1219:
250:
702:
1742:
440:
541:
285:
354:
136:
492:
1844:
1108:
128:
570:
390:
1477:
771:
460:
590:
934:
1482:
1255:
1829:
1061:
916:
892:
1722:
1575:
1373:
1570:
784:
1727:
873:
764:
302:
The need for both a positive and negative spectrum in the definition is why the spectral asymmetry usually occurs in the study of
1545:
1143:
707:
This sum is formally divergent, and the divergences must be accounted for and removed using standard regularization techniques.
1514:
788:
1504:
1499:
1492:
1428:
1312:
1737:
1248:
939:
995:
363:
is an integer, ranging over all positive and negative values. One may show in a straightforward manner that in this case
1363:
1222:
944:
929:
757:
1874:
1348:
959:
1794:
49:
1849:
1747:
1627:
1204:
964:
1158:
1082:
1343:
1199:
580:
Related to the spectral asymmetry is the vacuum expectation value of the energy associated with the operator, the
1854:
1717:
1550:
1535:
1307:
1015:
1436:
949:
1895:
1446:
1317:
1241:
1051:
852:
292:
61:
924:
395:
1809:
1784:
1602:
1591:
1302:
1148:
497:
296:
258:
245:{\displaystyle B=\lim _{t\to 0}{\frac {1}{2}}\sum _{n}\operatorname {sgn}(\omega _{n})\exp(-t|\omega _{n}|)}
1660:
1650:
1645:
320:
1353:
1179:
1123:
1087:
37:
1405:
465:
130:, an equal number of which are positive and negative, the spectral asymmetry may be defined as the sum
1819:
1798:
1712:
1597:
1560:
1162:
69:
1622:
1358:
1128:
1066:
780:
1900:
1752:
1681:
1612:
1418:
1153:
1020:
106:
546:
366:
1859:
1834:
1519:
1441:
1133:
41:
1864:
1565:
1413:
1368:
1292:
1138:
1056:
1025:
1005:
990:
985:
980:
817:
77:
53:
45:
1839:
1824:
1732:
1695:
1691:
1655:
1617:
1555:
1540:
1509:
1451:
1410:
1397:
1322:
1264:
1000:
954:
902:
897:
868:
749:
827:
697:{\displaystyle E=\lim _{t\to 0}{\frac {1}{2}}\sum _{n}|\omega _{n}|\exp(-t|\omega _{n}|)}
1789:
1768:
1686:
1676:
1487:
1394:
1327:
1287:
1189:
1041:
842:
581:
445:
303:
89:
57:
1889:
1194:
1118:
847:
832:
822:
288:
65:
1607:
1461:
1402:
1184:
837:
807:
85:
1233:
1804:
1389:
1113:
1103:
1010:
812:
52:. In physics, it has numerous applications, typically resulting in a fractional
17:
1297:
1046:
886:
882:
878:
101:
33:
1282:
1268:
1869:
1814:
29:
81:
21:
73:
40:. In mathematics, the spectral asymmetry arises in the study of
1237:
753:
729:
Two-phase models of baryons and the chiral
Casimir effect
593:
549:
500:
468:
448:
398:
369:
323:
261:
139:
109:
314:
As an example, consider an operator with a spectrum
1777:
1761:
1705:
1669:
1638:
1584:
1528:
1470:
1427:
1382:
1336:
1275:
1172:
1096:
1075:
1034:
973:
915:
861:
796:
1845:Spectral theory of ordinary differential equations
1109:Spectral theory of ordinary differential equations
696:
564:
535:
486:
454:
434:
384:
348:
279:
244:
122:
1743:Schröder–Bernstein theorems for operator algebras
601:
147:
727:Linas Vepstas, A.D. Jackson, A.S. Goldhaber,
1249:
765:
8:
718:Spectral asymmetry and Riemannian geometry I
28:is the asymmetry in the distribution of the
1256:
1242:
1234:
800:
772:
758:
750:
88:, and can be understood as describing the
68:is given by the spectral asymmetry of the
56:due to the asymmetry of the spectrum of a
686:
680:
671:
651:
645:
636:
630:
616:
604:
592:
548:
519:
499:
467:
447:
397:
368:
328:
322:
260:
234:
228:
219:
195:
176:
162:
150:
138:
114:
108:
72:. The spectral asymmetry of the confined
1062:Group algebra of a locally compact group
572:is therefore a periodic sawtooth curve.
435:{\displaystyle B(\alpha )=B(\alpha +m)}
76:fields is an important property of the
536:{\displaystyle B(\alpha )=1/2-\alpha }
280:{\displaystyle \operatorname {sgn}(x)}
1576:Spectral theory of normal C*-algebras
1374:Spectral theory of normal C*-algebras
349:{\displaystyle \omega _{n}=n+\alpha }
48:, and is given a deep meaning by the
7:
1571:Spectral theory of compact operators
716:MF Atiyah, VK Patodi and IM Singer,
1723:Cohen–Hewitt factorization theorem
14:
1728:Extensions of symmetric operators
487:{\displaystyle 0<\alpha <1}
1546:Positive operator-valued measure
1218:
1217:
1144:Topological quantum field theory
1830:Rayleigh–Faber–Krahn inequality
691:
687:
672:
662:
652:
637:
608:
559:
553:
510:
504:
429:
417:
408:
402:
379:
373:
274:
268:
239:
235:
220:
210:
201:
188:
154:
1:
1738:Limiting absorption principle
940:Uniform boundedness principle
738:Linas Vepstas, A.D. Jackson,
1364:Singular value decomposition
1795:Hearing the shape of a drum
1478:Decomposition of a spectrum
123:{\displaystyle \omega _{n}}
50:Atiyah-Singer index theorem
1917:
1383:Special Elements/Operators
1083:Invariant subspace problem
720:, Proc. Camb. Phil. Soc.,
565:{\displaystyle B(\alpha )}
385:{\displaystyle B(\alpha )}
1855:Superstrong approximation
1718:Banach algebra cohomology
1551:Projection-valued measure
1536:Borel functional calculus
1308:Projection-valued measure
1213:
803:
740:Justifying the Chiral Bag
1447:Spectrum of a C*-algebra
1318:Spectrum of a C*-algebra
1052:Spectrum of a C*-algebra
62:vacuum expectation value
1875:Wiener–Khinchin theorem
1810:Kuznetsov trace formula
1785:Almost Mathieu operator
1603:Banach function algebra
1592:Amenable Banach algebra
1349:Gelfand–Naimark theorem
1303:Noncommutative topology
1149:Noncommutative geometry
297:zeta function regulator
100:Given an operator with
1850:Sturm–Liouville theory
1748:Sherman–Takeda theorem
1628:Tomita–Takesaki theory
1403:Hermitian/Self-adjoint
1354:Gelfand representation
1205:Tomita–Takesaki theory
1180:Approximation property
1124:Calculus of variations
698:
566:
537:
488:
456:
436:
386:
350:
281:
246:
124:
1344:Gelfand–Mazur theorem
1200:Banach–Mazur distance
1163:Generalized functions
699:
567:
538:
489:
457:
437:
387:
351:
282:
247:
125:
84:, it is known as the
1820:Proto-value function
1799:Dirichlet eigenvalue
1713:Abstract index group
1598:Approximate identity
1561:Rigged Hilbert space
1437:Krein–Rutman theorem
1283:Involution/*-algebra
945:Kakutani fixed-point
930:Riesz representation
591:
584:, which is given by
547:
498:
466:
446:
396:
367:
321:
259:
137:
107:
70:Hamiltonian operator
1623:Von Neumann algebra
1359:Polar decomposition
1129:Functional calculus
1088:Mahler's conjecture
1067:Von Neumann algebra
781:Functional analysis
742:, Physics Reports,
60:. For example, the
1753:Unbounded operator
1682:Essential spectrum
1661:Schur–Horn theorem
1651:Bauer–Fike theorem
1646:Alon–Boppana bound
1639:Finite-Dimensional
1613:Nuclear C*-algebra
1457:Spectral asymmetry
1154:Riemann hypothesis
853:Topological vector
746:(1990) p. 109-143.
735:(1984) p. 280-284.
731:, Physics Letters
694:
635:
615:
562:
533:
484:
452:
432:
382:
346:
277:
242:
181:
161:
120:
42:elliptic operators
26:spectral asymmetry
1883:
1882:
1860:Transfer operator
1835:Spectral geometry
1520:Spectral abscissa
1500:Approximate point
1442:Normal eigenvalue
1231:
1230:
1134:Integral operator
911:
910:
626:
624:
600:
455:{\displaystyle m}
172:
170:
146:
46:compact manifolds
1908:
1865:Transform theory
1585:Special algebras
1566:Spectral theorem
1529:Spectral Theorem
1369:Spectral theorem
1258:
1251:
1244:
1235:
1221:
1220:
1139:Jones polynomial
1057:Operator algebra
801:
774:
767:
760:
751:
703:
701:
700:
695:
690:
685:
684:
675:
655:
650:
649:
640:
634:
625:
617:
614:
571:
569:
568:
563:
542:
540:
539:
534:
523:
493:
491:
490:
485:
461:
459:
458:
453:
442:for any integer
441:
439:
438:
433:
391:
389:
388:
383:
355:
353:
352:
347:
333:
332:
286:
284:
283:
278:
251:
249:
248:
243:
238:
233:
232:
223:
200:
199:
180:
171:
163:
160:
129:
127:
126:
121:
119:
118:
78:chiral bag model
1916:
1915:
1911:
1910:
1909:
1907:
1906:
1905:
1896:Spectral theory
1886:
1885:
1884:
1879:
1840:Spectral method
1825:Ramanujan graph
1773:
1757:
1733:Fredholm theory
1701:
1696:Shilov boundary
1692:Structure space
1670:Generalizations
1665:
1656:Numerical range
1634:
1618:Uniform algebra
1580:
1556:Riesz projector
1541:Min-max theorem
1524:
1510:Direct integral
1466:
1452:Spectral radius
1423:
1378:
1332:
1323:Spectral radius
1271:
1265:Spectral theory
1262:
1232:
1227:
1209:
1173:Advanced topics
1168:
1092:
1071:
1030:
996:Hilbert–Schmidt
969:
960:Gelfand–Naimark
907:
857:
792:
778:
713:
676:
641:
589:
588:
578:
545:
544:
543:. The graph of
496:
495:
464:
463:
462:, and that for
444:
443:
394:
393:
365:
364:
324:
319:
318:
312:
304:Dirac operators
299:, may be used.
257:
256:
224:
191:
135:
134:
110:
105:
104:
98:
12:
11:
5:
1914:
1912:
1904:
1903:
1898:
1888:
1887:
1881:
1880:
1878:
1877:
1872:
1867:
1862:
1857:
1852:
1847:
1842:
1837:
1832:
1827:
1822:
1817:
1812:
1807:
1802:
1792:
1790:Corona theorem
1787:
1781:
1779:
1775:
1774:
1772:
1771:
1769:Wiener algebra
1765:
1763:
1759:
1758:
1756:
1755:
1750:
1745:
1740:
1735:
1730:
1725:
1720:
1715:
1709:
1707:
1703:
1702:
1700:
1699:
1689:
1687:Pseudospectrum
1684:
1679:
1677:Dirac spectrