134 a(the) p 1881 5134 a(c) n(hoice) p 2127 5134 a(of) p 2221 5134 a(b) r(oundary) p 2593 5134 a(conditions.) p 3022 5134 a(Let) p Ft 609 5233 a(H) p Fr 685 5203 a(min) p Fp 678 5256 a(L) p Fv 827 5233 a(b) r(e) p 940 5233 a(the) p 1083 5233 a(op) r(erator) p 1418 5233 a(de\014ned) p 1704 5233 a(b) n(y) p Ft 859 5403 a(H) p Fr 935 5369 a(min) p Fp 928 5424 a(L) p Ft 1049 5403 a(') p Fv 1126 5403 a(=) p 1214 5403 a([\() p Ft(p) p Fs 1330 5403 a(\000) p Ft 1413 5403 a(A) p Fv(\)) p Fr 1507 5369 a(2) p Fv 1563 5403 a(+) p Ft 1646 5403 a(V) p Fv 1713 5403 a(]) p Ft(';) p 1993 5403 a(') p Fs 2071 5403 a(2) p Fm 2149 5403 a(D) p Fv(\() p Ft(H) p Fr 2326 5369 a(min) p Fp 2319 5424 a(L) p Fv 2440 5403 a(\)) p 2495 5403 a(=) p Ft 2583 5403 a(H) p Fr 2659 5369 a(2) 2652 5424 y(0) p Fv 2696 5403 a(\(\012) p Fp 2788 5415 a(L) p Fv 2838 5403 a(\)) p Ft(:) p Fy 1854 5652 a(2) p 90 rotate dyy eop %%Page: 3 3 3 2 bop Fv 609 548 a(It) p 706 548 a(is) p 796 548 a(w) n(ell-kno) n (wn) p 1234 548 a(that) p 1421 548 a(if) p Ft 1504 548 a(H) p Fp 1573 560 a(L) p Fv 1657 548 a(is) p 1748 548 a(a) p 1824 548 a(self-adjoin) n(t) p 2257 548 a(extension) p 2631 548 a(of) p Ft 2733 548 a(H) p Fr 2809 518 a(min) p Fp 2802 571 a(L) p Fv 2923 548 a(,) p 2982 548 a(then) p Ft 609 648 a(H) p Fp 685 617 a(N) 678 670 y(L) p Fs 771 648 a(\024) p Ft 859 648 a(H) p Fp 928 660 a(L) p Fs 1000 648 a(\024) p Ft 1088 648 a(H) p Fp 1164 617 a(D) 1157 670 y(L) p Fv 1224 648 a(.) p Fw 609 797 a(Corollary) p 1034 797 a(3.) p Fl 1150 797 a(Supp) l(ose) p 1462 797 a(Assumption) p 1915 797 a(A,) p 2028 797 a(and) p 2186 797 a(let) p Ft 2299 797 a(H) p Fp 2368 809 a(L) p Fl 2443 797 a(b) l(e) p 2542 797 a(a) p 2610 797 a(self-adjoint) p 3037 797 a(ex-) 609 896 y(tension) p 900 896 a(of) p Ft 997 896 a(H) p Fj 1073 866 a(min) p Fp 1066 919 a(L) p Fl 1222 896 a(for) p 1355 896 a(e) l(ach) p Ft 1542 896 a(L) p Fl(.) p 1662 896 a(Supp) l(ose) p Ft 1104 1119 a(k) p Fv 1150 1119 a(\() p Ft(\025) p Fv(\)) p 1286 1119 a(=) p 1405 1119 a(lim) p Fp 1374 1173 a(L) p Fk(!1) p Fv 1602 1063 a(1) p 1575 1100 96 4 v Ft 1575 1176 a(L) p Fp 1632 1152 a(d) p Fv 1681 1119 a(#) p Fn 1750 1052 a(\010) p Fv 1798 1119 a(eigen) n(v) p 2020 1119 a(alues) p 2228 1119 a(of) p Ft 2322 1119 a(H) p Fp 2391 1131 a(L) p Fs 2464 1119 a(\024) p Ft 2552 1119 a(\025) p Fn 2600 1052 a(\011) p Fl 609 1341 a(exists) p 839 1341 a(for) p 975 1341 a(almost) p 1243 1341 a(al) t(l) p Ft 1364 1341 a(\025) p Fl(.) p 1485 1341 a(Then) p Ft 1704 1341 a(k) p Fp 1750 1311 a(D) p Fv 1810 1341 a(\() p Ft(\025) p Fv(\)) p Fl 1955 1341 a(and) p Ft 2120 1341 a(k) p Fp 2166 1311 a(N) p Fv 2228 1341 a(\() p Ft(\025) p Fv(\)) p Fl 2374 1341 a(exist) p 2570 1341 a(for) p 2705 1341 a(almost) p 2973 1341 a(al) t(l) p Ft 3095 1341 a(\025) p Fl 609 1441 a(and) p Ft 772 1441 a(k) p Fp 818 1411 a(D) p Fv 878 1441 a(\() p Ft(\025) p Fv(\)) p 1018 1441 a(=) p Ft 1109 1441 a(k) p Fp 1155 1411 a(N) p Fv 1217 1441 a(\() p Ft(\025) p Fv(\)) p 1357 1441 a(=) p Ft 1448 1441 a(k) p Fv 1494 1441 a(\() p Ft(\025) p Fv(\)) p Fl 1638 1441 a(\(a.e.) p Ft 1847 1441 a(\025) p Fl(\).) p 1999 1441 a(Mor) l(e) l(over,) p 2391 1441 a(if) p Ft 2474 1441 a(k) p Fv 2520 1441 a(\() p Ft(\025) p Fv(\)) p Fl 2664 1441 a(is) p 2755 1441 a(c) l(ontinuous) 609 1540 y(at) p Ft 709 1540 a(\025) p Fl(,) p 812 1540 a(then) p Ft 997 1540 a(k) p Fp 1043 1510 a(D) p Fv 1103 1540 a(\() p Ft(\025) p Fv(\)) p 1239 1540 a(=) p Ft 1327 1540 a(k) p Fp 1373 1510 a(N) p Fv 1436 1540 a(\() p Ft(\025) p Fv(\)) p 1572 1540 a(=) p Ft 1659 1540 a(k) p Fv 1705 1540 a(\() p Ft(\025) p Fv(\)) p Fl(.) p Fv 734 1689 a(It) p 826 1689 a(is) p 912 1689 a(widely) p 1175 1689 a(b) r(eliev) n(ed) p 1498 1689 a(that) p 1681 1689 a(the) p 1826 1689 a(IDS) p 1996 1689 a(is) p 2082 1689 a(indep) r(enden) n(t) p 2555 1689 a(of) p 2652 1689 a(the) p 2797 1689 a(b) r(oundary) 609 1789 y(conditions,) p 1037 1789 a(and) p 1204 1789 a(often) p 1420 1789 a(it) p 1509 1789 a(is) p 1598 1789 a(implicitly) p 1975 1789 a(assumed.) p 2358 1789 a(Ho) n(w) n(ev) n(er,) p 2723 1789 a(it) p 2812 1789 a(is) p 2901 1789 a(pro) n(v) n(ed) 609 1889 y(only) p 794 1889 a(for) p 924 1889 a(Sc) n(hr\177) p 1084 1889 a(odinger) p 1381 1889 a(op) r(erators) p 1751 1889 a(with) p 1943 1889 a(ergo) r(dic) p 2234 1889 a(p) r(oten) n (tials) p 2620 1889 a(\(without) p 2964 1889 a(mag-) 609 1988 y(netic) p 811 1988 a(\014eld\)) p 1022 1988 a(using) p 1238 1988 a(sto) r(c) n(hastic) p 1621 1988 a(metho) r(d) p 1922 1988 a(\(cf.) p 2066 1988 a([4) o(],) p 2203 1988 a([2]\).) p 2383 1988 a(It) p 2471 1988 a(app) r(ears) p 2778 1988 a(that) p 2956 1988 a(op) r(er-) 609 2088 y(ator) p 785 2088 a(theoretical) p 1197 2088 a(pro) r(of) p 1415 2088 a(has) p 1563 2088 a(not) p 1712 2088 a(b) r(een) p 1908 2088 a(kno) n(wn,) p 2195 2088 a(and) p 2357 2088 a(our) p 2506 2088 a(result) p 2738 2088 a(can) p 2891 2088 a(b) r(e) p 3005 2088 a(also) 609 2188 y(applied) p 899 2188 a(to) p 1000 2188 a(deterministic) p 1498 2188 a(Sc) n(hr\177) p 1658 2188 a(odinger) p 1952 2188 a(op) r(erators,) p 2342 2188 a(pro) n(vided) p 2683 2188 a(that) p 2862 2188 a(the) p 3004 2188 a(IDS) 609 2287 y(can) p 765 2287 a(b) r(e) p 881 2287 a(de\014ned) p 1170 2287 a(in) p 1271 2287 a(the) p 1417 2287 a(sense) p 1633 2287 a(of) p 1731 2287 a(Corollary) p 2102 2287 a(3.) p 2213 2287 a(This) p 2406 2287 a(w) n(ork) p 2613 2287 a(is) p 2699 2287 a(partly) p 2950 2287 a(moti-) 609 2387 y(v) p 648 2387 a(ated) p 826 2387 a(b) n(y) p 935 2387 a(an) p 1043 2387 a(application) p 1465 2387 a(to) p 1560 2387 a(Sc) n(hr\177) p 1720 2387 a(odinger) p 2008 2387 a(op) r(erator) p 2336 2387 a(with) p 2518 2387 a(random) p 2816 2387 a(magnetic) 609 2487 y(\014eld) p 795 2487 a(\(cf.) p 946 2487 a([9],) p 1092 2487 a([5],) p 1238 2487 a([6) o(]\).) p 1436 2487 a(W) p 1514 2487 a(e) p 1585 2487 a(note) p 1775 2487 a(that) p 1961 2487 a(a) p 2036 2487 a(similar) p 2315 2487 a(idea) p 2496 2487 a(can) p 2654 2487 a(b) r(e) p 2773 2487 a(applied) p 3070 2487 a(to) 609 2586 y(discrete) p 910 2586 a(magnetic) p 1260 2586 a(Sc) n(hr\177) p 1420 2586 a(odinger) p 1710 2586 a(op) r(erators,) p 2097 2586 a(and) p 2254 2586 a(in) p 2347 2586 a(fact) p 2506 2586 a(it) p 2584 2586 a(is) p 2663 2586 a(m) n(uc) n(h) p 2880 2586 a(simpler) 609 2686 y(for) p 736 2686 a(the) p 878 2686 a(discrete) p 1182 2686 a(case.) p 1390 2686 a(W) p 1468 2686 a(e) p 1533 2686 a(refer) p 1724 2686 a(App) r(endix) p 2104 2686 a(of) p 2198 2686 a([5) o(]) p 2312 2686 a(for) p 2439 2686 a(a) p 2508 2686 a(pro) r(of) p 2724 2686 a(of) p Ft 2818 2686 a(k) p Fp 2864 2656 a(D) p Fv 2947 2686 a(=) p Ft 3035 2686 a(k) p Fp 3081 2656 a(N) p Fv 609 2785 a(whic) n(h) p 847 2785 a(emplo) n(y) p 1133 2785 a(this) p 1295 2785 a(idea.) 734 2885 y(This) p 926 2885 a(pap) r(er) p 1162 2885 a(is) p 1248 2885 a(organized) p 1625 2885 a(as) p 1730 2885 a(follo) n(ws:) p 2040 2885 a(W) p 2118 2885 a(e) p 2186 2885 a(giv) n(e) p 2359 2885 a(a) p 2431 2885 a(pro) r(of) p 2651 2885 a(of) p 2748 2885 a(Theorem) p 3102 2885 a(1) 609 2985 y(in) p 712 2985 a(Section) p 1008 2985 a(2.) p 1125 2985 a(A) p 1221 2985 a(k) n(ey) p 1376 2985 a(estimate) p 1714 2985 a(is) p 1803 2985 a(Prop) r(osition) p 2256 2985 a(5,) p 2355 2985 a(and) p 2522 2985 a(its) p 2643 2985 a(pro) r(of) p 2865 2985 a(is) p 2954 2985 a(giv) n(en) 609 3084 y(in) p 711 3084 a(Section) p 1007 3084 a(3.) p 1124 3084 a(The) p 1300 3084 a(pro) r(of) p 1522 3084 a(of) p 1622 3084 a(a) p 1697 3084 a(tec) n(hnical) p 2050 3084 a(lemma) p 2323 3084 a(\(Lemma) p 2657 3084 a(9\)) p 2763 3084 a(is) p 2852 3084 a(giv) n(en) p 3074 3084 a(in) 609 3184 y(App) r(endix) p 983 3184 a(A.) p 1090 3184 a(As) p 1207 3184 a(a) p 1270 3184 a(simple) p 1522 3184 a(\(almost) p 1816 3184 a(trivial\)) p 2090 3184 a(application) p 2512 3184 a(of) p 2601 3184 a(Corollary) p 2962 3184 a(3,) p 3049 3184 a(w) n(e) 609 3284 y(compute) p 946 3284 a(the) p 1089 3284 a(IDS) p 1256 3284 a(for) p 1383 3284 a(the) p 1526 3284 a(Landau) p 1827 3284 a(Hamiltonian) p 2304 3284 a(on) p Fq 2420 3284 a(R) p Fr 2474 3253 a(2) p Fv 2545 3284 a(in) p 2641 3284 a(App) r(endix) p 3022 3284 a(B.) p Fw 734 3383 a(Notation:) p Fv 1171 3383 a(W) p 1249 3383 a(e) p 1315 3383 a(denote) p 1584 3383 a(the) p 1729 3383 a(trace) p 1938 3383 a(ideal) p 2138 3383 a(of) p 2234 3383 a(order) p Ft 2453 3383 a(r) p Fv 2522 3383 a(b) n(y) p Fi 2639 3383 a(I) p Fp 2685 3395 a(r) p Fv 2722 3383 a(.) p 2786 3383 a(The) p 2959 3383 a(inner) 609 3483 y(pro) r(duct) p 916 3483 a(of) p Ft 1005 3483 a(L) p Fr 1062 3453 a(2) p Fv 1099 3483 a(-spaces) p 1376 3483 a(is) p 1454 3483 a(denoted) p 1763 3483 a(b) n(y) p Fs 1873 3483 a(h) p Ft(';) p 1996 3483 a( ) p Fs 2053 3483 a(i) p Fv 2109 3483 a(=) p Fn 2197 3416 a(R) p 2266 3437 55 4 v Ft 2266 3483 a(' ) s(dx) p Fv(,) p 2515 3483 a(and) p 2671 3483 a(the) p 2809 3483 a(norm) p Fs 3021 3483 a(k) p 3071 3483 a(\001) p 3102 3483 a(k) p Fv 609 3582 a(without) p 921 3582 a(an) n(y) p 1080 3582 a(subscript) p 1439 3582 a(is) p 1525 3582 a(the) p Ft 1670 3582 a(L) p Fr 1727 3552 a(2) p Fv 1764 3582 a(-norm.) p 2049 3582 a(#) p Ft(A) p Fv 2210 3582 a(denotes) p 2513 3582 a(the) p 2659 3582 a(cardinalit) n(y) p 3077 3582 a(of) 609 3682 y(a) p 678 3682 a(set) p Ft 808 3682 a(A) p Fv(.) p Fu 609 3956 a(2) p 811 3956 a(Pro) t(of) p 1182 3956 a(of) p 1335 3956 a(Theorem) p 1905 3956 a(1) p Fv 609 4138 a(A) n(t) p 730 4138 a(\014rst,) p 925 4138 a(w) n(e) p 1047 4138 a(recall) p 1269 4138 a(the) p 1413 4138 a(Krein's) p 1700 4138 a(theory) p 1961 4138 a(of) p 2056 4138 a(the) p 2200 4138 a(sp) r(ectral) p 2512 4138 a(shift) p 2700 4138 a(function) p 3026 4138 a(\(cf.) 609 4238 y([1],) p 748 4238 a([10) o(]\).) p Fw 609 4387 a(Theorem) p 1015 4387 a(4) p 1094 4387 a(\(Krein\).) p Fl 1474 4387 a(L) l(et) p Ft 1625 4387 a(A) p Fr 1687 4399 a(1) p Fl 1762 4387 a(and) p Ft 1931 4387 a(A) p Fr 1993 4399 a(2) p Fl 2068 4387 a(b) l(e) p 2177 4387 a(self-adjoint) p 2616 4387 a(op) l(er) l(ators) p 2984 4387 a(such) 609 4487 y(that) p Ft 778 4487 a(W) p Fv 891 4487 a(=) p Ft 978 4487 a(A) p Fr 1040 4499 a(1) p Fs 1094 4487 a(\000) p Ft 1175 4487 a(A) p Fr 1237 4499 a(2) p Fs 1297 4487 a(2) p Fi 1376 4487 a(I) p Fr 1422 4499 a(1) p Fl 1488 4487 a(is) p 1576 4487 a(a) p 1647 4487 a(tr) l(ac) l(e) p 1848 4487 a(class) p 2047 4487 a(op) l(er) l(ator.) p 2407 4487 a(Then) p 2622 4487 a(ther) l(e) p 2828 4487 a(exists) p 3054 4487 a(an) 609 4586 y(inte) l(gr) l(able) p 982 4586 a(function) p Ft 1308 4586 a(\030) p Fv 1348 4586 a(\() p Ft(\025) p Fv(\)) p Fl 1491 4586 a(on) p Fq 1610 4586 a(R) p Fl 1700 4586 a(such) p 1888 4586 a(that) p Fv 1218 4810 a(T) p 1271 4810 a(r) o(\() p Ft(f) p 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5403 a(the) p 1528 5403 a(tr) l(ac) l(e) p 1731 5403 a(norm.) p Fy 1854 5652 a(3) p 90 rotate dyy eop %%Page: 4 4 4 3 bop Fv 734 548 a(The) p 917 548 a(function) p Ft 1255 548 a(\030) p Fv 1295 548 a(\() p Ft(\025) p Fv(\)) p 1449 548 a(in) p 1558 548 a(the) p 1714 548 a(Theorem) p 2077 548 a(4) p 2159 548 a(is) p 2255 548 a(called) p 2503 548 a(the) p 2659 548 a(sp) r(ectral) p 2984 548 a(shift) 609 648 y(function.) p 967 648 a(W) p 1045 648 a(e) p 1110 648 a(often) p 1319 648 a(denote) p Ft 1587 648 a(\030) p Fv 1627 648 a(\() p Ft(\025) p Fv(\)) p 1763 648 a(=) p Ft 1851 648 a(\030) p Fv 1891 648 a(\() p Ft(\025) p Fv(;) p Ft 2008 648 a(A) p Fr 2070 660 a(1) p Ft 2108 648 a(;) p 2145 648 a(A) p Fr 2207 660 a(2) p Fv 2245 648 a(\).) p 2337 648 a(Let) p Ft 1477 821 a(M) p Fs 1589 821 a(\025) p Fv 1677 821 a(sup\() p Fs(\000) p Ft(V) p Fv 1966 821 a(\() p Ft(x) p Fv(\)\)) p 2128 821 a(+) p 2211 821 a(1) p Ft(;) p Fv 609 994 a(and) p 775 994 a(let) p Ft 899 994 a(m) p Fv 1002 994 a(=) p 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a(\000) p Fp(m) p Fv 2599 1267 a(on) p Ft 2714 1267 a(L) p Fr 2771 1233 a(2) p Fv 2808 1267 a(\(\012) p Fp 2900 1279 a(L) p Fv 2950 1267 a(\)) p Ft(:) p Fv 609 1440 a(W) p 687 1440 a(e) p 752 1440 a(pro) n(v) n(e) p 976 1440 a(the) p 1119 1440 a(next) p 1306 1440 a(prop) r(osition) p 1742 1440 a(in) p 1839 1440 a(Section) p 2130 1440 a(3.) p Fw 609 1586 a(Prop) s(osition) p 1125 1586 a(5.) p Ft 1241 1586 a(A) p Fr 1303 1598 a(1) p Fs 1365 1586 a(\000) p Ft 1453 1586 a(A) p Fr 1515 1598 a(2) p Fl 1590 1586 a(is) p 1686 1586 a(a) p 1766 1586 a(tr) l(ac) l(e) p 1976 1586 a(class) p 2183 1586 a(op) l(er) l(ator.) p 2566 1586 a(Mor) l(e) l(over,) p 2966 1586 a(ther) l(e) 609 1686 y(exists) p Ft 837 1686 a(C) p Fr 896 1698 a(1) p Ft 956 1686 a(>) p Fv 1044 1686 a(0) p Fl 1115 1686 a(such) p 1304 1686 a(that) p Fs 1230 1859 a(k) p Ft(A) p Fr 1334 1871 a(1) p Fs 1389 1859 a(\000) p Ft 1472 1859 a(A) p Fr 1534 1871 a(2) p Fs 1572 1859 a(k) p Ff 1614 1871 a(I) p Fh 1652 1879 a(1) p Fs 1711 1859 a(\024) p Ft 1799 1859 a(C) p Fr 1858 1871 a(1) p Ft 1895 1859 a(L) p Fp 1952 1825 a(d) p Fk(\000) p Fr(1) p Ft 2075 1859 a(;) p 2282 1859 a(L) p Fs 2362 1859 a(2) p Fq 2440 1859 a(N) p Ft 2494 1859 a(:) p Fv 734 2032 a(Com) n(bining) p 1167 2032 a(Theorem) p 1530 2032 a(4) p 1611 2032 a(with) p 1812 2032 a(Prop) r(osition) p 2270 2032 a(5,) p 2377 2032 a(w) n(e) p 2511 2032 a(no) n(w) p 2696 2032 a(pro) n(v) n(e) p 2931 2032 a(Theo-) 609 2132 y(rem) p 775 2132 a(1.) p Fl 609 2304 a(Pr) l(o) l(of) p 832 2304 a(of) p 930 2304 a(The) l(or) l(em) p 1274 2304 a(1.) p Fv 1388 2304 a(W) p 1466 2304 a(e) p 1529 2304 a(ma) n(y) p 1708 2304 a(supp) r(ose) p Ft 2018 2304 a(f) p Fs 2091 2304 a(2) p Ft 2169 2304 a(C) p Fr 2234 2274 a(1) 2228 2325 y(0) p Fv 2272 2304 a(\(\() p Fs(\000) p Ft(M) p Fv 2506 2304 a(+) p 2586 2304 a(1) p Ft(=) p Fv(2) p Ft(;) p Fs 2749 2304 a(1) p Fv(\)\).) p 2954 2304 a(Then) 609 2404 y(there) p 822 2404 a(exists) p Ft 1051 2404 a(g) p Fs 1116 2404 a(2) p Ft 1195 2404 a(C) p Fr 1260 2374 a(1) 1254 2425 y(0) p Fv 1297 2404 a(\(\() p Fs(\000) p Fv(1) p Ft(;) p Fv 1505 2404 a(0]\)) p 1629 2404 a(suc) n(h) p 1817 2404 a(that) p Ft 1433 2577 a(f) p Fv 1483 2577 a(\() p Ft(\025) p Fv(\)) p 1618 2577 a(=) p Ft 1706 2577 a(g) p Fv 1749 2577 a(\() p Fs(\000) p Fv(\() p Ft(\025) p Fv 1945 2577 a(+) p Ft 2028 2577 a(M) p Fv 2118 2577 a(\)) p Fk 2150 2543 a(\000) p Fp(m) p Fv 2264 2577 a(\)) p Ft(;) p Fv 609 2750 a(and) p 771 2750 a(hence) p Ft 1194 2924 a(f) p Fv 1244 2924 a(\() p Ft(H) p Fp 1352 2889 a(D) 1345 2944 y(L) p Fv 1412 2924 a(\)) p 1468 2924 a(=) p Ft 1555 2924 a(g) p Fv 1598 2924 a(\() p Ft(A) p Fr 1692 2936 a(1) p Fv 1730 2924 a(\)) p Ft(;) p 1965 2924 a(f) p Fv 2015 2924 a(\() p Ft(H) p Fp 2123 2889 a(N) 2116 2944 y(L) p Fv 2185 2924 a(\)) p 2241 2924 a(=) p Ft 2328 2924 a(g) p Fv 2371 2924 a(\() p Ft(A) p Fr 2465 2936 a(2) p Fv 2503 2924 a(\)) p Ft(:) p Fv 609 3097 a(Then) p 826 3097 a(b) n(y) p 941 3097 a(Theorem) p 1292 3097 a(4,) p 1384 3097 a(w) n(e) p 1507 3097 a(ha) n(v) n(e) 1027 3270 y(T) p 1080 3270 a(r) p Fn 1113 3203 a(\000) p Ft 1151 3270 a(f) p Fv 1201 3270 a(\() p Ft(H) p Fp 1309 3236 a(D) 1302 3291 y(L) p Fv 1369 3270 a(\)) p Fs 1419 3270 a(\000) p Ft 1502 3270 a(f) p Fv 1552 3270 a(\() p Ft(H) p Fp 1660 3236 a(N) 1653 3291 y(L) p Fv 1723 3270 a(\)) p Fn 1755 3203 a(\001) p Fv 1816 3270 a(=) p 1904 3270 a(T) p 1957 3270 a(r\() p Ft(g) p Fv 2065 3270 a(\() p Ft(A) p Fr 2159 3282 a(1) p Fv 2196 3270 a(\)) p Fs 2247 3270 a(\000) p Ft 2330 3270 a(g) p Fv 2373 3270 a(\() p Ft(A) p Fr 2467 3282 a(2) p Fv 2504 3270 a(\)\)) 1816 3446 y(=) p Fn 1904 3333 a(Z) p Ft 2001 3446 a(g) p Fk 2044 3412 a(0) p Fv 2067 3446 a(\() p Ft(\026) p Fv(\)) p Ft(\030) p Fv 2221 3446 a(\() p Ft(\026) p Fv(;) p Ft 2340 3446 a(A) p Fr 2402 3458 a(1) p Ft 2440 3446 a(;) p 2477 3446 a(A) p Fr 2539 3458 a(2) p Fv 2577 3446 a(\)) p Ft(d\026;) p Fv 609 3657 a(and) p 771 3657 a(also) p Fs 1295 3831 a(k) p Ft(\030) p Fs 1377 3831 a(k) p Fp 1419 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5403 a(H) p Fr 1781 5373 a(1) p Fv 1818 5403 a(\(\012) p Fp 1910 5415 a(L) p Fv 1960 5403 a(\).) p 2052 5403 a(W) p 2130 5403 a(e) p 2194 5403 a(recall) p 2415 5403 a(the) p 2557 5403 a(Krein's) p 2843 5403 a(form) n(ula:) p Fy 1854 5652 a(4) p 90 rotate dyy eop %%Page: 5 5 5 4 bop Fw 609 548 a(Lemma) p 948 548 a(6.) p Fl 1063 548 a(L) l(et) p Ft 1207 548 a(z) p Fs 1272 548 a(2) p Ft 1351 548 a(\032) p Fv(\() p Ft(H) p Fp 1502 518 a(D) 1495 571 y(L) p Fv 1562 548 a(\)) p Fs 1612 548 a(\\) p Ft 1686 548 a(\032) p Fv(\() p Ft(H) p Fp 1837 518 a(N) 1830 571 y(L) p Fv 1900 548 a(\)) p Fl(.) p 1996 548 a(Then) p Fv 768 728 a(\() p Ft(H) p Fp 876 694 a(D) 869 749 y(L) p Fs 955 728 a(\000) p Ft 1038 728 a(z) p Fv 1081 728 a(\)) p Fk 1113 694 a(\000) p Fr(1) p Fs 1220 728 a(\000) p Fv 1303 728 a(\() p Ft(H) p Fp 1411 694 a(N) 1404 749 y(L) p Fs 1493 728 a(\000) p Ft 1576 728 a(z) p Fv 1619 728 a(\)) p Fk 1651 694 a(\000) p Fr(1) p Fv 1762 728 a(=) p 1850 728 a(\() p Ft(H) p Fp 1958 694 a(N) 1951 749 y(L) p Fs 2039 728 a(\000) p Ft 2122 728 a(z) p Fv 2165 728 a(\)) p Fk 2197 694 a(\000) p Fr(1) p Ft 2286 728 a(\015) p Fk 2334 694 a(\003) p Ft 2372 728 a(\015) p 2443 728 a(@) p Fp 2487 740 a(\027) p Fv 2528 728 a(\() p Ft(H) p Fp 2636 694 a(D) 2629 749 y(L) p Fs 2715 728 a(\000) p Ft 2798 728 a(z) p Fv 2841 728 a(\)) p Fk 2873 694 a(\000) p Fr(1) p Ft 2961 728 a(;) p Fl 609 908 a(wher) l(e) p Ft 856 908 a(\027) p Fl 944 908 a(denotes) p 1255 908 a(the) p 1405 908 a(outer) p 1635 908 a(unit) p 1821 908 a(normal) p 2118 908 a(ve) l(ctor,) p 2404 908 a(and) p Ft 2577 908 a(@) p Fp 2621 920 a(\027) p Fl 2704 908 a(is) p 2806 908 a(the) p 2956 908 a(outer) 609 1008 y(normal) p 895 1008 a(derivative) p 1275 1008 a(on) p 1394 1008 a(the) p 1532 1008 a(b) l(oundary.) 609 1182 y(Pr) l(o) l(of.) p Fv 873 1182 a(This) p 1066 1182 a(is) p 1153 1182 a(w) n(ell-kno) n(wn,) p 1611 1182 a(but) p 1766 1182 a(w) n(e) p 1892 1182 a(pro) n(v) n(e) p 2119 1182 a(it) p 2205 1182 a(for) p 2335 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p Fp 2939 1427 a(D) 2932 1482 y(L) p Fv 2999 1462 a(\)) p Ft(:) p Fv 609 1642 a(Then) p 826 1642 a(it) p 909 1642 a(su\016ces) p 1191 1642 a(to) p 1293 1642 a(sho) n(w) p Fs 1003 1822 a(h) p Fv(\() p Ft(H) p Fp 1143 1788 a(N) 1136 1842 y(L) p Fs 1225 1822 a(\000) p Fv 1313 1822 a(\026) p Ft 1308 1822 a(z) p Fv 1351 1822 a(\)) p Ft(';) p 1474 1822 a( ) p Fs 1531 1822 a(i) p 1582 1822 a(\000) p 1665 1822 a(h) p Ft(';) p Fv 1788 1822 a(\() p Ft(H) p Fp 1896 1788 a(D) 1889 1842 y(L) p Fs 1975 1822 a(\000) p Ft 2058 1822 a(z) p Fv 2101 1822 a(\)) p Ft( ) p Fs 2190 1822 a(i) p Fv 2245 1822 a(=) p Fs 2333 1822 a(h) p Ft(\015) p 2413 1822 a(';) p 2504 1822 a(\015) p 2552 1822 a(@) p Fp 2596 1834 a(\027) p Ft 2637 1822 a( ) p Fs 2694 1822 a(i) p Ft(:) p Fv 2973 1822 a(\(3.1\)) 609 2002 y(The) p 780 2002 a(left) p 925 2002 a(hand) p 1133 2002 a(side) p 1299 2002 a(of) p 1394 2002 a(\(3.1\)) p 1592 2002 a(is) p Fn 767 2112 a(Z) p Fr 813 2300 a(\012) p Fo 860 2308 a(L) p 922 2152 474 4 v Fv 922 2225 a(\() p Ft(p) p Fs 1015 2225 a(\000) p Ft 1098 2225 a(A) p Fv(\)) p Fr 1192 2201 a(2) p Ft 1230 2225 a(') p Fv(\() p Ft(x) p Fv(\)) p Ft 1419 2225 a( ) p Fv 1476 2225 a(\() p Ft(x) p Fv(\)) p Ft(dx) p Fs 1697 2225 a(\000) p Fn 1780 2112 a(Z) p Fr 1827 2300 a(\012) p Fo 1874 2308 a(L) p 1936 2152 167 4 v Ft 1936 2225 a(') p Fv(\() p Ft(x) p Fv(\)) q(\() p Ft(p) p Fs 2195 2225 a(\000) p Ft 2278 2225 a(A) p Fv(\)) p Fr 2372 2190 a(2) p Ft 2409 2225 a( ) p Fv 2466 2225 a(\() p Ft(x) p Fv(\)) p Ft(dx) p Fv 859 2455 a(=) p Fs 947 2455 a(\000) p Fn 1026 2342 a(Z) p Fr 1072 2531 a(\012) p Fo 1119 2539 a(L) p 1181 2387 129 4 v Fs 1181 2455 a(4) p Ft(') p 1332 2455 a( ) s(dx) p Fs 1498 2455 a(\000) p Ft 1581 2455 a(i) p Fn 1624 2342 a(Z) p Fr 1670 2531 a(\012) p Fo 1717 2539 a(L) p 1779 2383 563 4 v Fv 1779 2455 a(\() p Ft(A) p Fs 1892 2455 a(\001) p Ft 1933 2455 a(@) p Fv 2001 2455 a(+) p Ft 2084 2455 a(@) p Fs 2151 2455 a(\001) p Ft 2192 2455 a(A) p Fv(\)) p Ft(') p 2364 2455 a( ) s(dx) p 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2190 2926 a(Namely) p 2460 2926 a(,) p 2516 2926 a(there) p 2733 2926 a(exists) p 2967 2926 a(\011\() p Ft(x) p Fv(\)) 609 3025 y(suc) n(h) p 796 3025 a(that) p Ft 1428 3208 a(p) p Fs 1489 3208 a(\000) p Fv 1594 3187 a(~) p Ft 1572 3208 a(A) p Fv 1657 3208 a(=) p Ft 1745 3208 a(e) p Fp 1784 3174 a(i) p Fr(\011) p Fv 1862 3208 a(\() p Ft(p) p Fs 1954 3208 a(\000) p Ft 2037 3208 a(A) p Fv(\)) p Ft(e) p Fk 2170 3174 a(\000) p Fp(i) p Fr(\011) p Ft 2301 3208 a(:) p Fv 609 3391 a(Then,) p 849 3391 a(for) p Ft 976 3391 a(') p Fs 1053 3391 a(2) p Ft 1132 3391 a(H) p Fr 1208 3360 a(1) p Fp(;A) p Fv 1315 3391 a(\() p Ft(Q) p Fp 1413 3403 a(n) p Fv 1458 3391 a(\),) p 1541 3391 a(w) n(e) p 1663 3391 a(ha) n(v) n(e) p Fs 745 3573 a(k) p Ft(\015) p Fp 830 3585 a(n) p Ft 874 3573 a(') p Fs(k) p Fp 970 3588 a(L) p Fh 1016 3572 a(2) p Fr 1048 3588 a(\() p Fp(@) t(Q) p Fo 1165 3596 a(n) p Fr 1206 3588 a(\)) p Fv 1259 3573 a(=) p Fs 1347 3573 a(k) p Ft(e) p Fp 1428 3539 a(i) p Fr(\011) p Ft 1506 3573 a(\015) p Fp 1549 3585 a(n) p Ft 1594 3573 a(e) p Fk 1633 3539 a(\000) p Fp(i) p Fr(\011) p Ft 1763 3573 a(') p Fs(k) p Fp 1859 3588 a(L) p Fh 1905 3572 a(2) p Fr 1937 3588 a(\() p Fp(@) t(Q) p Fo 2054 3596 a(n) p Fr 2095 3588 a(\)) p Fs 1259 3710 a(\024) p Ft 1347 3710 a(C) p Fs 1412 3710 a(k) p Ft(e) p Fk 1493 3676 a(\000) p Fp(i) p Fr(\011) p Ft 1623 3710 a(') p Fs(k) p Fp 1719 3725 a(H) p Fh 1777 3709 a(1) p Fr 1810 3725 a(\() p Fp(Q) p Fo 1888 3733 a(n) p Fr 1929 3725 a(\)) p Fs 1982 3710 a(\024) p Ft 2070 3710 a(C) p Fn 2135 3643 a(\000) p Fs 2173 3710 a(k) p Ft(pe) p Fk 2296 3676 a(\000) p Fp(i) p Fr(\011) p Ft 2425 3710 a(') p Fs(k) p Fv 2540 3710 a(+) p Fs 2623 3710 a(k) p Ft(') p Fs(k) p Fn 2761 3643 a(\001) p Fs 1259 3850 a(\024) p Ft 1347 3850 a(C) p Fk 1412 3816 a(0) p Fn 1435 3783 a(\000) p Fs 1473 3850 a(k) p Fv(\() p Ft(p) p Fs 1607 3850 a(\000) p Fv 1712 3829 a(~) p Ft 1690 3850 a(A) p Fv 1753 3850 a(\)) p Ft(e) p Fk 1824 3816 a(\000) p Fp(i) p Fr(\011) p Ft 1954 3850 a(') p Fs(k) p Fv 2068 3850 a(+) p Fs 2151 3850 a(k) p Ft(') p Fs(k) p Fn 2289 3783 a(\001) p Fv 1259 3987 a(=) p Ft 1347 3987 a(C) p Fk 1412 3953 a(0) p Fn 1435 3920 a(\000) p Fs 1473 3987 a(k) p Ft(e) p Fk 1554 3953 a(\000) p Fp(i) p Fr(\011) p Fv 1684 3987 a(\() p Ft(p) p Fs 1777 3987 a(\000) p Ft 1860 3987 a(A) p Fv(\)) p Ft(') p Fs(k) p Fv 2068 3987 a(+) p Fs 2151 3987 a(k) p Ft(') p Fs(k) p Fn 2289 3920 a(\001) p Fs 2350 3987 a(\024) p Ft 2438 3987 a(C) p Fk 2503 3953 a(00) p Fs 2545 3987 a(k) p Ft(') p Fs(k) p Fp 2683 4003 a(H) p Fh 2741 3987 a(1) p Fo(;A) p Fr 2836 4003 a(\() p Fp(Q) p Fo 2914 4011 a(n) p Fr 2955 4003 a(\)) p Ft 2985 3987 a(;) p Fv 609 4170 a(where) p 849 4170 a(all) p 965 4170 a(constan) n(ts) p 1332 4170 a(are) p 1471 4170 a(indep) r(enden) n(t) p 1941 4170 a(of) p Ft 2036 4170 a(n) p Fv 2113 4170 a(and) p Ft 2275 4170 a(L) p Fv(.) p 3083 4170 V 3087 4117 50 4 v 3087 4170 V 3136 4170 4 57 v Fw 609 4344 a(De\014nition) p 1055 4344 a(1.) p Fv 1171 4344 a(Let) p Ft 1314 4344 a(J) p Fv 1391 4344 a(=) p Ft 1479 4344 a(J) p Fv 1533 4344 a(\() p Ft(x) p Fv(\)) p 1667 4344 a(b) r(e) p 1774 4344 a(a) p 1837 4344 a(smo) r(oth) p 2125 4344 a(function) p 2444 4344 a(on) p 2554 4344 a(\012) p Fp 2614 4356 a(L) p Fv 2686 4344 a(with) p Ft 2869 4344 a(L) p Fv 2947 4344 a(b) r(eing) 609 4444 y(a) p 678 4444 a(parameter.) p 1108 4444 a(W) p 1186 4444 a(e) p 1251 4444 a(write) p Ft 1463 4444 a(J) p Fs 1540 4444 a(2) p Fi 1619 4444 a(S) p Fv 1715 4444 a(if) p 1791 4444 a(for) p 1918 4444 a(an) n(y) p Ft 2075 4444 a(\013) p Fv(,) p Fs 1244 4626 a(j) p Ft(@) p Fp 1316 4592 a(\013) 1311 4647 y(x) p Ft 1363 4626 a(J) p Fv 1417 4626 a(\() p Ft(x) p Fv(\)) p Fs(j) p 1575 4626 a(\024) p Ft 1663 4626 a(C) p Fp 1722 4638 a(\013) p Ft 1769 4626 a(;) p 1972 4626 a(x) p Fs 2043 4626 a(2) p Fv 2121 4626 a(\012) p Fp 2181 4638 a(L) p Ft 2231 4626 a(;) p 2268 4626 a(L) p Fs 2347 4626 a(2) p Fq 2426 4626 a(N) p Ft 2480 4626 a(;) p Fv 609 4809 a(and) p 771 4809 a(supp) p Ft 965 4809 a(J) p Fs 1042 4809 a(\032) p Fn 1129 4742 a(\010) p Ft 1178 4809 a(x) p Fs 1248 4809 a(2) p Fv 1327 4809 a(\012) p Fp 1387 4821 a(L) p Fn 1460 4738 a(\014) 1460 4788 y(\014) p Fv 1510 4809 a(dist) q(\() p Ft(x;) p 1761 4809 a(@) p Fv 1810 4809 a(\012) p Fp 1870 4821 a(L) p Fv 1920 4809 a(\)) p Fs 1975 4809 a(\024) p Fv 2062 4809 a(1) p Fn 2104 4742 a(\011) p Fv 2152 4809 a(.) 734 4958 y(F) p 781 4958 a(or) p Ft 895 4958 a(f) p Fs 987 4958 a(2) p Fi 1086 4958 a(S) p Fv(,) p 1220 4958 a(w) n(e) p 1355 4958 a(denote) p 1634 4958 a(the) p 1789 4958 a(m) n(ultiplication) p 2334 4958 a(op) r(erator) p 2681 4958 a(b) n(y) p 2808 4958 a(the) p 2963 4958 a(same) 609 5058 y(sym) n(b) r(ol.) p 949 5058 a(W) p 1027 5058 a(e) p 1100 5058 a(c) n(ho) r(ose) p Ft 1371 5058 a(J) p Fr 1417 5070 a(0) p Fs 1490 5058 a(2) p Fi 1582 5058 a(S) p Fv 1686 5058 a(suc) n(h) p 1881 5058 a(that) p Ft 2069 5058 a(J) p Fr 2115 5070 a(0) p Fv 2188 5058 a(=) p 2289 5058 a(1) p 2366 5058 a(in) p 2470 5058 a(a) p 2547 5058 a(neigh) n(b) r(orho) r(o) r(d) p 3077 5058 a(of) p Ft 609 5158 a(@) p Fv 658 5158 a(\012) p Fp 718 5170 a(L) p Fv 768 5158 a(.) p Fy 1854 5652 a(6) p 90 rotate dyy eop %%Page: 7 7 7 6 bop Fw 609 548 a(Lemma) p 948 548 a(9.) p Fl 1063 548 a(L) l(et) p Ft 1207 548 a(J) p Fs 1284 548 a(2) p Fi 1362 548 a(S) p Fl 1461 548 a(and) p 1622 548 a(let) p Ft 1739 548 a(`) p Fs 1796 548 a(2) p Fq 1875 548 a(N) p Fl 1929 548 a(.) p 1998 548 a(Then) p 2215 548 a(ther) l(e) p 2422 548 a(exist) p 2615 548 a(functions) p Fn 1191 653 a(\010) p Ft 1239 720 a(J) p Fp 1285 732 a(ij) p Fn 1367 649 a(\014) 1367 699 y(\014) p Ft 1417 720 a(i) p Fv 1469 720 a(=) p 1557 720 a(1) p Ft(;) p Fs 1636 720 a(\001) p 1673 720 a(\001) p 1710 720 a(\001) p Ft 1760 720 a(;) p 1797 720 a(`;) p 1869 720 a(j) p Fv 1930 720 a(=) p 2018 720 a(1) p Ft(;) p Fs 2097 720 a(\001) p 2134 720 a(\001) p 2171 720 a(\001) p Ft 2221 720 a(;) p 2258 720 a(K) p Fn 2335 653 a(\011) p Fs 2405 720 a(\032) p Fi 2493 720 a(S) p Fl 609 892 a(and) p 770 892 a(uniformly) p 1149 892 a(b) l(ounde) l(d) p 1465 892 a(family) p 1718 892 a(of) p 1815 892 a(op) l(er) l(ators) p Fn 1017 997 a(\010) p Ft 1065 1064 a(B) p Fp 1128 1076 a(ij) p Fn 1210 994 a(\014) 1210 1043 y(\014) p Ft 1260 1064 a(i) p Fv 1312 1064 a(=) p 1400 1064 a(1) p Ft(;) p Fs 1479 1064 a(\001) p 1516 1064 a(\001) p 1553 1064 a(\001) p Ft 1603 1064 a(;) p 1640 1064 a(`;) p 1712 1064 a(j) p Fv 1773 1064 a(=) p 1861 1064 a(1) p Ft(;) p Fs 1940 1064 a(\001) p 1977 1064 a(\001) p 2014 1064 a(\001) p Ft 2064 1064 a(;) p 2101 1064 a(K) p Fn 2178 997 a(\011) p Fs 2248 1064 a(\032) p Ft 2336 1064 a(B) p Fv 2403 1064 a(\() p Ft(L) p Fr 2492 1030 a(2) p Fv 2529 1064 a(\(\012) p Fp 2621 1076 a(L) p Fv 2671 1064 a(\)\)) p Fl 609 1236 a(such) p 798 1236 a(that) p Ft 1037 1470 a(J) p Fp 1091 1435 a(`) p Fv 1123 1470 a(\() p Ft(H) p Fp 1231 1435 a([) 1224 1490 y(L) p Fv 1293 1470 a(+) p Ft 1376 1470 a(M) p Fv 1466 1470 a(\)) p Fk 1498 1435 a(\000) p Fp(`) p Fv 1605 1470 a(=) p Fp 1722 1366 a(K) p Fn 1692 1391 a(X) p Fp 1695 1568 a(j) p Fr 1725 1568 a(=1) p Fp 1866 1366 a(`) p Fn 1827 1391 a(Y) p Fp 1826 1568 a(i) p Fr(=1) p Fn 1934 1402 a(\002) p Ft 1968 1470 a(J) p Fp 2014 1482 a(ij) p Fv 2073 1470 a(\() p Ft(H) p Fp 2181 1435 a([) 2174 1490 y(L) p Fv 2242 1470 a(+) p Ft 2325 1470 a(M) p Fv 2415 1470 a(\)) p Fk 2447 1435 a(\000) p Fr(1) p Ft 2536 1470 a(B) p Fp 2599 1482 a(ij) p Fn 2658 1402 a(\003) p Ft 2692 1470 a(;) p Fl 609 1730 a(wher) l(e) p Ft 844 1730 a([) p Fv 899 1730 a(=) p Ft 987 1730 a(B) p Fl 1083 1730 a(or) p Ft 1191 1730 a(N) p Fl 1267 1730 a(.) p 1330 1730 a(In) p 1438 1730 a(p) l(articular,) p Ft 1844 1730 a(J) p Fr 1890 1742 a(1) p Fp(j) p Fv 1981 1730 a(=) p Ft 2068 1730 a(J) p Fl 2122 1730 a(.) p Fv 734 1875 a(The) p 904 1875 a(pro) r(of) p 1121 1875 a(is) p 1205 1875 a(somewhat) p 1593 1875 a(long,) p 1796 1875 a(and) p 1957 1875 a(w) n(e) p 2079 1875 a(giv) n(e) p 2250 1875 a(it) p 2333 1875 a(in) p 2430 1875 a(App) r(endix.) p Fw 609 2021 a(Lemma) p 948 2021 a(10.) p Fl 1111 2021 a(Ther) l(e) p 1350 2021 a(exists) p 1577 2021 a(an) p 1696 2021 a(extension) p 2063 2021 a(op) l(er) l(ator) p Ft 1377 2193 a(\014) p Fv 1474 2193 a(:) p Ft 1544 2193 a(H) p Fr 1620 2158 a(2) p Fp(;A) p Fv 1726 2193 a(\(\012) p Fp 1818 2205 a(L) p Fv 1868 2193 a(\)) p Fs 1924 2193 a(7!) p Ft 2030 2193 a(H) p Fr 2106 2158 a(2) p Fp(;A) p Fv 2212 2193 a(\() p Fq(R) p Fp 2299 2158 a(d) p Fv 2343 2193 a(\)) p Fl 609 2365 a(such) p 798 2365 a(that) p 968 2365 a(the) p 1106 2365 a(op) l(er) l(ator) p 1432 2365 a(norm) p 1654 2365 a(is) p 1743 2365 a(uniformly) p 2122 2365 a(b) l(ounde) l(d) p 2438 2365 a(in) p Ft 2540 2365 a(L) p Fl(.) 609 2537 y(Pr) l(o) l(of.) p Fv 873 2537 a(This) p 1070 2537 a(result) p 1308 2537 a(is) p 1399 2537 a(w) n(ell-kno) n(wn) p 1837 2537 a(if) p Ft 1920 2537 a(A) p Fv 2017 2537 a(=) p 2116 2537 a(0.) p 2239 2537 a(See) p 2393 2537 a([8]) p 2516 2537 a(Chapter) p 2846 2537 a(IV,) p 2996 2537 a(Sec-) 609 2637 y(tion) p 780 2637 a(3.2.) p 946 2637 a(W) p 1024 2637 a(e) p 1089 2637 a(sk) n(etc) n(h) p 1341 2637 a(the) p 1484 2637 a(necessary) p 1852 2637 a(mo) r(di\014cation.) 734 2736 y(W) p 812 2736 a(e) p 885 2736 a(construct) p 1259 2736 a(an) p 1383 2736 a(extension) p 1758 2736 a(op) r(erator) p 2101 2736 a(in) p 2207 2736 a(a) p 2284 2736 a(neigh) n(b) r(orho) r(o) r(d) p 2814 2736 a(of) p 2917 2736 a(lattice) 609 2836 y(p) r(oin) n(ts) p 851 2836 a(in) p Ft 940 2836 a(@) p Fv 989 2836 a(\012) p Fp 1049 2848 a(L) p Fv 1119 2836 a(with) p 1300 2836 a(a) p 1362 2836 a(\014xed) p 1555 2836 a(size.) p 1742 2836 a(Then) p 1951 2836 a(w) n(e) p 2066 2836 a(paste) p 2275 2836 a(them) p 2480 2836 a(using) p 2690 2836 a(the) p 2825 2836 a(partition) 609 2936 y(of) p 702 2936 a(unit) n(y) p 884 2936 a(.) p 943 2936 a(In) p 1045 2936 a(eac) n(h) p 1230 2936 a(op) r(en) p 1428 2936 a(set,) p 1579 2936 a(w) n(e) p 1699 2936 a(c) n(hange) p 1972 2936 a(the) p 2113 2936 a(gauge) p 2346 2936 a(so) p 2446 2936 a(that) p Ft 2623 2936 a(A) p Fv 2711 2936 a(is) p 2793 2936 a(uniformly) 609 3035 y(b) r(ounded) p 945 3035 a(\(as) p 1076 3035 a(in) p 1169 3035 a(the) p 1309 3035 a(pro) r(of) p 1522 3035 a(of) p 1614 3035 a(Lemma) p 1907 3035 a(8\).) p 2039 3035 a(Then) p 2253 3035 a(the) p 2392 3035 a(extension) p 2756 3035 a(op) r(erator) p 3088 3035 a(is) 609 3135 y(constructed) p 1059 3135 a(in) p 1157 3135 a(the) p 1302 3135 a(usual) p 1521 3135 a(w) n(a) n(y) p 1691 3135 a(in) p 1789 3135 a(this) p 1953 3135 a(op) r(en) p 2155 3135 a(set,) p 2310 3135 a(and) p 2473 3135 a(the) p 2618 3135 a(op) r(erator) p 2954 3135 a(norm) 609 3234 y(is) p 704 3234 a(uniformly) p 1094 3234 a(b) r(ounded.) p 1500 3234 a(Then) p 1729 3234 a(w) n(e) p 1862 3234 a(c) n(hange) p 2148 3234 a(the) p 2303 3234 a(gauge) p 2550 3234 a(bac) n(k,) p 2781 3234 a(and) p 2954 3234 a(paste) 609 3334 y(together) p 936 3334 a(to) p 1037 3334 a(construct) p 1402 3334 a(the) p 1545 3334 a(global) p 1789 3334 a(extension) p 2155 3334 a(op) r(erator.) p 2522 3334 a(W) p 2600 3334 a(e) p 2665 3334 a(note) p 2849 3334 a(that) p 3028 3334 a(the) 609 3434 y(deriv) p 787 3434 a(ativ) n(e) p 993 3434 a(of) p 1090 3434 a(the) p 1236 3434 a(partition) p 1585 3434 a(of) p 1683 3434 a(unit) n(y) p 1903 3434 a(can) p 2058 3434 a(b) r(e) p 2174 3434 a(also) p 2343 3434 a(tak) n(en) p 2572 3434 a(to) p 2677 3434 a(b) r(e) p 2793 3434 a(uniformly) 609 3533 y(b) r(ounded.) p 3083 3533 4 57 v 3087 3481 50 4 v 3087 3533 V 3136 3533 4 57 v Fw 609 3706 a(Lemma) p 948 3706 a(11.) p Fl 1111 3706 a(L) l(et) p Ft 1253 3706 a(a) p Fs 1320 3706 a(2) p Fq 1398 3706 a(N) p Fl 1486 3706 a(such) p 1673 3706 a(that) p Ft 1840 3706 a(a) p 1907 3706 a(>) p 1995 3706 a(d=) p Fv(4) p Fl 2149 3706 a(\(as) p 2287 3706 a(in) p 2387 3706 a(the) p 2523 3706 a(last) p 2676 3706 a(se) l(ction\),) p 3012 3706 a(and) 609 3805 y(let) p Ft 726 3805 a(J) p Fs 803 3805 a(2) p Fi 881 3805 a(S) p Fl(.) p 1014 3805 a(Then) p Ft 1230 3805 a(J) p Fv 1284 3805 a(\() p Ft(H) p Fp 1392 3775 a([) 1385 3828 y(L) p Fv 1454 3805 a(+) p Ft 1537 3805 a(M) p Fv 1627 3805 a(\)) p Fk 1659 3775 a(\000) p Fr(1) p Fs 1771 3805 a(2) p Fi 1849 3805 a(I) p Fr 1895 3817 a(2) p Fp(a) p Fl 1998 3805 a(and) p Fs 1043 3987 a(k) p Ft(J) p Fv 1139 3987 a(\() p Ft(H) p Fp 1247 3952 a([) 1240 4007 y(L) p Fv 1308 3987 a(+) p Ft 1391 3987 a(M) p Fv 1481 3987 a(\)) p Fk 1513 3952 a(\000) p Fr(1) p Fs 1602 3987 a(k) p Ff 1644 3999 a(I) p Fh 1682 4007 a(2) p Fo(a) p Fs 1774 3987 a(\024) p Ft 1861 3987 a(C) p 1926 3987 a(L) p Fr 1983 3952 a(\() p Fp(d) p Fk(\000) p Fr(1\)) p Fp(=) p Fr(2) p Fp(a) p Ft 2262 3987 a(;) p 2468 3987 a(L) p Fs 2548 3987 a(2) p Fq 2626 3987 a(N) p Ft 2680 3987 a(;) p Fl 609 4159 a(wher) l(e) p Ft 844 4159 a([) p Fv 899 4159 a(=) p Ft 987 4159 a(D) p Fl 1087 4159 a(or) p Ft 1194 4159 a(N) p Fl 1270 4159 a(.) 609 4331 y(Pr) l(o) l(of.) p Fv 873 4331 a(Let) p Ft 1022 4331 a(\014) p Fv 1101 4331 a(as) p 1203 4331 a(in) p 1300 4331 a(Lemma) p 1596 4331 a(10.) p 1739 4331 a(Then) p Ft 782 4503 a(J) p Fv 836 4503 a(\() p Ft(H) p Fp 944 4469 a([) 937 4524 y(L) p Fv 1005 4503 a(+) p Ft 1088 4503 a(M) p Fv 1178 4503 a(\)) p Fk 1210 4469 a(\000) p Fr(1) p Fv 1322 4503 a(=) p Ft 1410 4503 a(J) p Fv 1464 4503 a(\() p Ft(H) p Fv 1591 4503 a(+) p Ft 1674 4503 a(M) p Fv 1764 4503 a(\)) p Fk 1796 4469 a(\000) p Fr(1) p Fn 1885 4436 a(\002) p Fv 1919 4503 a(\() p Ft(H) p Fv 2046 4503 a(+) p Ft 2129 4503 a(M) p Fv 2219 4503 a(\)) p Ft(\014) p Fv 2302 4503 a(\() p Ft(H) p Fp 2410 4469 a([) 2403 4524 y(L) p Fv 2471 4503 a(+) p Ft 2554 4503 a(M) p Fv 2644 4503 a(\)) p Fk 2676 4469 a(\000) p Fr(1) p Fn 2765 4436 a(\003) p Fv 2973 4503 a(\(3.3\)) 609 4688 y(and) p 763 4688 a(\() p Ft(H) p Fv 890 4688 a(+) p Ft 973 4688 a(M) p Fv 1063 4688 a(\)) p Ft(\014) p Fv 1146 4688 a(\() p Ft(H) p Fp 1254 4657 a([) 1247 4710 y(L) p Fv 1316 4688 a(+) p Ft 1399 4688 a(M) p Fv 1489 4688 a(\)) p Fk 1521 4657 a(\000) p Fr(1) p Fv 1630 4688 a(is) p 1706 4688 a(uniformly) p 2078 4688 a(b) r(ounded.) p 2446 4688 a(By) p 2569 4688 a(the) p 2705 4688 a(diamagnetic) 609 4787 y(inequalit) n(y) p 994 4787 a(\(cf.) p 1149 4787 a([3) o(]) p 1264 4787 a(\(1.8\)\),) p 1518 4787 a(w) n(e) p 1640 4787 a(ha) n(v) n(e) p Fs 867 4959 a(j) p Ft(J) p Fv 944 4959 a(\() p Ft(H) p Fv 1071 4959 a(+) p Ft 1154 4959 a(M) p Fv 1244 4959 a(\)) p Fk 1276 4925 a(\000) p Fr(1) p Ft 1365 4959 a(') p Fs(j) p 1465 4959 a(\024) p 1553 4959 a(j) p Ft(J) p Fs 1630 4959 a(j) p Fv(\() p Ft(H) p Fr 1754 4971 a(0) p Fv 1810 4959 a(+) p 1893 4959 a(1\)) p Fk 1967 4925 a(\000) p Fr(1) p Fs 2056 4959 a(j) p Ft(') p Fs(j) p Fv 2322 4959 a(for) p Ft 2450 4959 a(') p Fs 2527 4959 a(2) p Ft 2605 4959 a(L) p Fr 2662 4925 a(2) p Fv 2699 4959 a(\() p Fq(R) p Fp 2785 4925 a(d) p Fv 2830 4959 a(\)) p Ft(;) p Fv 609 5131 a(where) p Ft 847 5131 a(H) p Fr 916 5143 a(0) p Fv 977 5131 a(=) p Fs 1064 5131 a(\0004) p Fv(.) p 1262 5131 a(By) p 1390 5131 a(Theorem) p 1739 5131 a(4.1) p 1871 5131 a(of) p 1964 5131 a([7) o(],) p 2101 5131 a(w) n(e) p 2221 5131 a(learn) p Fs 2427 5131 a(j) p Ft(J) p Fs 2504 5131 a(j) p Fv(\() p Ft(H) p Fr 2628 5143 a(0) p Fv 2680 5131 a(+) p 2760 5131 a(1\)) p Fk 2834 5101 a(\000) p Fr(1) p Fs 2946 5131 a(2) p Fi 3024 5131 a(I) p Fr 3070 5143 a(2) p Fp(a) p Fv 609 5231 a(and) p Fs 796 5403 a(kj) p Ft(J) p Fs 915 5403 a(j) p Fv(\() p Ft(H) p Fr 1039 5415 a(0) p Fv 1095 5403 a(+) p 1178 5403 a(1\)) p Fk 1252 5369 a(\000) p Fr(1) p Fs 1341 5403 a(k) p Ff 1383 5415 a(I) p Fh 1421 5423 a(2) p Fo(a) p Fs 1512 5403 a(\024) p 1600 5403 a(k) p Ft(J) p Fs 1696 5403 a(k) p Fp 1738 5418 a(L) p Fh 1784 5401 a(2) p Fo(a) p Fs 1851 5403 a(k) p Fv(\() p Ft(\030) p Fr 1965 5369 a(2) p Fv 2021 5403 a(+) p 2104 5403 a(1\)) p Fk 2178 5369 a(\000) p Fr(1) p Fs 2267 5403 a(k) p Fp 2309 5418 a(L) p Fh 2355 5401 a(2) p Fo(a) p Fs 2445 5403 a(\024) p Ft 2533 5403 a(C) p 2598 5403 a(L) p Fr 2655 5369 a(\() p Fp(d) p Fk(\000) p Fr(1\)) p Fp(=) p Fr(2) p Fp(a) p Ft 2934 5403 a(:) p Fy 1854 5652 a(7) p 90 rotate dyy eop %%Page: 8 8 8 7 bop Fv 609 548 a(Then) p 824 548 a(w) n(e) p 945 548 a(apply) p 1172 548 a(Theorem) p 1521 548 a(2.13) p 1694 548 a(of) p 1787 548 a([7]) p 1901 548 a(to) p 2001 548 a(observ) n(e) p Ft 2295 548 a(J) p Fv 2349 548 a(\() p Ft(H) p Fv 2472 548 a(+) p Ft 2552 548 a(M) p Fv 2642 548 a(\)) p Fk 2674 518 a(\000) p Fr(1) p Fs 2786 548 a(2) p Fi 2864 548 a(I) p Fr 2910 560 a(2) p Fp(a) p Fv 3010 548 a(and) p Fs 1277 731 a(k) p Ft(J) p Fv 1373 731 a(\() p Ft(H) p Fv 1499 731 a(+) p Ft 1582 731 a(M) p Fv 1672 731 a(\)) p Fk 1704 696 a(\000) p Fr(1) p Fs 1793 731 a(k) p Ff 1835 743 a(I) p Fh 1873 751 a(2) p Fo(a) p Fs 1964 731 a(\024) p Ft 2052 731 a(C) p 2117 731 a(L) p Fr 2174 696 a(\() p Fp(d) p Fk(\000) p Fr(1\)) p Fp(=) p Fr(2) p Fp(a) p Ft 2453 731 a(:) p Fv 609 913 a(This) p 799 913 a(and) p 960 913 a(\(3.3\)) p 1159 913 a(imply) p 1391 913 a(the) p 1534 913 a(conclusion.) p 3083 913 4 57 v 3087 861 50 4 v 3087 913 V 3136 913 4 57 v Fl 609 1088 a(Pr) l(o) l(of) p 832 1088 a(of) p 930 1088 a(Pr) l(op) l(osition) p 1371 1088 a(5.) p Fv 1484 1088 a(Let) p 1633 1088 a(1) p Fs 1698 1088 a(\024) p Ft 1785 1088 a(k) p Fs 1854 1088 a(\024) p Ft 1942 1088 a(m) p Fv 2042 1088 a(and) p 2204 1088 a(w) n(e) p 2326 1088 a(consider) p Ft 1070 1270 a(P) p Fp 1123 1282 a(k) p Fv 1187 1270 a(=) p 1275 1270 a(\() p Ft(H) p Fp 1383 1236 a(N) 1376 1291 y(L) p Fv 1464 1270 a(+) p Ft 1547 1270 a(M) p Fv 1637 1270 a(\)) p Fk 1669 1236 a(\000) p Fp(m) p Fk(\000) p Fr(1+) p Fp(k) p Ft 1957 1270 a(\015) p Fk 2005 1236 a(\003) p Ft 2042 1270 a(\015) p 2090 1270 a(@) p Fp 2134 1282 a(\027) p Fv 2175 1270 a(\() p Ft(H) p Fp 2283 1236 a(D) 2276 1291 y(L) p Fv 2362 1270 a(+) p Ft 2445 1270 a(M) p Fv 2535 1270 a(\)) p Fk 2567 1236 a(\000) p Fp(k) p Ft 2660 1270 a(:) p Fv 609 1453 a(By) p 739 1453 a(\(3.2\),) p 961 1453 a(it) p 1044 1453 a(su\016ces) p 1326 1453 a(to) p 1428 1453 a(estimate) p Fs 1760 1453 a(k) p Ft(P) p Fp 1855 1465 a(k) p Fs 1896 1453 a(k) p Ff 1938 1465 a(I) p Fh 1976 1473 a(1) p Fv 2012 1453 a(.) p 2072 1453 a(By) p 2202 1453 a(Lemma) p 2499 1453 a(9,) p Ft 609 1636 a(\015) p 657 1636 a(@) p Fp 701 1648 a(\027) p Fv 742 1636 a(\() p Ft(H) p Fp 850 1601 a(D) 843 1656 y(L) p Fv 929 1636 a(+) p Ft 1012 1636 a(M) p Fv 1102 1636 a(\)) p Fk 1134 1601 a(\000) p Fp(k) p Fv 1250 1636 a(=) p Ft 1337 1636 a(\015) p 1385 1636 a(@) p Fp 1429 1648 a(\027) p Ft 1470 1636 a(J) p Fp 1524 1601 a(k) p Fr 1516 1656 a(0) p Fv 1565 1636 a(\() p Ft(H) p Fp 1673 1601 a(D) 1666 1656 y(L) p Fv 1752 1636 a(+) p Ft 1835 1636 a(M) p Fv 1925 1636 a(\)) p Fk 1957 1601 a(\000) p Fp(k) p Fv 1250 1851 a(=) p Ft 1337 1851 a(\015) p 1385 1851 a(@) p Fp 1429 1863 a(\027) p Fv 1470 1851 a(\() p Ft(H) p Fp 1578 1817 a(D) 1571 1872 y(L) p Fv 1657 1851 a(+) p Ft 1740 1851 a(M) p Fv 1830 1851 a(\)) p Fk 1862 1817 a(\000) p Fr(1) p Fp 1995 1747 a(K) p Fn 1965 1772 a(X) p Fp 1967 1949 a(j) p Fr 1997 1949 a(=1) p Ft 2099 1851 a(B) p Fr 2162 1863 a(1) p Fp(j) 2279 1747 y(k) p Fn 2244 1772 a(Y) p Fp 2243 1949 a(i) p Fr(=2) p Fn 2351 1784 a(\002) p Ft 2386 1851 a(J) p Fp 2432 1863 a(ij) p Fv 2490 1851 a(\() p Ft(H) p Fp 2598 1817 a(D) 2591 1872 y(L) p Fv 2677 1851 a(+) p Ft 2760 1851 a(M) p Fv 2850 1851 a(\)) p Fk 2882 1817 a(\000) p Fr(1) p Ft 2971 1851 a(B) p Fp 3034 1863 a(ij) p Fn 3092 1784 a(\003) p Ft 3127 1851 a(:) p Fv 609 2122 a(Hence,) p 879 2122 a(using) p 1096 2122 a(Lemma) p 1393 2122 a(11) p 1503 2122 a(and) p 1665 2122 a([7) o(]) p 1780 2122 a(Theorem) p 2131 2122 a(2.8,) p 2288 2122 a(w) n(e) p 2410 2122 a(\014nd) p Fn 1100 2266 a(\015) 1100 2316 y(\015) 1100 2366 y(\015) 1100 2416 y(\015) p Fp 1182 2283 a(k) p Fn 1147 2308 a(Y) p Fp 1146 2485 a(i) p Fr(=2) p Fn 1254 2319 a(\002) p Ft 1288 2387 a(J) p Fp 1334 2399 a(ij) p Fv 1393 2387 a(\() p Ft(H) p Fp 1501 2352 a(D) 1494 2407 y(L) p Fv 1580 2387 a(+) p Ft 1663 2387 a(M) p Fv 1753 2387 a(\)) p Fk 1785 2352 a(\000) p Fr(1) p Ft 1874 2387 a(B) p Fp 1937 2399 a(ij) p Fn 1995 2319 a(\003) 2030 2266 y(\015) 2030 2316 y(\015) 2030 2366 y(\015) 2030 2416 y(\015) p Ff 2076 2470 a(I) p Fo 2114 2478 a(p) p Fs 2176 2387 a(\024) p Ft 2264 2387 a(C) p 2329 2387 a(L) p Fr 2386 2352 a(\() p Fp(d) p Fk(\000) p Fr(1\)) p Fp(=p) p Ft 2629 2387 a(;) p Fv 609 2652 a(where) p Ft 849 2652 a(p) p Fk 891 2622 a(\000) p Fr(1) p Fv 1003 2652 a(=) p 1091 2652 a(\() p Ft(k) p Fs 1187 2652 a(\000) p Fv 1270 2652 a(1\)) p Ft(=) p Fv(2) p Ft(a) p Fv(.) p 1531 2652 a(Similarly) p 1850 2652 a(,) p 1900 2652 a(w) n(e) p 2023 2652 a(ha) n(v) n(e) p Ft 609 2924 a(\015) p Fv 657 2924 a(\() p Ft(H) p Fp 765 2890 a(N) 758 2945 y(L) p Fv 846 2924 a(+) p Ft 929 2924 a(M) p Fv 1019 2924 a(\)) p Fk 1051 2890 a(\000) p Fp(m) p Fk(\000) p Fr(1+) p Fp(k) p Fv 1362 2924 a(=) p Ft 1449 2924 a(\015) p Fv 1497 2924 a(\() p Ft(H) p Fp 1605 2890 a(N) 1598 2945 y(L) p Fv 1686 2924 a(+) p Ft 1770 2924 a(M) p Fv 1860 2924 a(\)) p Fk 1892 2890 a(\000) p Fr(1) p Fp 2025 2820 a(a) p Fe 2061 2795 a(0) p Fn 1994 2845 a(X) p Fp 1997 3022 a(j) p Fr 2027 3022 a(=1) p Ft 2128 2924 a(B) p Fk 2195 2890 a(0) p Fr 2191 2945 a(1) p Fp(j) 2273 2820 y(m) p Fr(+1) p Fk(\000) p Fp(k) p Fn 2336 2845 a(Y) p Fp 2335 3022 a(i) p Fr(=2) p Fn 2505 2857 a(\002) p Ft 2539 2924 a(J) p Fk 2593 2890 a(0) p Fp 2585 2945 a(ij) p Fv 2644 2924 a(\() p Ft(H) p Fp 2752 2890 a(N) 2745 2945 y(L) p Fv 2833 2924 a(+) p Ft 2916 2924 a(M) p Fv 3006 2924 a(\)) p Fk 3038 2890 a(\000) p Fr(1) p Ft 3127 2924 a(B) p Fk 3194 2890 a(0) p Fp 3190 2945 a(ij) p Fn 3249 2857 a(\003) p Fv 609 3191 a(and) p Fn 1101 3314 a(\015) 1101 3364 y(\015) 1101 3414 y(\015) 1101 3464 y(\015) p Fp 1182 3331 a(k) p Fn 1147 3356 a(Y) p Fp 1147 3532 a(i) p Fr(=2) p Fn 1254 3367 a(\002) p Ft 1289 3434 a(J) p Fk 1343 3400 a(0) p Fp 1335 3455 a(ij) p Fv 1393 3434 a(\() p Ft(H) p Fp 1501 3400 a(N) 1494 3455 y(L) p Fv 1583 3434 a(+) p Ft 1666 3434 a(M) p Fv 1756 3434 a(\)) p Fk 1788 3400 a(\000) p Fr(1) p Ft 1877 3434 a(B) p Fk 1944 3400 a(0) p Fp 1940 3455 a(ij) p Fn 1998 3367 a(\003) 2033 3314 y(\015) 2033 3364 y(\015) 2033 3414 y(\015) 2033 3464 y(\015) p Ff 2079 3517 a(I) p Fo 2117 3525 a(q) p Fs 2177 3434 a(\024) p Ft 2265 3434 a(C) p 2330 3434 a(L) p Fr 2387 3400 a(\() p Fp(d) p Fk(\000) p Fr(1\)) p Fp(=q) p Ft 2629 3434 a(;) p Fv 609 3700 a(where) p Ft 849 3700 a(q) p Fk 889 3670 a(\000) p Fr(1) p Fv 1001 3700 a(=) p 1089 3700 a(\() p Ft(m) p Fs 1213 3700 a(\000) p Ft 1296 3700 a(k) p Fv 1342 3700 a(\)) p Ft(=) p Fv(2) p Ft(a) p Fv(.) p 1561 3700 a(Recall) p 1811 3700 a(that) p 1991 3700 a(w) n(e) p 2113 3700 a(ha) n(v) n(e) p 2304 3700 a(c) n(hosen) p Ft 2570 3700 a(m) p Fv 2671 3700 a(so) p 2773 3700 a(that) 1513 3865 y(1) p 1513 3902 42 4 v Ft 1513 3978 a(p) p Fv 1583 3921 a(+) 1676 3865 y(1) p 1676 3902 V Ft 1677 3978 a(q) p Fv 1751 3921 a(=) p Ft 1848 3865 a(m) p Fs 1940 3865 a(\000) p Fv 2023 3865 a(1) p 1848 3902 216 4 v 1914 3978 a(2) p Ft(a) p Fv 2097 3921 a(=) p 2185 3921 a(1) p Ft(:) p Fv 609 4148 a(Hence,) p 883 4148 a(com) n(bining) p 1286 4148 a(these) p 1502 4148 a(with) p 1694 4148 a(Lemma) p 1994 4148 a(7) p 2067 4148 a(and) p 2232 4148 a([7) o(]) p 2350 4148 a(Theorem) p 2704 4148 a(2.8,) p 2866 4148 a(w) n(e) p 2991 4148 a(con-) 609 4248 y(clude) p Ft 826 4248 a(P) p Fp 879 4260 a(k) p Fs 943 4248 a(2) p Fi 1022 4248 a(I) p Fr 1068 4260 a(1) p Fv 1133 4248 a(and) p Fs 1078 4430 a(k) p Ft(P) p Fp 1173 4442 a(k) p Fs 1214 4430 a(k) p Ff 1256 4442 a(I) p Fh 1294 4450 a(1) p Fs 1353 4430 a(\024) p Ft 1441 4430 a(C) p 1506 4430 a(L) p Fr 1563 4396 a(\() p Fp(d) p Fk(\000) p Fr(1\)) p Fp(=q) p Fs 1823 4430 a(\002) p Ft 1906 4430 a(C) p 1971 4430 a(L) p Fr 2028 4396 a(\() p Fp(d) p Fk(\000) p Fr(1\)) p Fp(=p) p Fv 2295 4430 a(=) p Ft 2383 4430 a(C) p Fk 2448 4396 a(0) p Ft 2471 4430 a(L) p Fp 2528 4396 a(d) p Fk(\000) p Fr(1) p Ft 2651 4430 a(:) p Fv 609 4613 a(This) p 799 4613 a(comp) r(etes) p 1161 4613 a(the) p 1304 4613 a(pro) r(of.) p 3083 4613 4 57 v 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2769 5169 a(Kato-class) 609 5269 y(p) r(oten) n(tial) p 961 5269 a(\(cf.) p 1108 5269 a(e.g.,) p 1285 5269 a([3]) p 1402 5269 a(Section) p 1694 5269 a(1.2\).) p 1896 5269 a(Let) p Ft 2047 5269 a(H) p Fv 2151 5269 a(b) r(e) p 2266 5269 a(a) p 2336 5269 a(self-adjoin) n(t) p 2765 5269 a(realization) 609 5368 y(of) p 704 5368 a(the) p 847 5368 a(di\013eren) n(tial) p 1262 5368 a(op) r(erator) p 1597 5368 a(in) p 1694 5368 a(the) p 1837 5368 a(quadratic) p 2208 5368 a(form) p 2405 5368 a(sense.) p Fy 1854 5652 a(8) p 90 rotate dyy eop %%Page: 9 9 9 8 bop Fv 734 548 a(Let) p Ft 874 548 a(M) p 987 548 a(>) p Fs 1074 548 a(\000) p Fv 1153 548 a(inf) p Ft 1268 548 a(\033) p Fv 1318 548 a(\() p Ft(H) p Fv 1426 548 a(\),) p 1502 548 a(and) p 1656 548 a(\014x) p Ft 1765 548 a(J) p Fr 1811 560 a(0) p Fs 1871 548 a(2) p Ft 1950 548 a(C) p Fk 2015 518 a(1) p Fv 2085 548 a(\(\012\)) p 2229 548 a(suc) n(h) p 2408 548 a(that) p 2580 548 a(sup) p Fs 2718 548 a(j) p Ft(@) p Fp 2790 518 a(\013) 2785 568 y(x) p Ft 2837 548 a(J) p Fr 2883 560 a(0) p Fv 2921 548 a(\() p Ft(x) p Fv(\)) p Fs(j) p Ft 3079 548 a(<) p Fs 609 648 a(1) p Fv 731 648 a(for) p 868 648 a(an) n(y) p 1036 648 a(m) n(ulti-index) p Ft 1489 648 a(\013) p Fv(.) p 1635 648 a(Let) p Fi 1794 648 a(S) p Fv 1902 648 a(b) r(e) p 2025 648 a(the) p 2179 648 a(linear) p 2420 648 a(subspace) p 2778 648 a(of) p Ft 2883 648 a(C) p Fk 2948 617 a(1) p Fv 3019 648 a(\(\012\)) 609 747 y(spanned) p 932 747 a(b) n(y) p 1048 747 a(the) p 1191 747 a(deriv) p 1369 747 a(ativ) n(es) p 1604 747 a(of) p Ft 1699 747 a(J) p Fr 1745 759 a(0) p Fv 1782 747 a(,) p 1833 747 a(i.e.,) p Fi 1056 905 a(S) p Fv 1148 905 a(=) p 1235 905 a(Span) p Fn 1415 837 a(\010) p Ft 1464 905 a(@) p Fp 1513 870 a(\013) 1508 925 y(x) p Ft 1560 905 a(J) p Fr 1606 917 a(0) p Fn 1666 834 a(\014) 1666 884 y(\014) p Ft 1717 905 a(\013) p Fv 1793 905 a(:) p 1839 905 a(m) n(ulti-index) p Fn 2254 837 a(\011) p Fs 2326 905 a(\032) p Ft 2414 905 a(C) p Fk 2479 870 a(1) p Fv 2549 905 a(\(\012\)) p Ft(:) p Fv 609 1062 a(F) p 656 1062 a(or) p 758 1062 a(simplicit) n(y) p 1102 1062 a(,) p 1153 1062 a(w) n(e) p 1275 1062 a(denote) p Ft 1543 1062 a(R) p Fv 1629 1062 a(=) p 1717 1062 a(\() p Ft(H) p Fv 1844 1062 a(+) p Ft 1927 1062 a(M) p Fv 2017 1062 a(\)) p Fk 2049 1032 a(\000) p Fr(1) p Fv 2138 1062 a(,) p Ft 2188 1062 a(T) p Fv 2271 1062 a(=) p Ft 2359 1062 a(p) p Fs 2419 1062 a(\000) p Ft 2502 1062 a(A) p Fv(.) p 2625 1062 a(W) p 2703 1062 a(e) p 2768 1062 a(write) p Ft 1015 1219 a(Q) p Fr 1081 1231 a(1) p Fv 1141 1219 a(=) p Ft 1228 1219 a(R) q(;) p 1412 1219 a(Q) p Fr 1478 1231 a(2) p Fv 1538 1219 a(=) p Ft 1626 1219 a(T) p 1687 1219 a(R) q(;) p 1869 1219 a(Q) p Fr 1935 1231 a(3) p Fv 1995 1219 a(=) p Ft 2083 1219 a(R) q(T) p 2203 1219 a(;) p 2322 1219 a(Q) p Fr 2388 1231 a(4) p Fv 2448 1219 a(=) p Ft 2535 1219 a(T) p 2596 1219 a(R) q(T) p 2716 1219 a(:) p Fv 609 1377 a(Let) p Fi 767 1377 a(B) p Fv 877 1377 a(b) r(e) p 999 1377 a(the) p 1151 1377 a(subspace) p 1507 1377 a(of) p Ft 1610 1377 a(B) p Fv 1677 1377 a(\() p Ft(L) p Fr 1766 1347 a(2) p Fv 1804 1377 a(\(\012\)\)) p 1997 1377 a(spanned) p 2330 1377 a(b) n(y) p 2454 1377 a(the) p 2606 1377 a(p) r(olynomials) p 3077 1377 a(of) p Ft 609 1476 a(Q) p Fr 675 1488 a(1) p Ft 712 1476 a(;) p Fs 749 1476 a(\001) p 786 1476 a(\001) p 823 1476 a(\001) p Ft 873 1476 a(;) p 910 1476 a(Q) p Fr 976 1488 a(4) p Fv 1041 1476 a(and) p 1202 1476 a(elemen) n(ts) p 1542 1476 a(of) p Fi 1636 1476 a(S) p Fv(.) p 1765 1476 a(W) p 1843 1476 a(e) p 1908 1476 a(also) p 2075 1476 a(write) p Fi 1088 1711 a(B) p Fp 1161 1723 a(k) p Fv 1226 1711 a(=) p Fn 1313 1594 a(\032) p Fp 1406 1607 a(m) p Fn 1375 1632 a(X) p Fp 1378 1809 a(j) p Fr 1408 1809 a(=1) p Fp 1545 1607 a(k) p Fn 1510 1632 a(Y) p Fp 1509 1809 a(i) p Fr(=1) p Ft 1631 1711 a(J) p Fp 1677 1723 a(ij) p Ft 1735 1711 a(R) q(B) p Fp 1862 1723 a(ij) p Fn 1943 1591 a(\014) 1943 1640 y(\014) 1943 1690 y(\014) 1943 1740 y(\014) p Ft 1994 1711 a(J) p Fp 2040 1723 a(ij) p Fs 2122 1711 a(2) p Fi 2200 1711 a(S) p Ft(;) p 2306 1711 a(B) p Fp 2369 1723 a(ij) p Fs 2450 1711 a(2) p Fi 2529 1711 a(B) p Fn 2602 1594 a(\033) p Fv 609 1952 a(for) p Ft 736 1952 a(k) p Fs 805 1952 a(2) p Fq 884 1952 a(N) p Fv 938 1952 a(.) p 1003 1952 a(Our) p 1174 1952 a(goal) p 1350 1952 a(is) p 1433 1952 a(the) p 1576 1952 a(follo) n(wing.) p Fw 609 2091 a(Prop) s(osition) p 1125 2091 a(12.) p Fl 1289 2091 a(F) p 1337 2091 a(or) p 1444 2091 a(any) p Ft 1603 2091 a(k) p Fs 1672 2091 a(2) p Fq 1750 2091 a(N) p Fl 1840 2091 a(and) p Ft 2001 2091 a(J) p Fs 2078 2091 a(2) p Fi 2157 2091 a(S) p Fl(,) p Ft 2281 2091 a(J) p Fp 2335 2061 a(k) p Ft 2375 2091 a(R) p Fp 2439 2061 a(k) p Fs 2503 2091 a(2) p Fi 2581 2091 a(B) p Fp 2654 2103 a(k) p Fl 2696 2091 a(.) p Fv 734 2231 a(Lemma) p 1030 2231 a(9) p 1099 2231 a(follo) n(ws) p 1372 2231 a(immediately) p 1845 2231 a(from) p 2041 2231 a(Prop) r(osition) p 2488 2231 a(12.) 734 2330 y(W) p 812 2330 a(e) p 879 2330 a(pro) n(v) n(e) p 1106 2330 a(the) p 1252 2330 a(assertion) p 1602 2330 a(b) n(y) p 1720 2330 a(induction.) p 2130 2330 a(Supp) r(ose) p 2458 2330 a(the) p 2604 2330 a(assertion) p 2954 2330 a(holds) 609 2430 y(for) p Ft 737 2430 a(k) p Fs 806 2430 a(\024) p Ft 895 2430 a(K) p Fv 972 2430 a(,) p 1022 2430 a(and) p 1184 2430 a(w) n(e) p 1307 2430 a(sho) n(w) p Ft 1513 2430 a(J) p Fp 1567 2400 a(K) p Fr 1627 2400 a(+1) p Ft 1715 2430 a(R) p Fp 1779 2400 a(K) p Fr 1839 2400 a(+1) p Fs 1950 2430 a(2) p Fi 2029 2430 a(B) p Fp 2102 2442 a(K) p Fr 2162 2442 a(+1) p Fv 2251 2430 a(.) p 2312 2430 a(By) p 2443 2430 a(the) p 2586 2430 a(assumption,) p 3049 2430 a(w) n(e) 609 2530 y(can) p 761 2530 a(write) p Ft 1140 2747 a(J) p Fp 1194 2713 a(K) p Ft 1258 2747 a(R) p Fp 1322 2713 a(K) p Fv 1409 2747 a(=) p Fp 1527 2644 a(m) p Fn 1496 2669 a(X) p Fp 1499 2845 a(j) p Fr 1529 2845 a(=1) p Fp 1654 2644 a(K) p Fn 1631 2669 a(Y) p Fp 1630 2845 a(i) p Fr(=1) p Ft 1751 2747 a(J) p Fp 1797 2759 a(ij) p Ft 1856 2747 a(R) q(B) p Fp 1983 2759 a(ij) p Fv 2064 2747 a(=) p Fp 2182 2644 a(m) p Fn 2152 2669 a(X) p Fp 2154 2845 a(j) p Fr 2184 2845 a(=1) p Ft 2286 2747 a(P) p Fp 2339 2759 a(j) p Fs 2397 2747 a(2) p Fi 2475 2747 a(B) p Fp 2548 2759 a(K) p Fv 609 3002 a(with) p 803 3002 a(some) p Ft 1015 3002 a(J) p Fp 1061 3014 a(ij) p Fs 1150 3002 a(2) p Fi 1236 3002 a(S) p Fv 1337 3002 a(and) p Ft 1503 3002 a(B) p Fp 1566 3014 a(ij) p Fs 1655 3002 a(2) p Fi 1741 3002 a(B) p Fv(.) p 1888 3002 a(Since) p Ft 2109 3002 a(J) p Fp 2163 2972 a(K) p Fr 2223 2972 a(+1) p Ft 2311 3002 a(R) p Fp 2375 2972 a(K) p Fr 2435 2972 a(+1) p Fv 2553 3002 a(=) p Ft 2648 3002 a(J) p Fn 2702 2934 a(\000) 2740 2939 y(P) p Ft 2842 3002 a(P) p Fp 2895 3014 a(j) p Fn 2930 2934 a(\001) p Ft 2968 3002 a(R) p Fv 3032 3002 a(,) p 3088 3002 a(it) 609 3101 y(su\016ces) p 891 3101 a(to) p 993 3101 a(sho) n(w) p Ft 1097 3319 a(J) p 1151 3319 a(P) p 1216 3319 a(R) p Fs 1303 3319 a(2) p Fi 1381 3319 a(B) p Fp 1454 3331 a(K) p Fr 1514 3331 a(+1) p Fv 1686 3319 a(for) p Ft 1868 3319 a(P) p Fv 1956 3319 a(=) p Fp 2068 3215 a(K) p Fn 2045 3240 a(Y) p Fp 2044 3417 a(i) p Fr(=1) p Ft 2165 3319 a(J) p Fp 2211 3331 a(i) p Ft 2239 3319 a(R) q(B) p Fp 2366 3331 a(i) p Fs 2416 3319 a(2) p Fi 2495 3319 a(B) p Fp 2568 3331 a(K) p Ft 2632 3319 a(:) p Fv 2952 3319 a(\(A.1\)) 734 3549 y(W) p 812 3549 a(e) p 877 3549 a(use) p 1020 3549 a(the) p 1163 3549 a(follo) n(wing) p 1514 3549 a(notations:) 1269 3685 y(~) p Fi 1253 3706 a(B) p Fr 1326 3718 a(1) p Fv 1387 3706 a(=) p Fn 1474 3639 a(\010) 1523 3644 y(P) p Ft 1610 3706 a(B) p Fp 1673 3718 a(i) p Ft 1701 3706 a(P) p Fp 1754 3718 a(i) p Fn 1805 3636 a(\014) 1805 3686 y(\014) p Ft 1856 3706 a(B) p Fp 1919 3718 a(i) p Fs 1969 3706 a(2) p Fi 2048 3706 a(B) p Ft(;) p 2158 3706 a(P) p Fp 2211 3718 a(i) p Fs 2262 3706 a(2) p Fi 2341 3706 a(B) p Fr 2414 3718 a(1) p Fn 2451 3639 a(\011) p Fi 1298 3840 a(G) p Fv 1387 3840 a(=) p Fn 1474 3773 a(\010) 1523 3778 y(P) p Ft 1610 3840 a(B) p Fp 1673 3852 a(i) p Ft 1701 3840 a(J) p Fp 1747 3852 a(i) p Fn 1798 3770 a(\014) 1798 3819 y(\014) p Ft 1848 3840 a(B) p Fp 1911 3852 a(i) p Fs 1962 3840 a(2) p Fi 2041 3840 a(B) p Ft(;) p 2151 3840 a(J) p Fp 2197 3852 a(i) p Fs 2248 3840 a(2) p Fi 2326 3840 a(S) p Fn 2395 3773 a(\011) p Ft 2443 3840 a(:) p Fw 609 3997 a(Lemma) p 948 3997 a(13.) p Fl 1111 3997 a(L) l(et) p Ft 1255 3997 a(J) p Fl 1309 3997 a(,) p Ft 1364 3997 a(J) p Fr 1410 4009 a(1) p Fs 1470 3997 a(2) p Fi 1548 3997 a(S) p Fl(.) p 1681 3997 a(Then) p Ft 1897 3997 a(J) p 1951 3997 a(J) p Fr 1997 4009 a(1) p Ft 2035 3997 a(R) p Fs 2121 3997 a(2) p Fi 2200 3997 a(B) p Fr 2273 4009 a(1) p Fi 2310 3997 a(G) p Fv 2394 3997 a(+) p Fi 2477 3997 a(B) p Fr 2550 4009 a(2) p Fl 2588 3997 a(.) 609 4167 y(Pr) l(o) l(of.) p Fv 873 4167 a(In) p 979 4167 a(the) p 1124 4167 a(follo) n(wing) p 1477 4167 a(computations,) p Ft 2020 4167 a(J) p Fk 2066 4179 a(\003) p Fv 2134 4167 a(denotes) p 2436 4167 a(some) p 2646 4167 a(elemen) n(t) p 2955 4167 a(of) p Fi 3052 4167 a(S) p Fv(.) 609 4266 y(Then) p 826 4266 a(w) n(e) p 948 4266 a(ha) n(v) n(e) p Ft 866 4424 a(J) p Fr 912 4436 a(1) p Ft 949 4424 a(R) p Fv 1036 4424 a(=) p Ft 1124 4424 a(R) q(J) p Fr 1234 4436 a(1) p Fv 1289 4424 a(+) p Ft 1372 4424 a(R) p Fv 1436 4424 a([) p Ft(H) r(;) p 1567 4424 a(J) p Fr 1613 4436 a(1) p Fv 1650 4424 a(]) p Ft(R) p Fv 1760 4424 a(=) p Ft 1848 4424 a(R) q(J) p Fr 1958 4436 a(1) p Fv 2013 4424 a(+) p Ft 2096 4424 a(R) p Fv 2160 4424 a(\() p Ft(J) p Fr 2238 4436 a(2) p Fv 2294 4424 a(+) p Ft 2377 4424 a(T) p 2438 4424 a(J) p Fr 2484 4436 a(3) p Fv 2520 4424 a(\)) p Ft(R) p Fv 1036 4548 a(=) p Ft 1124 4548 a(R) q(J) p Fr 1234 4560 a(1) p Fv 1289 4548 a(+) p Ft 1372 4548 a(R) q(J) p Fr 1482 4560 a(2) p Ft 1519 4548 a(R) p Fv 1601 4548 a(+) p Ft 1684 4548 a(R) q(T) p 1809 4548 a(R) q(J) p Fr 1919 4560 a(3) p Fv 1973 4548 a(+) p Ft 2056 4548 a(R) q(T) p 2181 4548 a(R) p Fv 2245 4548 a([) p Ft(H) r(;) p 2376 4548 a(J) p Fr 2422 4560 a(3) p Fv 2458 4548 a(]) p Ft(R) p Fv 1036 4673 a(=) p Ft 1124 4673 a(R) q(J) p Fr 1234 4685 a(1) p Fv 1289 4673 a(+) p Ft 1372 4673 a(R) q(J) p Fr 1482 4685 a(2) p Ft 1519 4673 a(R) p Fv 1601 4673 a(+) p Ft 1684 4673 a(R) q(T) p 1809 4673 a(R) q(J) p Fr 1919 4685 a(3) p Fv 1973 4673 a(+) p Ft 2056 4673 a(R) q(T) p 2181 4673 a(R) q(J) p Fr 2291 4685 a(4) p Ft 2327 4673 a(R) p Fv 2409 4673 a(+) p Ft 2492 4673 a(R) q(T) p 2617 4673 a(R) q(T) p 2742 4673 a(J) p Fr 2788 4685 a(5) p Ft 2823 4673 a(R) p Fv 1036 4806 a(=) p Ft 1124 4806 a(R) p Fn 1188 4739 a(\000) p Ft 1225 4806 a(J) p Fr 1271 4818 a(1) p Fv 1327 4806 a(+) p Ft 1410 4806 a(Q) p Fr 1476 4818 a(2) p Ft 1513 4806 a(J) p Fr 1559 4818 a(3) p Fv 1615 4806 a(+) p 1698 4806 a(\() p Ft(J) p Fr 1776 4818 a(2) p Ft 1813 4806 a(R) p Fv 1895 4806 a(+) p Ft 1978 4806 a(Q) p Fr 2044 4818 a(2) p Ft 2081 4806 a(J) p Fr 2127 4818 a(4) p Ft 2165 4806 a(R) p Fv 2247 4806 a(+) p Ft 2330 4806 a(Q) p Fr 2396 4818 a(4) p Ft 2433 4806 a(J) p Fr 2479 4818 a(5) p Ft 2516 4806 a(R) p Fv 2580 4806 a(\)) p Fn 2612 4739 a(\001) p Fv 2952 4806 a(\(A.2\)) 609 4964 y(and) p 771 4964 a(hence) p Ft 813 5121 a(J) p 867 5121 a(J) p Fr 913 5133 a(1) p Ft 950 5121 a(R) p Fv 1037 5121 a(=) p 1124 5121 a(\() p Ft(J) p 1210 5121 a(R) p Fv 1274 5121 a(\)\() p Ft(J) p Fr 1384 5133 a(1) p Fv 1441 5121 a(+) p Ft 1524 5121 a(Q) p Fr 1590 5133 a(2) p Ft 1626 5121 a(J) p Fr 1672 5133 a(3) p Fv 1710 5121 a(\)) p 1761 5121 a(+) p 1844 5121 a(\() p Ft(J) p 1930 5121 a(R) p Fv 1994 5121 a(\)\() p Ft(J) p Fr 2104 5133 a(2) p Ft 2141 5121 a(R) p Fv 2224 5121 a(+) p Ft 2307 5121 a(Q) p Fr 2373 5133 a(2) p Ft 2409 5121 a(J) p Fr 2455 5133 a(4) p Ft 2493 5121 a(R) p Fv 2575 5121 a(+) p Ft 2658 5121 a(Q) p Fr 2724 5133 a(4) p Ft 2761 5121 a(J) p Fr 2807 5133 a(5) p Ft 2844 5121 a(R) p Fv 2908 5121 a(\)) p Fs 1037 5246 a(2) p Fi 1115 5246 a(B) p Fr 1188 5258 a(1) p Fs 1244 5246 a(\001) p Fi 1286 5246 a(G) p Fv 1369 5246 a(+) p Fi 1452 5246 a(B) p Fr 1525 5258 a(2) p Ft 1563 5246 a(:) p 3083 5403 4 57 v 3087 5350 50 4 v 3087 5403 V 3136 5403 4 57 v Fy 1854 5652 a(9) p 90 rotate dyy eop %%Page: 10 10 10 9 bop Fw 609 548 a(Lemma) p 948 548 a(14.) p Fl 1111 548 a(L) l(et) p Ft 1255 548 a(J) p Fr 1301 560 a(1) p Fs 1361 548 a(2) p Fi 1439 548 a(S) p Fl(.) p 1572 548 a(Then) p Ft 1788 548 a(J) p Fr 1834 560 a(1) p Ft 1871 548 a(Q) p Fp 1937 560 a(i) p Fs 1988 548 a(2) p Fi 2066 548 a(G) p Fv 2150 548 a(+) 2249 526 y(~) p Fi 2233 548 a(B) p Fr 2306 560 a(1) p Fl 2373 548 a(for) p Ft 2506 548 a(i) p Fv 2557 548 a(=) p 2645 548 a(1) p Ft(;) p Fs 2724 548 a(\001) p 2761 548 a(\001) p 2798 548 a(\001) p Ft 2848 548 a(;) p Fv 2885 548 a(4) p Fl(.) 609 719 y(Pr) l(o) l(of.) p Fv 873 719 a(W) p 951 719 a(e) p 1016 719 a(note) p Ft 1025 885 a(R) q(J) p Fv 1165 885 a(=) p Ft 1253 885 a(J) p 1307 885 a(R) p Fv 1389 885 a(+) p Ft 1472 885 a(R) q(J) p Fp 1582 897 a(a) p Ft 1622 885 a(R) p Fv 1704 885 a(+) p Ft 1787 885 a(R) q(T) p 1912 885 a(J) p Fp 1958 897 a(b) p Ft 1990 885 a(R) p Fs 2077 885 a(2) p Fv 2171 863 a(~) p Fi 2155 885 a(B) p Fr 2228 897 a(1) p Fv 2349 885 a(if) p Ft 2481 885 a(J) p Fs 2558 885 a(2) p Fi 2636 885 a(S) p Ft(:) p Fv 2952 885 a(\(A.3\)) p Ft 609 1066 a(J) p Fr 655 1078 a(1) p Ft 692 1066 a(Q) p Fr 758 1078 a(1) p Fs 820 1066 a(2) p Fi 899 1066 a(G) p Fv 984 1066 a(+) 1083 1045 y(~) p Fi 1067 1066 a(B) p Fr 1140 1078 a(1) p Fv 1206 1066 a(follo) n(ws) p 1480 1066 a(immediately) p 1953 1066 a(from) p 2150 1066 a(\(A.2\).) p 2404 1066 a(Using) p 2638 1066 a(\(A.2\),) p 2882 1066 a(w) n(e) p 3005 1066 a(also) 609 1166 y(ha) n(v) n(e) p Ft 690 1332 a(J) p Fr 736 1344 a(1) p Ft 773 1332 a(Q) p Fr 839 1344 a(2) p Fv 899 1332 a(=) p Ft 987 1332 a(J) p Fr 1033 1344 a(1) p Ft 1070 1332 a(T) p 1131 1332 a(R) p Fv 1217 1332 a(=) p Ft 1304 1332 a(T) p 1365 1332 a(J) p Fr 1411 1344 a(1) p Ft 1448 1332 a(R) p Fv 1530 1332 a(+) p Ft 1613 1332 a(J) p Fr 1659 1344 a(6) p Ft 1696 1332 a(R) p Fv 899 1456 a(=) p 987 1456 a(\() p Ft(T) p 1080 1456 a(R) p Fv 1144 1456 a(\)\() p Ft(J) p Fr 1254 1468 a(1) p Fv 1309 1456 a(+) p Ft 1392 1456 a(Q) p Fr 1458 1468 a(2) p Ft 1495 1456 a(J) p Fr 1541 1468 a(3) p Fv 1578 1456 a(\)) p 1629 1456 a(+) p 1712 1456 a(\() p Ft(T) p 1805 1456 a(R) p Fv 1869 1456 a(\)\() p Ft(J) p Fr 1979 1468 a(2) p Ft 2016 1456 a(R) p Fv 2098 1456 a(+) p Ft 2181 1456 a(Q) p Fr 2247 1468 a(2) p Ft 2284 1456 a(J) p Fr 2330 1468 a(4) p Ft 2367 1456 a(R) p Fv 2449 1456 a(+) p Ft 2532 1456 a(Q) p Fr 2598 1468 a(4) p Ft 2635 1456 a(J) p Fr 2681 1468 a(5) p Ft 2718 1456 a(R) p Fv 2782 1456 a(\)) p 2833 1456 a(+) p Ft 2916 1456 a(J) p Fr 2962 1468 a(6) p Ft 2999 1456 a(R) p Fs 899 1597 a(2) p Fi 977 1597 a(G) p Fv 1061 1597 a(+) 1160 1575 y(~) p Fi 1144 1597 a(B) p Fr 1217 1609 a(1) p Ft 1255 1597 a(:) p Fv 609 1763 a(Ab) r(out) p Ft 867 1763 a(J) p Fr 913 1775 a(1) p Ft 951 1763 a(Q) p Fr 1017 1775 a(3) p Fv 1054 1763 a(,) p 1104 1763 a(w) n(e) p 1227 1763 a(ha) n(v) n(e) p Ft 776 1928 a(J) p Fr 822 1940 a(1) p Ft 859 1928 a(Q) p Fr 925 1940 a(3) p Fv 985 1928 a(=) p Ft 1073 1928 a(J) p Fr 1119 1940 a(1) p Ft 1156 1928 a(R) q(T) p Fv 1303 1928 a(=) p Ft 1390 1928 a(R) p Fn 1454 1861 a(\000) p Ft 1492 1928 a(J) p Fr 1538 1940 a(1) p Fv 1594 1928 a(+) p Ft 1677 1928 a(Q) p Fr 1743 1940 a(2) p Ft 1780 1928 a(J) p Fr 1826 1940 a(3) p Fv 1881 1928 a(+) p 1965 1928 a(\() p Ft(J) p Fr 2043 1940 a(2) p Ft 2080 1928 a(R) p Fv 2162 1928 a(+) p Ft 2245 1928 a(Q) p Fr 2311 1940 a(2) p Ft 2348 1928 a(J) p Fr 2394 1940 a(4) p Ft 2431 1928 a(R) p Fv 2514 1928 a(+) p Ft 2597 1928 a(Q) p Fr 2663 1940 a(4) p Ft 2699 1928 a(J) p Fr 2745 1940 a(5) p Ft 2783 1928 a(R) p Fv 2847 1928 a(\)) p Fn 2879 1861 a(\001) p Ft 2917 1928 a(T) p Fv 985 2053 a(=) p 1073 2053 a(\() p Ft(R) q(T) p Fv 1230 2053 a(\)) p Ft(J) p Fr 1308 2065 a(1) p Fv 1363 2053 a(+) p Ft 1446 2053 a(R) q(J) p Fr 1556 2065 a(7) p Fv 1611 2053 a(+) p 1694 2053 a(\() p Ft(R) q(T) p Fv 1851 2053 a(\)\() p Ft(R) q(T) p Fv 2040 2053 a(\)) p Ft(J) p Fr 2118 2065 a(3) p Fv 2173 2053 a(+) p 2256 2053 a(\() p Ft(R) q(T) p Fv 2413 2053 a(\)\() p Ft(R) q(J) p Fr 2587 2065 a(8) p Fv 2623 2053 a(\)) 1146 2178 y(+) p 1230 2178 a(\() p Ft(R) q(J) p Fr 1372 2190 a(2) p Fv 1409 2178 a(\)\() p Ft(R) q(T) p Fv 1598 2178 a(\)) p 1648 2178 a(+) p 1731 2178 a(\() p Ft(R) q(T) p Fv 1888 2178 a(\)\() p Ft(R) q(J) p Fr 2062 2190 a(4) p Fv 2098 2178 a(\)\() p Ft(R) q(T) p Fv 2287 2178 a(\)) p 2337 2178 a(+) p Ft 2420 2178 a(R) q(T) p 2545 2178 a(R) q(T) p 2670 2178 a(J) p Fr 2716 2190 a(5) p Ft 2751 2178 a(R) q(T) p 2871 2178 a(:) p Fv 609 2359 a(No) n(w) p 795 2359 a(the) p 935 2359 a(righ) n(t) p 1133 2359 a(hand) p 1337 2359 a(side) p 1501 2359 a(terms) p 1729 2359 a(are) p 1865 2359 a(elemen) n(ts) p 2201 2359 a(of) p Fi 2293 2359 a(G) p Fv 2370 2359 a(+) 2463 2337 y(~) p Fi 2447 2359 a(B) p Fr 2520 2371 a(1) p Fv 2582 2359 a(b) n(y) p 2694 2359 a(\(A.3\)) p 2910 2359 a(except) 609 2459 y(for) p 736 2459 a(the) p 879 2459 a(last) p 1037 2459 a(term.) p 1267 2459 a(The) p 1438 2459 a(last) p 1595 2459 a(term) p 1794 2459 a(is) p Ft 807 2625 a(R) q(T) p 932 2625 a(R) q(T) p 1057 2625 a(J) p Fr 1103 2637 a(5) p Ft 1138 2625 a(R) q(T) p Fv 1285 2625 a(=) p 1372 2625 a(\() p Ft(R) q(T) p Fv 1529 2625 a(\)\() p Ft(R) q(J) p Fr 1703 2637 a(5) p Fv 1740 2625 a(\)\() p Ft(T) p 1865 2625 a(R) q(T) p Fv 1990 2625 a(\)) p 2039 2625 a(+) p 2122 2625 a(\() p Ft(R) q(T) p Fv 2279 2625 a(\)\() p Ft(R) q(J) p Fr 2453 2637 a(9) p Fv 2490 2625 a(\)\() p Ft(R) q(T) p Fv 2679 2625 a(\)) p Fs 2733 2625 a(2) p Fv 2828 2603 a(~) p Fi 2812 2625 a(B) p Fr 2885 2637 a(1) p Ft 2922 2625 a(:) p Fv 609 2791 a(Th) n(us) p Ft 819 2791 a(J) p Fr 865 2803 a(1) p Ft 903 2791 a(Q) p Fr 969 2803 a(3) p Fs 1029 2791 a(2) p Fi 1107 2791 a(G) p Fv 1191 2791 a(+) 1290 2769 y(~) p Fi 1274 2791 a(B) p Fr 1347 2803 a(1) p Fv 1384 2791 a(.) p 1444 2791 a(Similarly) p 1763 2791 a(,) p 1814 2791 a(w) n(e) p 1936 2791 a(compute) p Ft 1225 2956 a(J) p Fr 1271 2968 a(1) p Ft 1309 2956 a(Q) p Fr 1375 2968 a(4) p Fv 1435 2956 a(=) p Ft 1522 2956 a(J) p Fr 1568 2968 a(1) p Ft 1606 2956 a(T) p 1667 2956 a(R) q(T) p Fv 1812 2956 a(=) p Ft 1900 2956 a(T) p 1961 2956 a(J) p Fr 2007 2968 a(1) p Ft 2043 2956 a(R) q(T) p Fv 2186 2956 a(+) p Ft 2269 2956 a(J) p Fr 2315 2968 a(10) p Ft 2385 2956 a(R) q(T) p 2505 2956 a(:) p Fv 609 3122 a(The) p 780 3122 a(second) p 1047 3122 a(term) p 1246 3122 a(in) p 1342 3122 a(the) p 1485 3122 a(righ) n(t) p 1685 3122 a(hand) p 1893 3122 a(side) p 2059 3122 a(is) p Ft 2142 3122 a(J) p Fr 2188 3134 a(10) p Ft 2259 3122 a(Q) p Fr 2325 3134 a(3) p Fs 2384 3122 a(2) p Fi 2463 3122 a(G) p Fv 2546 3122 a(+) 2644 3101 y(~) p Fi 2628 3122 a(B) p Fr 2701 3134 a(1) p Fv 2739 3122 a(.) p 2799 3122 a(By) p 2929 3122 a(\(A.2\),) 609 3222 y(the) p 752 3222 a(\014rst) p 924 3222 a(term) p 1122 3222 a(is) p Ft 711 3388 a(T) p 772 3388 a(J) p Fr 818 3400 a(1) p Ft 854 3388 a(R) q(T) p Fv 1001 3388 a(=) p Ft 1088 3388 a(T) p 1149 3388 a(R) p Fn 1213 3321 a(\000) p Ft 1250 3388 a(J) p Fr 1296 3400 a(1) p Fv 1352 3388 a(+) p Ft 1435 3388 a(Q) p Fr 1501 3400 a(2) p Ft 1538 3388 a(J) p Fr 1584 3400 a(3) p Fv 1640 3388 a(+) p 1723 3388 a(\() p Ft(J) p Fr 1801 3400 a(2) p Ft 1838 3388 a(R) p Fv 1920 3388 a(+) p Ft 2003 3388 a(Q) p Fr 2069 3400 a(2) p Ft 2106 3388 a(J) p Fr 2152 3400 a(4) p Ft 2189 3388 a(R) p Fv 2272 3388 a(+) p Ft 2355 3388 a(Q) p Fr 2421 3400 a(4) p Ft 2457 3388 a(J) p Fr 2503 3400 a(5) p Ft 2541 3388 a(R) p Fv 2605 3388 a(\)) p Fn 2637 3321 a(\001) p Ft 2675 3388 a(T) p Fv 1001 3512 a(=) p 1088 3512 a(\() p Ft(T) p 1181 3512 a(R) q(T) p Fv 1306 3512 a(\)) p Ft(J) p Fr 1384 3524 a(1) p Fv 1438 3512 a(+) p 1521 3512 a(\() p Ft(T) p 1614 3512 a(R) p Fv 1678 3512 a(\)) p Ft(J) p Fr 1756 3524 a(11) p Fv 1845 3512 a(+) p 1928 3512 a(\() p Ft(T) p 2021 3512 a(R) q(T) p Fv 2146 3512 a(\)\() p Ft(R) q(T) p Fv 2335 3512 a(\)) p Ft(J) p Fr 2413 3524 a(3) p Fv 2466 3512 a(+) p 2549 3512 a(\() p Ft(T) p 2642 3512 a(R) p Fv 2706 3512 a(\)\() p Ft(T) p 2831 3512 a(R) p Fv 2895 3512 a(\)) p Ft(J) p Fr 2973 3524 a(12) p Fv 1162 3653 a(+) p 1245 3653 a(\() p Ft(T) p 1338 3653 a(R) p Fv 1402 3653 a(\)\() p Ft(J) p Fr 1512 3665 a(2) p Ft 1549 3653 a(Q) p Fr 1615 3665 a(3) p Fv 1670 3653 a(+) p Ft 1753 3653 a(Q) p Fr 1819 3665 a(2) p Ft 1856 3653 a(J) p Fr 1902 3665 a(4) p Ft 1940 3653 a(Q) p Fr 2006 3665 a(3) p Fv 2061 3653 a(+) p Ft 2144 3653 a(Q) p Fr 2210 3665 a(4) p Ft 2247 3653 a(J) p Fr 2293 3665 a(5) p Ft 2330 3653 a(Q) p Fr 2396 3665 a(3) p Fv 2433 3653 a(\)) p Fs 2488 3653 a(2) p Fi 2567 3653 a(G) p Fv 2650 3653 a(+) 2749 3631 y(~) p Fi 2733 3653 a(B) p Fr 2806 3665 a(1) p Ft 2844 3653 a(;) p Fv 609 3834 a(where) p 849 3834 a(w) n(e) p 971 3834 a(ha) n(v) n(e) p 1163 3834 a(used) p Ft 1352 3834 a(J) p Fk 1398 3846 a(\003) p Ft 1437 3834 a(Q) p Fr 1503 3846 a(3) p Fs 1563 3834 a(2) p Fi 1641 3834 a(G) p Fv 1725 3834 a(+) 1824 3812 y(~) p Fi 1808 3834 a(B) p Fr 1881 3846 a(1) p Fv 1918 3834 a(.) p 3083 3834 4 57 v 3087 3781 50 4 v 3087 3834 V 3136 3834 4 57 v Fl 609 4005 a(Pr) l(o) l(of) p 832 4005 a(of) p 930 4005 a(Pr) l(op) l(osition) p 1371 4005 a(12) p 1486 4005 a(\(c) l(ontinue) l(d\).) p Fv 1968 4005 a(By) p 2098 4005 a(Lemma) p 2395 4005 a(13,) p 2529 4005 a(w) n(e) p 2651 4005 a(ha) n(v) n(e) p Ft 709 4171 a(J) p Fv 763 4171 a(\() p Ft(J) p Fr 841 4183 a(1) p Ft 879 4171 a(R) q(B) p Fv 1010 4171 a(\)) p 1065 4171 a(=) p 1152 4171 a(\() p Ft(B) p Fr 1247 4183 a(1) p Ft 1285 4171 a(G) p Fr 1350 4183 a(1) p Fv 1406 4171 a(+) p Ft 1489 4171 a(B) p Fr 1552 4183 a(2) p Fv 1589 4171 a(\)) p Ft(B) t(;) p Fv 1808 4171 a(with) p Ft 2053 4171 a(B) p Fr 2116 4183 a(1) p Fs 2176 4171 a(2) p Fi 2255 4171 a(B) p Fr 2328 4183 a(1) p Ft 2365 4171 a(;) p 2402 4171 a(B) p Fr 2465 4183 a(2) p Fs 2525 4171 a(2) p Fi 2604 4171 a(B) p Fr 2677 4183 a(2) p Ft 2714 4171 a(;) p 2751 4171 a(G) p Fr 2816 4183 a(1) p Fs 2877 4171 a(2) p Fi 2955 4171 a(G) p Ft(;) p Fv 609 4337 a(if) p Ft 679 4337 a(J) p Fv 733 4337 a(,) p Ft 778 4337 a(J) p Fr 824 4349 a(1) p Fs 884 4337 a(2) p Fi 963 4337 a(S) p Fv 1053 4337 a(and) p Ft 1207 4337 a(B) p Fs 1297 4337 a(2) p Fi 1376 4337 a(B) p Fv(.) p 1507 4337 a(By) p 1631 4337 a(Lemma) p 1920 4337 a(14,) p Ft 2049 4337 a(G) p Fr 2114 4349 a(1) p Ft 2151 4337 a(Q) p Fp 2217 4349 a(j) p Fs 2275 4337 a(2) p Fi 2353 4337 a(G) p Fv 2423 4337 a(+) 2509 4315 y(~) p Fi 2493 4337 a(B) p Fr 2566 4349 a(1) p Fv 2625 4337 a(\() p Ft(j) p Fv 2719 4337 a(=) p 2807 4337 a(1) p Ft(;) p Fs 2886 4337 a(\001) p 2923 4337 a(\001) p 2960 4337 a(\001) p Ft 3010 4337 a(;) p Fv 3047 4337 a(4\),) 609 4446 y(and) p 762 4446 a(b) n(y) p 870 4446 a(iterating) p 1198 4446 a(this) p 1352 4446 a(form) n(ula,) p 1673 4446 a(w) n(e) p 1787 4446 a(learn) p Ft 1987 4446 a(G) p Fr 2052 4458 a(1) p Ft 2090 4446 a(B) p Fs 2180 4446 a(2) p Fi 2258 4446 a(G) p Fv 2325 4446 a(+) 2408 4424 y(~) p Fi 2392 4446 a(B) p Fr 2465 4458 a(1) p Fv 2503 4446 a(.) p 2560 4446 a(Hence) p 2799 4446 a(w) n(e) p 2913 4446 a(obtain) p Ft 1454 4612 a(J) p Fv 1508 4612 a(\() p Ft(J) p Fr 1586 4624 a(1) p Ft 1623 4612 a(R) q(B) p Fv 1754 4612 a(\)) p Fs 1809 4612 a(2) p Fi 1888 4612 a(B) p Fr 1961 4624 a(1) p Fi 1999 4612 a(G) p Fv 2082 4612 a(+) p Fi 2165 4612 a(B) p Fr 2238 4624 a(2) p Ft 2276 4612 a(:) p Fv 609 4778 a(Iterating) p 953 4778 a(this) p 1115 4778 a(form) n(ula) p 1420 4778 a(again,) p 1664 4778 a(w) n(e) p 1786 4778 a(learn) p Ft 1287 5020 a(J) p Fn 1341 4903 a(\032) p Fp 1427 4917 a(K) p Fn 1404 4942 a(Y) p Fp 1404 5118 a(i) p Fr(=1) p Ft 1525 5020 a(J) p Fp 1571 5032 a(i) p Ft 1599 5020 a(R) q(B) p Fp 1726 5032 a(i) p Fn 1753 4903 a(\033) p Fs 1838 5020 a(2) p Fi 1917 5020 a(B) p Fp 1990 5032 a(K) p Fi 2054 5020 a(G) p Fv 2138 5020 a(+) p Fi 2221 5020 a(B) p Fp 2294 5032 a(K) p Fr 2354 5032 a(+1) p Ft 2442 5020 a(;) p Fv 609 5283 a(and) p 776 5283 a(hence) p Ft 1011 5283 a(J) p Fn 1065 5215 a(\010) 1114 5220 y(Q) p Fp 1192 5241 a(K) 1192 5308 y(i) p Fr(=1) p Ft 1317 5283 a(J) p Fp 1363 5295 a(i) p Ft 1391 5283 a(R) q(B) p Fp 1518 5295 a(i) p Fn 1545 5215 a(\011) p Ft 1594 5283 a(R) p Fs 1689 5283 a(2) p Fi 1775 5283 a(B) p Fp 1848 5295 a(K) p Fr 1908 5295 a(+1) p Fv 1997 5283 a(.) p 2072 5283 a(This) p 2266 5283 a(completes) p 2654 5283 a(the) p 2802 5283 a(induction) 609 5382 y(step) p 785 5382 a(\(A.1\),) p 1027 5382 a(and) p 1188 5382 a(Prop) r(osition) p 1635 5382 a(12) p 1746 5382 a(is) p 1830 5382 a(pro) n(v) n(ed.) p 3083 5382 V 3087 5330 50 4 v 3087 5382 V 3136 5382 4 57 v Fy 1831 5652 a(10) p 90 rotate dyy eop %%Page: 11 11 11 10 bop Fu 609 548 a(App) t(endix) p 1214 548 a(B.) p 1378 548 a(In) l(tegrated) p 2020 548 a(densit) l(y) p 2472 548 a(of) p 2612 548 a(states) p 2980 548 a(for) 609 697 y(the) p 842 697 a(Landau) p 1323 697 a(Hamiltonian) p 2097 697 a(on) p Fd 2328 697 a(R) p Fv 2409 654 a(2) p Fu 2460 697 a(.) p Fv 609 879 a(In) p 722 879 a(this) p 894 879 a(app) r(endix,) p 1288 879 a(w) n(e) p 1420 879 a(compute) p 1766 879 a(the) p 1918 879 a(IDS) p 2095 879 a(for) p 2231 879 a(the) p 2383 879 a(Landau) p 2694 879 a(Hamiltonian) 609 979 y(on) p Fq 728 979 a(R) p Fr 782 949 a(2) p Fv 826 979 a(.) p 897 979 a(It) p 991 979 a(app) r(ears) p 1302 979 a(that) p 1486 979 a(the) p 1633 979 a(result) p 1868 979 a(\(and) p 2066 979 a(the) p 2213 979 a(follo) n(wing) p 2567 979 a(computation\)) p 3088 979 a(is) 609 1078 y(w) n(ell-kno) n(wn) p 1043 1078 a(among) p 1314 1078 a(ph) n(ysicists,) p 1715 1078 a(and) p 1880 1078 a(in) p 1980 1078 a(fact) p 2146 1078 a(the) p 2292 1078 a(author) p 2563 1078 a(w) n(as) p 2725 1078 a(informed) p 3077 1078 a(of) 609 1178 y(this) p 767 1178 a(b) n(y) p 878 1178 a(T.) p 984 1178 a(Koma) p 1225 1178 a(\(Dept.) p 1482 1178 a(Ph) n(ysics,) p 1799 1178 a(Gakush) n(uin) p 2211 1178 a(Univ) n(ersit) n(y\).) p 2674 1178 a(Ho) n(w) n(ev) n(er,) p 3028 1178 a(the) 609 1278 y(justi\014cation) p 1060 1278 a(requires) p 1369 1278 a(the) p 1508 1278 a(fact) p 1668 1278 a(that) p 1845 1278 a(the) p 1984 1278 a(IDS) p 2147 1278 a(is) p 2227 1278 a(indep) r(enden) n(t) p 2694 1278 a(of) p 2785 1278 a(the) p 2924 1278 a(c) n(hoice) 609 1377 y(of) p 704 1377 a(the) p 847 1377 a(b) r(oundary) p 1221 1377 a(condition.) 734 1477 y(W) p 812 1477 a(e) p 890 1477 a(consider) p Ft 1227 1477 a(H) p Fv 1348 1477 a(=) p 1458 1477 a(\() p Ft(p) p Fs 1559 1477 a(\000) p Ft 1651 1477 a(A) p Fv(\)) p Fr 1745 1447 a(2) p Fv 1823 1477 a(on) p Ft 1952 1477 a(L) p Fr 2009 1447 a(2) p Fv 2046 1477 a(\() p Fq(R) p Fr 2132 1447 a(2) p Fv 2175 1477 a(\),) p 2275 1477 a(where) p Ft 2528 1477 a(A) p Fv 2636 1477 a(=) p 2746 1477 a(\(0) p Ft(;) p 2857 1477 a(B) t(x) p Fv(\)) p 3044 1477 a(for) 609 1577 y(\() p Ft(x;) p 725 1577 a(y) p Fv 769 1577 a(\)) p Fs 825 1577 a(2) p Fq 903 1577 a(R) p Fr 957 1546 a(2) p Fv 1028 1577 a(with) p Ft 1217 1577 a(B) p Fs 1307 1577 a(6) p Fv(=) p 1395 1577 a(0.) p 1497 1577 a(F) p 1544 1577 a(or) p 1646 1577 a(simplicit) n(y) p 1990 1577 a(,) p 2040 1577 a(w) n(e) p 2163 1577 a(supp) r(ose) p Ft 2475 1577 a(B) p 2565 1577 a(>) p Fv 2653 1577 a(0.) p 2754 1577 a(W) p 2832 1577 a(e) p 2897 1577 a(de\014ne) p Ft 1333 1759 a(H) p Fp 1402 1771 a(L) p Fv 1475 1759 a(=) p 1563 1759 a(\() p Ft(p) p Fs 1655 1759 a(\000) p Ft 1738 1759 a(A) p Fv(\)) p Fr 1832 1725 a(2) p Fv 2036 1759 a(on) p Ft 2151 1759 a(L) p Fr 2208 1725 a(2) p Fv 2245 1759 a(\(\012) p Fp 2337 1771 a(L) p Fv 2387 1759 a(\)) 609 1942 y(with) p Fm 609 2125 a(Q) p Fv(\() p Ft(H) p Fp 765 2137 a(L) p Fv 815 2125 a(\)) p 871 2125 a(=) p Fm 958 2125 a(D) p Fv(\(\() p Ft(H) p Fp 1160 2137 a(L) p Fv 1210 2125 a(\)) p Fr 1242 2090 a(1) p Fp(=) p Fr(2) p Fv 1347 2125 a(\)) 715 2259 y(=) p Fn 803 2192 a(\010) p Ft 851 2259 a(') p Fs 929 2259 a(2) p Ft 1007 2259 a(H) p Fr 1083 2225 a(1) p Fv 1120 2259 a(\(\012) p Fp 1212 2271 a(L) p Fv 1262 2259 a(\)) p Fn 1317 2189 a(\014) 1317 2239 y(\014) p Ft 1368 2259 a(') p Fv(\() p Ft(x;) p Fv 1538 2259 a(0\)) p 1636 2259 a(=) p Ft 1724 2259 a(') p Fv(\() p Ft(x;) p 1894 2259 a(L) p Fv(\)) p Ft(;) p 2020 2259 a(') p Fv(\(0) p Ft(;) p 2185 2259 a(y) p Fv 2229 2259 a(\)) p 2284 2259 a(=) p Ft 2372 2259 a(') p Fv(\() p Ft(L;) p 2552 2259 a(y) p Fv 2596 2259 a(\)) p 2651 2259 a(=) p 2739 2259 a(0) p Ft(;) p Fv 2818 2259 a(a.e.) p Ft 2978 2259 a(x;) p 3062 2259 a(y) p Fn 3106 2192 a(\011) p Ft 3155 2259 a(;) p Fv 609 2442 a(i.e.,) p Ft 777 2442 a(') p Fs 870 2442 a(2) p Fm 963 2442 a(D) p Fv(\() p Ft(H) p Fp 1133 2454 a(L) p Fv 1183 2442 a(\)) p 1252 2442 a(satis\014es) p 1567 2442 a(the) p 1719 2442 a(Diric) n(hlet) p 2071 2442 a(b) r(oundary) p 2454 2442 a(condition) p 2827 2442 a(on) p Fs 2952 2442 a(f) p Ft(x) p Fv 3079 2442 a(=) 609 2542 y(0) p Ft(;) p 688 2542 a(L) p Fs(g) p Fv(,) p 836 2542 a(and) p 998 2542 a(the) p 1141 2542 a(p) r(erio) r(dic) p 1459 2542 a(b) r(oundary) p 1833 2542 a(condition) p 2197 2542 a(on) p Fs 2313 2542 a(f) p Ft(y) p Fv 2421 2542 a(=) p 2509 2542 a(0) p Ft(;) p 2588 2542 a(L) p Fs(g) p Fv(.) p 2745 2542 a(W) p 2823 2542 a(e) p 2888 2542 a(set) p Ft 1200 2724 a(N) p Fp 1267 2736 a(L) p Fv 1316 2724 a(\() p Ft(E) p Fv 1414 2724 a(\)) p 1470 2724 a(=) p 1558 2724 a(#) p Fs(f) p Fv(eigen) n(v) p 1891 2724 a(alues) p 2098 2724 a(of) p Ft 2192 2724 a(H) p Fp 2261 2736 a(L) p Fs 2334 2724 a(\024) p Ft 2422 2724 a(E) p Fs 2488 2724 a(g) p Ft(;) p Fv 609 2907 a(and) p 771 2907 a(w) n(e) p 893 2907 a(compute) p Ft 1429 3090 a(k) p Fv 1475 3090 a(\() p Ft(E) p Fv 1573 3090 a(\)) p 1628 3090 a(=) p 1747 3090 a(lim) p Fp 1716 3143 a(L) p Fk(!1) p Ft 1908 3090 a(L) p Fk 1965 3055 a(\000) p Fr(2) p Ft 2054 3090 a(N) p Fp 2121 3102 a(L) p Fv 2170 3090 a(\() p Ft(E) p Fv 2268 3090 a(\)) p Ft(:) 609 3305 y(k) p Fv 655 3305 a(\() p Ft(E) p Fv 753 3305 a(\)) p 823 3305 a(is) p 916 3305 a(the) p 1068 3305 a(IDS) p 1245 3305 a(b) n(y) p 1369 3305 a(virtue) p 1621 3305 a(of) p 1725 3305 a(Corollary) p 2102 3305 a(3.) p 2231 3305 a(In) p 2344 3305 a(order) p 2571 3305 a(to) p 2682 3305 a(compute) p 3028 3305 a(the) 609 3405 y(eigen) n(v) p 831 3405 a(alues) p 1051 3405 a(of) p Ft 1157 3405 a(H) p Fp 1226 3417 a(L) p Fv 1276 3405 a(,) p 1341 3405 a(w) n(e) p 1476 3405 a(use) p 1631 3405 a(the) p 1786 3405 a(separation) p 2200 3405 a(of) p 2306 3405 a(v) p 2345 3405 a(ariables.) p 2719 3405 a(A) p 2821 3405 a(complete) 609 3504 y(system) p 885 3504 a(of) p 979 3504 a(eigenfunctions) p 1521 3504 a(of) p Fs 1616 3504 a(\000) p Ft(@) p Fr 1730 3474 a(2) p Fp 1725 3525 a(y) p Fv 1794 3504 a(is) p 1878 3504 a(giv) n(en) p 2094 3504 a(b) n(y) p Ft 1148 3705 a(f) p Fp 1189 3717 a(n) p Fv 1234 3705 a(\() p Ft(y) p Fv 1310 3705 a(\)) p 1365 3705 a(=) p Ft 1453 3705 a(c) p 1512 3705 a(e) p Fp 1551 3671 a(i) p Fr(2) p Fp(\031) r(ny) r(=L) p Ft 1809 3705 a(;) p 2012 3705 a(y) p Fs 2078 3705 a(2) p Fv 2157 3705 a([0) p Ft(;) p 2259 3705 a(L) p Fv(]) p Ft(;) p 2376 3705 a(n) p Fs 2448 3705 a(2) p Fq 2526 3705 a(Z) p Ft 2587 3705 a(;) p Fv 609 3888 a(where) p Ft 849 3888 a(c) p Fv 908 3888 a(=) p Ft 996 3888 a(L) p Fk 1053 3858 a(\000) p Fr(1) p Fp(=) p Fr(2) p Fv 1236 3888 a(is) p 1320 3888 a(a) p 1389 3888 a(normalization) p 1915 3888 a(constan) n(t.) p 2282 3888 a(Let) p Fi 1087 4071 a(H) p Fp 1147 4083 a(n) p Fv 1215 4071 a(=) p Fn 1303 4003 a(\010) p Ft 1351 4071 a(g) p Fv 1394 4071 a(\() p Ft(x) p Fv(\)) p Ft(f) p Fp 1546 4083 a(n) p Fv 1592 4071 a(\() p Ft(y) p Fv 1668 4071 a(\)) p Fs 1723 4071 a(2) p Ft 1801 4071 a(L) p Fr 1858 4036 a(2) p Fv 1895 4071 a(\(\012) p Fp 1987 4083 a(L) p Fv 2037 4071 a(\)) p Fn 2092 4000 a(\014) 2092 4050 y(\014) p Ft 2143 4071 a(g) p Fs 2209 4071 a(2) p Ft 2287 4071 a(L) p Fr 2344 4036 a(2) p Fv 2381 4071 a(\([0) p Ft(;) p 2515 4071 a(L) p Fv(]) p Fn 2595 4003 a(\011) p Ft 2643 4071 a(:) p Fv 609 4253 a(Then) p Ft 826 4253 a(L) p Fr 883 4223 a(2) p Fv 920 4253 a(\(\012) p Fp 1012 4265 a(L) p Fv 1062 4253 a(\)) p 1117 4253 a(=) p Fs 1205 4253 a(\010) p Fp 1270 4265 a(n) p Fk(2) p Fg(Z) p Fi 1412 4253 a(H) p Fp 1472 4265 a(n) p Fv 1517 4253 a(,) p 1567 4253 a(and) p 1729 4253 a(w) n(e) p 1851 4253 a(ha) n(v) n(e) p Ft 917 4486 a(H) p Fp 986 4498 a(L) p Ft 1050 4486 a(') p Fp 1104 4498 a(n) p Fv 1172 4486 a(=) p Fn 1260 4369 a(\024) p Fs 1304 4486 a(\000) p Ft(@) p Fr 1418 4451 a(2) p Fp 1413 4506 a(x) p Fv 1473 4486 a(+) p Fn 1556 4394 a(\020) p Fv 1615 4430 a(2) p Ft(\031) s(n) p 1615 4467 142 4 v 1658 4543 a(L) p Fs 1785 4486 a(\000) p Ft 1868 4486 a(B) t(x) p Fn 1982 4394 a(\021) p Fr 2033 4411 a(2) p Fn 2070 4369 a(\025) p Ft 2114 4486 a(') p Fp 2168 4498 a(n) p Fv 2379 4486 a(for) p Ft 2506 4486 a(') p Fp 2560 4498 a(n) p Fs 2629 4486 a(2) p Fi 2707 4486 a(H) p Fp 2767 4498 a(n) p Ft 2812 4486 a(:) p Fv 609 4714 a(Hence) p 856 4714 a(w) n(e) p 978 4714 a(need) p 1172 4714 a(only) p 1354 4714 a(to) p 1456 4714 a(compute) p 1792 4714 a(the) p 1935 4714 a(eigen) n(v) p 2157 4714 a(alues) p 2365 4714 a(of) p Ft 1038 4933 a(H) p Fp 1107 4945 a(L;n) p Fv 1240 4933 a(=) p Fs 1328 4933 a(\000) p Ft(@) p Fr 1442 4899 a(2) p Fp 1437 4954 a(x) p Fv 1497 4933 a(+) p Fn 1580 4841 a(\020) p Fv 1640 4877 a(2) p Ft(\031) s(n) p 1640 4914 V 1682 4990 a(L) p Fs 1810 4933 a(\000) p Ft 1893 4933 a(B) t(x) p Fn 2007 4841 a(\021) p Fr 2057 4858 a(2) p Fv 2260 4933 a(on) p Ft 2375 4933 a(L) p Fr 2432 4899 a(2) p Fv 2469 4933 a(\([0) p Ft(;) p 2603 4933 a(L) p Fv(]\)) 609 5149 y(with) p Fm 798 5149 a(Q) p Fv(\() p Ft(H) p Fp 954 5161 a(L;n) p Fv 1065 5149 a(\)) p 1120 5149 a(=) p Ft 1208 5149 a(H) p Fr 1284 5119 a(1) 1277 5170 y(0) p Fv 1321 5149 a(\([0) p Ft(;) p 1455 5149 a(L) p Fv(]\).) p 1627 5149 a(Since) p Fn 1277 5285 a(\020) p Fv 1336 5321 a(2) p Ft(\031) s(n) p 1336 5358 V 1379 5434 a(L) p Fs 1506 5377 a(\000) p Ft 1589 5377 a(B) t(x) p Fn 1703 5285 a(\021) p Fr 1754 5302 a(2) p Fv 1814 5377 a(=) p Ft 1901 5377 a(B) p Fr 1968 5342 a(2) p Fn 2006 5285 a(\020) p Ft 2055 5377 a(x) p Fs 2121 5377 a(\000) p Fv 2214 5321 a(2) p Ft(\031) s(n) p 2214 5358 V 2223 5434 a(B) t(L) p Fn 2366 5285 a(\021) p Fr 2416 5302 a(2) p Ft 2453 5377 a(;) p Fy 1831 5652 a(11) p 90 rotate dyy eop %%Page: 12 12 12 11 bop Ft 609 548 a(H) p Fp 678 560 a(L;n) p Fv 807 548 a(is) p 881 548 a(a) p 941 548 a(one-dimensional) p 1541 548 a(harmonic) p 1897 548 a(oscillator) p 2243 548 a(with) p 2423 548 a(the) p 2557 548 a(cen) n(ter) p 2795 548 a(at) p 2887 548 a(2) p Ft(\031) s(n=B) t(L) p Fv(,) 609 648 y(restricted) p 979 648 a(to) p 1081 648 a([0) p Ft(;) p 1183 648 a(L) p Fv(].) p 1322 648 a(If) 1560 863 y(0) p Ft 1624 863 a(<<) p Fv 1786 807 a(2) p Ft(\031) s(n) p 1786 844 142 4 v 1795 920 a(B) t(L) 1961 863 y(<<) p 2113 863 a(L;) p Fv 609 1063 a(then) p Ft 841 1239 a(\033) p Fv 891 1239 a(\() p Ft(H) p Fp 992 1251 a(L;n) p Fv 1103 1239 a(\)) p Fs 1158 1239 a(\030) p Ft 1246 1239 a(\033) p Fv 1296 1239 a(\() p Fs(\000) p Ft(@) p Fr 1442 1205 a(2) p Fp 1437 1260 a(x) p Fv 1498 1239 a(+) p Ft 1581 1239 a(B) p Fr 1648 1205 a(2) p Ft 1685 1239 a(x) p Fr 1732 1205 a(2) p Fv 1770 1239 a(\)) p 1825 1239 a(=) p Fs 1913 1239 a(f) p Fv(\(2) p Ft(`) p Fv 2081 1239 a(+) p 2164 1239 a(1\)) p Ft(B) p Fs 2319 1239 a(j) p Ft 2356 1239 a(`) p Fv 2414 1239 a(=) p 2501 1239 a(0) p Ft(;) p Fv 2580 1239 a(1) p Ft(;) p Fv 2659 1239 a(2) p Ft(;) p 2738 1239 a(:) p 2775 1239 a(:) p 2812 1239 a(:) p Fs 2847 1239 a(g) p Ft(:) p Fv 609 1415 a(On) p 748 1415 a(the) p 891 1415 a(other) p 1108 1415 a(hand,) p 1338 1415 a(if) 1369 1570 y(2) p Ft(\031) s(n) p 1369 1607 V 1378 1683 a(B) t(L) 1544 1626 y(<<) p Fv 1696 1626 a(0) p 1820 1626 a(or) 1987 1570 y(2) p Ft(\031) s(n) p 1987 1607 V 1996 1683 a(B) t(L) 2162 1626 y(>>) p 2314 1626 a(L;) p Fv 609 1831 a(then) p 798 1831 a(inf) p Ft 913 1831 a(\033) p Fv 963 1831 a(\() p Ft(H) p Fp 1064 1843 a(L;n) p Fv 1175 1831 a(\)) p Ft 1231 1831 a(>>) p Fv 1383 1831 a(0.) p 1484 1831 a(Hence) p 1731 1831 a(w) n(e) p 1853 1831 a(exp) r(ect) p Ft 1026 2052 a(N) p Fp 1093 2064 a(L) p Fv 1143 2052 a(\() p Ft(E) p Fv 1241 2052 a(\)) p Fs 1296 2052 a(\030) p Ft 1384 2052 a(L) p Fn 1441 1960 a(\020) p Fv 1516 1996 a(2) p Ft(\031) p 1500 2033 124 4 v 1500 2109 a(B) t(L) p Fn 1634 1960 a(\021) p Fk 1683 1977 a(\000) p Fr(1) p Fv 1772 2052 a(#) p Fn 1841 1984 a(\010) p Ft 1890 2052 a(`) p Fs 1948 2052 a(\025) p Fv 2035 2052 a(0) p Fn 2100 1981 a(\014) 2100 2031 y(\014) p Fv 2150 2052 a(\(2) p Ft(`) p Fv 2277 2052 a(+) p 2360 2052 a(1\)) p Ft(B) p Fs 2524 2052 a(\024) p Ft 2612 2052 a(E) p Fn 2678 1984 a(\011) p Fv 1296 2255 a(=) p Ft 1384 2255 a(L) p Fr 1441 2220 a(2) p Ft 1500 2199 a(B) p 1488 2236 92 4 v Fv 1488 2312 a(2) p Ft(\031) 1589 2255 y(F) p Fv 1654 2255 a(\() p Ft(E) p 1752 2255 a(=B) p Fv 1861 2255 a(\)) p Ft(;) p Fv 609 2454 a(where) p 849 2454 a(w) n(e) p 971 2454 a(set) p Ft 968 2685 a(F) p Fv 1033 2685 a(\() p Ft(\025) p Fv(\)) p 1169 2685 a(=) p Fn 1257 2543 a(\() p Fv 1324 2628 a(0) p Ft(;) p Fv 1554 2628 a(if) p Ft 1630 2628 a(\025) p 1702 2628 a(<) p Fv 1790 2628 a(1) p Ft 1831 2628 a(;) 1324 2748 y(`;) p Fv 1554 2748 a(if) p 1630 2748 a(2) p Ft(`) p Fs 1725 2748 a(\000) p Fv 1808 2748 a(1) p Fs 1873 2748 a(\024) p Ft 1960 2748 a(\025) p 2032 2748 a(<) p Fv 2119 2748 a(2) p Ft(`) p Fv 2214 2748 a(+) p 2297 2748 a(1) p 2366 2748 a(with) p Ft 2555 2748 a(`) p Fs 2613 2748 a(2) p Fq 2691 2748 a(N) p Ft 2751 2748 a(:) p Fv 609 2931 a(Th) n(us) p 819 2931 a(w) n(e) p 942 2931 a(arriv) n(e) p 1177 2931 a(at) p 1279 2931 a(a) p 1348 2931 a(conjecture) p Ft 1298 3145 a(k) p Fv 1344 3145 a(\() p Ft(E) p Fv 1442 3145 a(\)) p 1498 3145 a(=) p Ft 1607 3089 a(B) p 1595 3126 V Fv 1595 3202 a(2) p Ft(\031) 1697 3145 y(F) p Fv 1762 3145 a(\() p Ft(E) p 1860 3145 a(=B) p Fv 1969 3145 a(\)) p Ft(;) p 2204 3145 a(E) p Fs 2293 3145 a(2) p Fq 2372 3145 a(R) p Ft(:) p Fv 609 3345 a(In) p 713 3345 a(fact) p 877 3345 a(w) n(e) p 999 3345 a(can) p 1151 3345 a(pro) n(v) n(e) p 1375 3345 a(the) p 1518 3345 a(follo) n(wing.) p Fw 609 3492 a(Prop) s(osition) p 1125 3492 a(15.) p Fl 1289 3492 a(L) l(et) p Ft 1432 3492 a(E) p 1531 3492 a(=) p Fs 1521 3492 a(2) p 1600 3492 a(f) p Fv(\(2) p Ft(`) p Fv 1768 3492 a(+) p 1851 3492 a(1\)) p Ft(B) p Fs 2006 3492 a(j) p Ft 2043 3492 a(`) p Fv 2100 3492 a(=) p 2188 3492 a(0) p Ft(;) p Fv 2267 3492 a(1) p Ft(;) p 2346 3492 a(:) p 2383 3492 a(:) p 2420 3492 a(:) p Fs 2456 3492 a(g) p Fl(.) p 2561 3492 a(Then) p Ft 1166 3711 a(k) p Fv 1212 3711 a(\() p Ft(E) p Fv 1310 3711 a(\)) p 1365 3711 a(=) p 1484 3711 a(lim) p Fp 1453 3764 a(L) p Fk(!1) p Ft 1645 3711 a(L) p Fk 1702 3676 a(\000) p Fr(2) p Ft 1790 3711 a(N) p Fp 1857 3723 a(L) p Fv 1907 3711 a(\() p Ft(E) p Fv 2005 3711 a(\)) p 2060 3711 a(=) p Ft 2170 3654 a(B) p 2158 3691 V Fv 2158 3767 a(2) p Ft(\031) 2260 3711 y(F) p Fv 2325 3711 a(\() p Ft(E) p 2423 3711 a(=B) p Fv 2532 3711 a(\)) p Ft(:) p Fl 609 3920 a(Pr) l(o) l(of.) p Fv 873 3920 a(W) p 951 3920 a(e) p 1016 3920 a(set) p Ft 1146 3920 a(`) p Fr 1181 3932 a(0) p Fs 1241 3920 a(2) p Fq 1319 3920 a(Z) p Fv 1402 3920 a(so) p 1504 3920 a(that) p Ft 1684 3920 a(E) p Fs 1773 3920 a(2) p Fv 1852 3920 a(\(\(2) p Ft(`) p Fr 1993 3932 a(0) p Fs 2048 3920 a(\000) p Fv 2131 3920 a(1\)) p Ft(B) t(;) p Fv 2309 3920 a(\(2) p Ft(`) p Fr 2418 3932 a(0) p Fv 2473 3920 a(+) p 2556 3920 a(1\)) p Ft(B) p Fv 2697 3920 a(\).) p 2789 3920 a(Let) 1328 4096 y(0) p Ft 1392 4096 a(<) p 1480 4096 a(") p 1542 4096 a(<) p Fv 1629 4096 a(dist\() p Ft(E) p 1861 4096 a(;) p Fs 1898 4096 a(f) p Fv(\(2) p Ft(`) p Fr 2049 4108 a(0) p Fs 2104 4096 a(\006) p Fv 2187 4096 a(1\)) p Ft(B) p Fs 2328 4096 a(g) p Fv(\)) p Ft(;) p Fv 609 4272 a(and) p 772 4272 a(w) n(e) p 895 4272 a(denote) p 1163 4272 a(the) p Ft 1307 4272 a(j) p Fv 1346 4272 a(-th) p 1481 4272 a(eigen) n(v) p 1703 4272 a(alue) p 1879 4272 a(of) p Ft 1974 4272 a(H) p Fp 2043 4284 a(L;n) p Fv 2182 4272 a(b) n(y) p Ft 2299 4272 a(e) p Fp 2338 4233 a(j) 2338 4297 y(L;n) p Fv 2448 4272 a(.) p 2511 4272 a(Then) p 2728 4272 a(there) p 2942 4272 a(exists) p Ft 609 4372 a(M) p Fr 690 4384 a(1) p Ft 750 4372 a(>) p Fv 838 4372 a(0) p 907 4372 a(\(indep) r(enden) n(t) p 1410 4372 a(of) p Ft 1504 4372 a(L) p Fv(\)) p 1621 4372 a(suc) n(h) p 1808 4372 a(that) p 1988 4372 a(if) p 2064 4372 a(2) p Ft(\031) s(n=B) t(L) p Fs 2394 4372 a(2) p Fv 2472 4372 a([) p Ft(M) p Fr 2576 4384 a(1) p Ft 2613 4372 a(;) p 2650 4372 a(L) p Fs 2725 4372 a(\000) p Ft 2808 4372 a(M) p Fr 2889 4384 a(1) p Fv 2926 4372 a(]) p 2976 4372 a(then) p Fn 1018 4478 a(\014) 1018 4528 y(\014) p Ft 1045 4548 a(e) p Fp 1084 4509 a(j) 1084 4573 y(L;n) p Fs 1213 4548 a(\000) p Fv 1296 4548 a(\(2) p Ft(j) p Fs 1427 4548 a(\000) p Fv 1510 4548 a(1\)) p Ft(B) p Fn 1651 4478 a(\014) 1651 4528 y(\014) p Ft 1702 4548 a(<) p 1790 4548 a(") p Fv 1994 4548 a(for) p Ft 2121 4548 a(j) p Fv 2183 4548 a(=) p 2271 4548 a(1) p Ft(;) p 2350 4548 a(:) p 2387 4548 a(:) p 2424 4548 a(:) p 2460 4548 a(;) p 2497 4548 a(`) p Fr 2532 4560 a(0) p Fv 2587 4548 a(+) p 2670 4548 a(1) p Ft(:) p Fv 609 4725 a(This) p 799 4725 a(condition) p 1163 4725 a(implies) 666 4936 y(#) p Fs 749 4936 a(f) p Fv 790 4936 a(eigen) n(v) p 1012 4936 a(alues) p 1219 4936 a(of) p Ft 1314 4936 a(H) p Fp 1383 4948 a(L;n) p Fs 1517 4936 a(\024) p Ft 1604 4936 a(E) p Fs 1670 4936 a(g) p Fv 1735 4936 a(=) p Ft 1822 4936 a(`) p Fr 1857 4948 a(0) p Fv 1977 4936 a(if) 2146 4879 y(2) p Ft(\031) s(n) p 2146 4917 142 4 v 2155 4993 a(B) t(L) p Fs 2321 4936 a(2) p Fv 2399 4936 a([) p Ft(M) p Fr 2503 4948 a(1) p Ft 2540 4936 a(;) p 2577 4936 a(L) p Fs 2652 4936 a(\000) p Ft 2735 4936 a(M) p Fr 2816 4948 a(1) p Fv 2853 4936 a(]) p Ft(:) p Fv 2955 4936 a(\(B.1\)) 609 5135 y(On) p 748 5135 a(the) p 891 5135 a(other) p 1108 5135 a(hand,) p 1338 5135 a(there) p 1551 5135 a(exists) p Ft 1780 5135 a(M) p Fr 1861 5147 a(2) p Ft 1921 5135 a(>) p Fv 2008 5135 a(0) p 2077 5135 a(suc) n(h) p 2265 5135 a(that) 855 5346 y(inf) p Ft 970 5346 a(\033) p Fv 1020 5346 a(\() p Ft(H) p Fp 1121 5358 a(L;n) p Fv 1232 5346 a(\)) p Fs 1288 5346 a(\025) p Ft 1375 5346 a(E) p Fv 1460 5346 a(+) p 1543 5346 a(1) p 1751 5346 a(if) 1892 5290 y(2) p Ft(\031) s(n) p 1892 5327 V 1901 5403 a(B) t(L) 2076 5346 y(=) p Fs 2067 5346 a(2) p Fv 2145 5346 a([) p Fs(\000) p Ft(M) p Fr 2314 5358 a(2) p Ft 2350 5346 a(;) p 2387 5346 a(L) p Fv 2462 5346 a(+) p Ft 2545 5346 a(M) p Fr 2626 5358 a(2) p Fv 2663 5346 a(]) p Ft(:) p Fv 2955 5346 a(\(B.2\)) p Fy 1831 5652 a(12) p 90 rotate dyy eop %%Page: 13 13 13 12 bop Fv 609 548 a(W) p 687 548 a(e) p 752 548 a(also) p 919 548 a(note) p 1103 548 a(that) 652 731 y(#) p Fs 735 731 a(f) p Fv(e.v.'s) p 987 731 a(of) p Ft 1082 731 a(H) p Fp 1151 743 a(L;n) p Fs 1284 731 a(\024) p Ft 1372 731 a(E) p Fs 1438 731 a(g) p 1503 731 a(\024) p Fv 1590 731 a(#) p Fn 1673 663 a(\010) p Fv 1722 731 a(e.v.'s) p 1932 731 a(of) p Fs 2045 731 a(\000) p Ft 2128 731 a(@) p Fr 2177 696 a(2) p Fp 2172 751 a(x) p Fv 2232 731 a(+) p Ft 2315 731 a(B) p Fr 2382 696 a(2) p Ft 2420 731 a(x) p Fr 2467 696 a(2) p Fs 2528 731 a(\024) p Ft 2615 731 a(E) p Fn 2681 663 a(\011) p Fv 2753 731 a(=) p Ft 2840 731 a(`) p Fr 2875 743 a(0) p Fv 2955 731 a(\(B.3\)) 609 913 y(for) p 736 913 a(an) n(y) p Ft 893 913 a(n) p Fs 966 913 a(2) p Fq 1044 913 a(Z) p Fv(.) p 1160 913 a(Com) n(bining) p 1582 913 a(these,) p 1817 913 a(w) n(e) p 1940 913 a(compute) p Ft 635 1108 a(N) p Fp 702 1120 a(L) p Fv 752 1108 a(\() p Ft(E) p Fv 850 1108 a(\)) p 905 1108 a(=) p Fn 995 1029 a(X) p Fp 993 1207 a(n) p Fk(2) p Fg(Z) p Fv 1131 1108 a(#) p Fs 1214 1108 a(f) p Fv(eigen) n(v) p 1478 1108 a(alues) p 1685 1108 a(of) p Ft 1780 1108 a(H) p Fp 1849 1120 a(L;n) p Fs 1982 1108 a(\024) p Ft 2070 1108 a(E) p Fs 2136 1108 a(g) p Fv 905 1341 a(=) p Fn 1007 1262 a(X) p Fp 993 1440 a(n) p Fk(2) p Fp(I) p Fh 1108 1448 a(1) p Fv 1154 1341 a(#) p Fs(f\001) p 1302 1341 a(\001) p 1339 1341 a(\001) p 1376 1341 a(g) p Fv 1436 1341 a(+) p Fn 1533 1262 a(X) p Fp 1519 1440 a(n) p Fk(2) p Fp(I) p Fh 1634 1448 a(2) p Fv 1680 1341 a(#) p Fs(f\001) p 1828 1341 a(\001) p 1865 1341 a(\001) p 1902 1341 a(g) p Fv 1962 1341 a(+) p Fn 2058 1262 a(X) p Fp 2045 1440 a(n) p Fk(2) p Fp(I) p Fh 2160 1448 a(3) p Fv 2206 1341 a(#) p Fs(f\001) p 2354 1341 a(\001) p 2391 1341 a(\001) p 2427 1341 a(g) p Fv 2492 1341 a(=) p Ft 2580 1341 a(N) p Fr 2647 1353 a(1) p Fv 2702 1341 a(+) p Ft 2785 1341 a(N) p Fr 2852 1353 a(2) p Fv 2907 1341 a(+) p Ft 2990 1341 a(N) p Fr 3057 1353 a(3) p Ft 3094 1341 a(;) p Fv 609 1605 a(where) p Ft 950 1806 a(I) p Fr 986 1818 a(1) p Fv 1047 1806 a(=) p Fn 1135 1714 a(n) p Ft 1190 1806 a(n) p Fn 1263 1711 a(\014) 1263 1761 y(\014) 1263 1810 y(\014) p Fv 1324 1750 a(2) p Ft(\031) s(n) p 1324 1787 142 4 v 1333 1863 a(B) t(L) p Fs 1498 1806 a(2) p Fv 1577 1806 a([) p Ft(M) p Fr 1681 1818 a(1) p Ft 1718 1806 a(;) p 1755 1806 a(L) p Fs 1830 1806 a(\000) p Ft 1913 1806 a(M) p Fr 1994 1818 a(1) p Fv 2030 1806 a(]) p Fn 2053 1714 a(o) p Ft 2109 1806 a(;) 950 2006 y(I) p Fr 986 2018 a(2) p Fv 1047 2006 a(=) p Fn 1135 1914 a(n) p Ft 1190 2006 a(n) p Fn 1263 1911 a(\014) 1263 1960 y(\014) 1263 2010 y(\014) p Fv 1324 1950 a(2) p Ft(\031) s(n) p 1324 1987 V 1333 2063 a(B) t(L) p Fs 1498 2006 a(2) p Fv 1577 2006 a(\() p Fs(\0001) p Ft(;) p Fs 1794 2006 a(\000) p Ft(M) p Fr 1940 2018 a(2) p Fv 1976 2006 a(]) p Fs 2018 2006 a([) p Fv 2091 2006 a([) p Ft(L) p Fv 2189 2006 a(+) p Ft 2272 2006 a(M) p Fr 2353 2018 a(2) p Ft 2390 2006 a(;) p Fs 2427 2006 a(1) p Fv(\)) p Fn 2542 1914 a(o) p Ft 2598 2006 a(;) 950 2206 y(I) p Fr 986 2218 a(3) p Fv 1047 2206 a(=) p Fn 1135 2114 a(n) p Ft 1190 2206 a(n) p Fn 1263 2110 a(\014) 1263 2160 y(\014) 1263 2210 y(\014) p Fv 1324 2150 a(2) p Ft(\031) s(n) p 1324 2187 V 1333 2263 a(B) t(L) p Fs 1498 2206 a(2) p Fv 1577 2206 a(\() p Fs(\000) p Ft(M) p Fr 1755 2218 a(2) p Ft 1792 2206 a(;) p 1829 2206 a(M) p Fr 1910 2218 a(1) p Fv 1946 2206 a(\)) p Fs 1997 2206 a([) p Fv 2071 2206 a(\() p Ft(L) p Fs 2178 2206 a(\000) p Ft 2261 2206 a(M) p Fr 2342 2218 a(1) p Ft 2379 2206 a(;) p 2416 2206 a(L) p Fv 2491 2206 a(+) p Ft 2574 2206 a(M) p Fr 2655 2218 a(2) p Fv 2691 2206 a(\)) p Fn 2723 2114 a(o) p Ft 2779 2206 a(:) p Fv 609 2548 a(Since) 962 2764 y(#) p Ft(I) p Fr 1067 2776 a(1) p Fv 1128 2764 a(=) p Fn 1215 2671 a(\020) p Fv 1291 2707 a(2) p Ft(\031) p 1275 2744 124 4 v 1275 2821 a(B) t(L) p Fn 1408 2671 a(\021) p Fk 1458 2689 a(\000) p Fr(1) p Fv 1547 2764 a(\() p Ft(L) p Fs 1654 2764 a(\000) p Fv 1737 2764 a(2) p Ft(M) p Fr 1860 2776 a(1) p Fv 1897 2764 a(\)) p 1947 2764 a(+) p Fm 2030 2764 a(O) p Fv(\(1\)) p 2218 2764 a(=) p Ft 2316 2707 a(B) t(L) p Fr 2440 2677 a(2) p 2316 2744 161 4 v Fv 2350 2821 a(2) p Ft(\031) p Fv 2505 2764 a(+) p Fm 2588 2764 a(O) p Fv(\() p Ft(L) p Fv(\)) p Ft(;) p Fv 609 2974 a(w) n(e) p 731 2974 a(learn) p 939 2974 a(from) p 1135 2974 a(\(B.1\)) p 1351 2974 a(that) p Ft 1490 3210 a(N) p Fr 1557 3222 a(1) p Fv 1617 3210 a(=) p Ft 1705 3210 a(`) p Fr 1740 3222 a(0) p Ft 1787 3154 a(B) t(L) p Fr 1911 3124 a(2) p 1787 3191 V Fv 1821 3267 a(2) p Ft(\031) p Fv 1976 3210 a(+) p Fm 2059 3210 a(O) p Fv(\() p Ft(L) p Fv(\)) p Ft(:) p Fv 609 3420 a(By) p 739 3420 a(\(B.2\),) p 978 3420 a(it) p 1061 3420 a(is) p 1145 3420 a(easy) p 1327 3420 a(to) p 1429 3420 a(see) p 1563 3420 a(that) p Ft 1743 3420 a(N) p Fr 1810 3432 a(2) p Fv 1870 3420 a(=) p 1958 3420 a(0.) p 2059 3420 a(A) n(t) p 2179 3420 a(last,) p 2359 3420 a(using) p 2577 3420 a(\(B.3\),) p 2815 3420 a(w) n(e) p 2937 3420 a(ha) n(v) n(e) p Ft 808 3645 a(N) p Fr 875 3657 a(3) p Fs 935 3645 a(\024) p Ft 1022 3645 a(`) p Fr 1057 3657 a(0) p Fs 1112 3645 a(\002) p Fv 1196 3645 a(#) p Ft(I) p Fr 1301 3657 a(3) p Fv 1362 3645 a(=) p Ft 1449 3645 a(`) p Fr 1484 3657 a(0) p Fn 1521 3553 a(n\020) p Fv 1652 3589 a(2) p Ft(\031) p 1636 3626 124 4 v 1636 3702 a(B) t(L) p Fn 1769 3553 a(\021) p Fk 1819 3570 a(\000) p Fr(1) p Fv 1908 3645 a(\() p Ft(M) p Fr 2021 3657 a(1) p Fv 2077 3645 a(+) p Ft 2160 3645 a(M) p Fr 2241 3657 a(2) p Fv 2278 3645 a(\)) p 2328 3645 a(+) p Fm 2411 3645 a(O) p Fv(\(1\)) p Fn 2576 3553 a(o) p Fv 2655 3645 a(=) p Fm 2742 3645 a(O) p Fv(\() p Ft(L) p Fv(\)) p Ft(:) p Fv 609 3851 a(Th) n(us) p 819 3851 a(w) n(e) p 942 3851 a(obtain) p Ft 1407 4055 a(N) p Fp 1474 4067 a(L) p Fv 1523 4055 a(\() p Ft(E) p Fv 1621 4055 a(\)) p 1677 4055 a(=) p Ft 1764 4055 a(L) p Fr 1821 4021 a(2) p Ft 1880 3999 a(B) p 1868 4036 92 4 v Fv 1868 4112 a(2) p Ft(\031) 1970 4055 y(`) p Fr 2005 4067 a(0) p Fv 2060 4055 a(+) p Fm 2143 4055 a(O) p Fv(\() p Ft(L) p Fv(\)) p Ft(;) p Fv 609 4261 a(and) p 771 4261 a(this) p 932 4261 a(completes) p 1316 4261 a(the) p 1459 4261 a(pro) r(of.) p 3083 4261 4 57 v 3087 4208 50 4 v 3087 4261 V 3136 4261 4 57 v Fl 609 4435 a(R) l(emark) p 917 4435 a(.) p Fv 984 4435 a(The) p 1155 4435 a(IDS) p 1322 4435 a(for) p 1449 4435 a(the) p 1592 4435 a(free) p 1751 4435 a(Sc) n(hr\177) p 1911 4435 a(odinger) p 2206 4435 a(op) r(erator) p 2541 4435 a(is) p 2624 4435 a(giv) n(en) p 2841 4435 a(b) n(y) p Ft 1437 4656 a(k) p Fr 1480 4668 a(0) p Fv 1518 4656 a(\() p Ft(E) p Fv 1616 4656 a(\)) p 1672 4656 a(=) p Ft 1782 4600 a(E) p 1769 4637 92 4 v Fv 1769 4713 a(4) p Ft(\031) 1871 4656 y(;) p 2074 4656 a(E) p 2163 4656 a(>) p Fv 2251 4656 a(0) p Ft(:) p Fv 609 4866 a(If) p 695 4866 a(w) n(e) p 820 4866 a(compare) p Ft 1155 4866 a(k) p Fv 1201 4866 a(\() p Ft(E) p Fv 1299 4866 a(\)) p 1362 4866 a(with) p Ft 1554 4866 a(k) p Fr 1597 4878 a(0) p Fv 1634 4866 a(\() p Ft(E) p Fv 1732 4866 a(\),) p 1819 4866 a(w) n(e) p 1944 4866 a(\014nd) p 2113 4866 a(that) p Ft 2296 4866 a(k) p Fv 2342 4866 a(\() p Ft(E) p Fv 2440 4866 a(\)) p Fs 2500 4866 a(\025) p Ft 2592 4866 a(k) p Fr 2635 4878 a(0) p Fv 2672 4866 a(\() p Ft(E) p Fv 2770 4866 a(\)) p 2833 4866 a(do) r(es) p 3023 4866 a(not) 609 4966 y(alw) n(a) n(ys) p 885 4966 a(hold.) p 1131 4966 a(Namely) p 1401 4966 a(,) p 1464 4966 a(the) p 1616 4966 a(IDS) p 1793 4966 a(is) p Fl 1887 4966 a(not) p Fv 2049 4966 a(decreasing) p 2461 4966 a(function) p 2796 4966 a(of) p Ft 2900 4966 a(B) p Fv 3005 4966 a(\(see) 609 5066 y(Figure) p 871 5066 a(1\).) p Fy 1831 5652 a(13) p 90 rotate dyy eop %%Page: 14 14 14 13 bop 813 1778 a @beginspecial 85 @llx 501 @lly 526 @urx 726 @ury 2551 @rwi @clip @setspecial %%BeginDocument: ids4.eps %!PS-Adobe-3.0 EPSF-3.0 %%BoundingBox: 85 501 526 726 %%Title: (IDS \(DR\)) %%Creator: (\203N\203\211\203\212\203X\203\217\201[\203N\203X: AdobePS 8.6 \(219\)) %%CreationDate: (1:15 PM 2000\224N 3\214\216 14\223\372 \211\316\227j\223\372) %%For: (Shu Nakamura) %%Pages: 1 %%DocumentFonts: Helvetica Helvetica-Oblique Symbol %%DocumentNeededResources: font Helvetica Helvetica-Oblique Symbol %%DocumentSuppliedResources: %%DocumentData: Clean7Bit %%PageOrder: Ascend %%Orientation: Portrait %%DocumentMedia: Default 595 842 0 () () %RBINumCopies: 1 %RBINupNess: 1 1 %ADO_ImageableArea: 27 31 571 811 %%EndComments %%BeginDefaults %%ViewingOrientation: 1 0 0 1 %%EndDefaults userdict/dscInfo 5 dict dup begin /Title(IDS \(DR\))def /Creator(\203N\203\211\203\212\203X\203\217\201[\203N\203X: AdobePS 8.6 \(219\))def /CreationDate(1:15 PM 2000\224N 3\214\216 14\223\372 \211\316\227j\223\372)def /For(Shu Nakamura)def /Pages 1 def end put %%BeginProlog /md 204 dict def md begin/currentpacking where {pop /sc_oldpacking currentpacking def true setpacking}if %%BeginFile: lw8_feature-1.01 %%Copyright: Copyright 1990-1998 Adobe Systems Incorporated and Apple Computer Incorporated. All Rights Reserved. /bd{bind def}bind def /ld{load def}bd /xs{exch store}bd /Z{0 def}bd /T true def /F false def /level2 /languagelevel where { pop languagelevel 2 ge }{ F }ifelse def /odictstk Z /oopstk Z /fcl { count oopstk sub dup 0 gt { {pop}repeat }{ pop }ifelse countdictstack odictstk sub dup 0 gt { {end}repeat }{ pop }ifelse }bd /sfcl2 { /odictstk countdictstack store count/oopstk xs }bd /efcl2 { stopped{$error/newerror F put}if fcl }bd /noload Z /startnoload { {/noload save store}if }bd /endnoload { {noload restore}if }bd /setcopies{ level2 { 1 dict begin/NumCopies exch def currentdict end setpagedevice }{ userdict/#copies 3 -1 roll put }ifelse }def level2 startnoload /ststpgdev{}def /dopgdev{}def /stpgdev{}def /buf Z /didstop T def /sfcl { /didstop T store /odictstk countdictstack store count/oopstk xs currentfile cvx stopped { $error/newerror F put didstop { save/didstop xs /buf vmstatus exch sub exch pop dup 0 lt{pop 0}if dup 64000 gt{pop 64000}if string store { currentfile buf readline { (}efcl)eq{exit}if }{ /UnexpectedEOF errordict/rangecheck get exec }ifelse }loop didstop restore }if }if fcl }bd /efcl { /didstop F store exec stop }bd level2 endnoload level2 not startnoload /setpagedevice where{pop/realstpgdev/setpagedevice ld}if /SC_topddict Z /SC_spdict Z /$spusrdict F def /dopgdev { userdict/setpagedevice undef $spusrdict { userdict/setpagedevice/realstpgdev load put /$spusrdict F store }if SC_topddict realstpgdev }bd /stpgdev { SC_topddict dup 3 -1 roll { SC_spdict 2 index known { SC_spdict 2 index get dup 3 -1 roll { put dup }forall pop put dup }{ put dup }ifelse }forall pop pop }bd /ststpgdev { /setpagedevice where { userdict eq { /$spusrdict T store }if }if userdict/setpagedevice/stpgdev load put /SC_topddict 0 dict store /SC_spdict 3 dict begin /InputAttributes 0 dict def /Policies 0 dict def /OutputAttributes 0 dict def currentdict end store }def /sfcl/sfcl2 ld /efcl/efcl2 ld level2 not endnoload %%EndFile %%BeginFile: lw8_basic-3.0 /xdf{exch def}bd /:L/lineto /lw/setlinewidth /:M/moveto /rl/rlineto /rm/rmoveto /:C/curveto /:T/translate /:K/closepath /:mf/makefont /gS/gsave /gR/grestore /np/newpath 12{ld}repeat /framewidth -1 def /QDframwid -1 def /numframes Z /mTS matrix def /$m matrix def /av 86 def /por T def /normland F def /psb-nosave{}def /pse-nosave{}def /us Z /psb{/us save store}bd /pse{us restore}bd /level3 /languagelevel where { pop languagelevel 3 ge }{ F }ifelse def level2 startnoload /setjob { statusdict/jobname 3 -1 roll put }bd /devg/DeviceGray def /devr/DeviceRGB def /devc/DeviceCMYK def level2 endnoload level2 not startnoload /setjob { 1 dict begin/JobName xdf currentdict end setuserparams }bd /devg[/DeviceGray]def /devr[/DeviceRGB]def /devc[/DeviceCMYK]def level2 not endnoload /pm Z /mT Z /sD Z /initializepage { mT $m currentmatrix mTS concatmatrix pop /pm save store mT concat }bd /endp { pm restore }bd /adjRect { dup 2 mul 6 2 roll 4 index sub exch 5 -1 roll sub exch 4 2 roll 4 index add exch 5 -1 roll add exch 4 2 roll }bd /frame1up { gS mTS setmatrix QDframwid lw /setstrokeadjust where{pop T setstrokeadjust}if clippath pathbbox 2 index sub exch 3 index sub exch currentlinewidth framewidth mul adjRect numframes dup 0 lt{pop 0}if { 4 copy rS currentlinewidth framewidth mul 4 mul adjRect }repeat pop pop pop pop gR }bd /$c devr def /rectclip where { pop/rC/rectclip ld }{ /rC { np 4 2 roll :M 1 index 0 rl 0 exch rl neg 0 rl :K clip np }bd }ifelse /rectfill where { pop/rF/rectfill ld }{ /rF { gS np 4 2 roll :M 1 index 0 rl 0 exch rl neg 0 rl fill gR }bd }ifelse /rectstroke where { pop/rS/rectstroke ld }{ /rS { gS np 4 2 roll :M 1 index 0 rl 0 exch rl neg 0 rl :K stroke gR }bd }ifelse %%EndFile level3 startnoload %%BeginFile: lw8_safeclipL12-1.0 /rectclip where { pop/rCa/rectclip ld }{ /rCa { np 0 1 index length 4 idiv { 2 copy 4 getinterval aload pop 4 2 roll :M 1 index 0 rl 0 exch rl neg 0 rl :K 4 add }repeat clip np pop pop }bd }ifelse /savedstack Z /subsavedstack Z /execstring Z /saferCa { /execstring xs /odictstk countdictstack store /oopstk 0 store count 0 ne { savedstack 0 eq{ count 100 gt{count}{100}ifelse array/savedstack xs }{ count savedstack length gt { count array/savedstack xs }if }ifelse count savedstack 0 3 -1 roll getinterval astore/subsavedstack xs }{ /subsavedstack 0 store }ifelse execstring cvx stopped { $error/newerror F put newpath }if fcl subsavedstack 0 ne{ subsavedstack aload pop }if }bd %%EndFile level3 endnoload level3 not startnoload %%BeginFile: lw8_safeclipL3-1.0 /saferCa/pop ld %%EndFile level3 not endnoload %%BeginFile: lw8_level1_colorspace-2.0 /G/setgray ld /:F1/setgray ld /:F/setrgbcolor ld /:F4/setcmykcolor where { pop /setcmykcolor ld }{ { 3 { dup 3 -1 roll add dup 1 gt{pop 1}if 1 exch sub 4 1 roll }repeat pop setrgbcolor }bd }ifelse /:Fx { counttomark {0{G}0{:F}{:F4}} exch get exec pop }bd /$cs Z /:rg{devr :ss}bd /:sc{$cs :ss}bd /:dc { dup type/arraytype eq{0 get}if dup/DeviceCMYK eq { pop devc }{ /DeviceGray eq { devg }{ devr }ifelse }ifelse /$cs xdf }bd /:sgl{}def /:dr{}bd /:fCRD{pop}bd /:ckcs{}bd /:ss{/$c xdf}bd %%EndFile %%BeginFile: lw8_uniform_graphics-2.0 /@a { np :M 0 rl :L 0 exch rl 0 rl :L fill }bd /@b { np :M 0 rl 0 exch rl :L 0 rl 0 exch rl fill }bd /@c { moveto 0 rlineto stroke }bd /@w { moveto 0 exch rlineto stroke }bd /arct where { pop }{ /arct { arcto pop pop pop pop }bd }ifelse /x1 Z /x2 Z /y1 Z /y2 Z /rad Z /@q { /rad xs /y2 xs /x2 xs /y1 xs /x1 xs np x2 x1 add 2 div y1 :M x2 y1 x2 y2 rad arct x2 y2 x1 y2 rad arct x1 y2 x1 y1 rad arct x1 y1 x2 y1 rad arct fill }bd /@s { /rad xs /y2 xs /x2 xs /y1 xs /x1 xs np x2 x1 add 2 div y1 :M x2 y1 x2 y2 rad arct x2 y2 x1 y2 rad arct x1 y2 x1 y1 rad arct x1 y1 x2 y1 rad arct :K stroke }bd /@i { np 0 360 arc fill }bd /@j { gS np :T scale 0 0 .5 0 360 arc fill gR }bd /@e { np 0 360 arc :K stroke }bd /@f { np $m currentmatrix pop :T scale 0 0 .5 0 360 arc :K $m setmatrix stroke }bd /@k { gS np :T 0 0 :M 0 0 5 2 roll arc fill gR }bd /@l { gS np :T 0 0 :M 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p Fw 1124 2709 a(4) p Fv(,) p 1222 2709 a(833{870) p 1541 2709 a(\(1993\).) 651 2842 y([2]) p 784 2842 a(Carmona,) p 1221 2842 a(R.,) p 1410 2842 a(Lacroix,) p 1787 2842 a(J.:) p 1999 2842 a(Sp) r(ectral) p 2368 2842 a(Theory) p 2700 2842 a(of) p 2838 2842 a(Random) 784 2942 y(Sc) n(hr\177) p 944 2942 a(odinger) p 1239 2942 a(Op) r(erators.) p 1652 2942 a(Birkh\177) p 1857 2942 a(auser) p 2074 2942 a(1990.) 651 3075 y([3]) p 784 3075 a(Cycon,) p 1065 3075 a(H.) p 1180 3075 a(L.,) p 1307 3075 a(F) p 1354 3075 a(ro) r(ese,) p 1590 3075 a(R.) p 1704 3075 a(G.,) p 1845 3075 a(Kirsc) n(h,) p 2131 3075 a(W.,) p 2292 3075 a(Simon,) p 2571 3075 a(B.:) p 2716 3075 a(Sc) n(hr\177) p 2876 3075 a(odinger) 784 3174 y(Op) r(erators,) p 1193 3174 a(\(with) p 1410 3174 a(Applications) p 1887 3174 a(to) p 1984 3174 a(Quan) n(tum) p 2350 3174 a(Mec) n(hanics) p 2747 3174 a(and) p 2903 3174 a(Global) 784 3274 y(Geometry\),) p 1225 3274 a(Springer) p 1558 3274 a(V) p 1613 3274 a(erlag) p 1816 3274 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2626 3839 27 4 v 2652 3839 a(arc:99-420) p Fv(\).) 784 3938 y(T) p 837 3938 a(o) p 906 3938 a(app) r(ear) p 1181 3938 a(in) p 1278 3938 a(Ann.) p 1483 3938 a(Henri) p 1711 3938 a(P) n(oincar) n (\023) p 1990 3938 a(e.) 651 4071 y([6]) p 784 4071 a(Nak) p 927 4071 a(am) n(ura,) p 1204 4071 a(S.:) p 1331 4071 a(Lifshitz) p 1628 4071 a(tail) p 1772 4071 a(for) p 1896 4071 a(Sc) n(hr\177) p 2056 4071 a(odinger) p 2348 4071 a(op) r(erator) p 2680 4071 a(with) p 2866 4071 a(random) 784 4171 y(magnetic) p 1139 4171 a(\014led.) p 1342 4171 a(Preprin) n(t) p 1673 4171 a(1999) p 1867 4171 a(Dec.) p 2055 4171 a(\() p Fa(mp-arc) p Fv 2376 4171 a(99-488\).) 651 4303 y([7]) p 784 4303 a(Simon,) p 1073 4303 a(B.:) p 1234 4303 a(T) p 1287 4303 a(race) p 1472 4303 a(Ideals) p 1719 4303 a(and) p 1890 4303 a(their) p 2099 4303 a(Applications.) p 2614 4303 a(London) p 2924 4303 a(Math.) 784 4403 y(So) r(c.) p 961 4403 a(Lecture) p 1263 4403 a(Note) p 1463 4403 a(Ser.) p Fw 1629 4403 a(35) p Fv(,) p 1776 4403 a(Cam) n(brigde) p 2198 4403 a(Univ.) p 2424 4403 a(Press,) p 2666 4403 a(1979.) 651 4536 y([8]) p 784 4536 a(Stein,) p 1032 4536 a(E.:) p 1193 4536 a(Singular) p 1531 4536 a(In) n(tegrals) p 1884 4536 a(and) p 2056 4536 a(Di\013eren) n(tiabilit) n(y) p 2665 4536 a(Prop) r(erties) p 3077 4536 a(of) 784 4636 y(F) p 831 4636 a(unctions.) p 1187 4636 a(Princeton) p 1566 4636 a(Math.) p 1813 4636 a(Ser.) p Fw 1979 4636 a(30) p Fv(,) p 2126 4636 a(Princeton) p 2505 4636 a(Univ.) p 2731 4636 a(Press) p 2950 4636 a(1970.) 651 4768 y([9]) p 784 4768 a(Ueki,) p 1008 4768 a(N.:) p 1166 4768 a(Simple) p 1444 4768 a(examples) p 1808 4768 a(of) p 1909 4768 a(Lifshitz) p 2214 4768 a(tails) p 2401 4768 a(in) p 2504 4768 a(Gaussian) p 2866 4768 a(random) 784 4868 y(magnetic) p 1139 4868 a(\014elds.) p 1375 4868 a(T) p 1428 4868 a(o) p 1497 4868 a(app) r(ear) p 1772 4868 a(in) p 1868 4868 a(Ann.) p 2074 4868 a(Henri) p 2302 4868 a(P) n(oincar) n(\023) p 2581 4868 a(e) p 2643 4868 a(\(2000\).) 609 5001 y([10]) p 784 5001 a(Y) p 839 5001 a(afaev,) p 1077 5001 a(D.) p 1188 5001 a(R.:) p 1330 5001 a(Mathematical) p 1856 5001 a(Scattering) p 2249 5001 a(Theory) p 2504 5001 a(.) p 2551 5001 a(American) p 2924 5001 a(Math.) 784 5100 y(So) r(c.,) p 984 5100 a(Pro) n(vidence,) p 1433 5100 a(RI,) p 1575 5100 a(1992.) p Fy 1831 5652 a(14) p 90 rotate dyy eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF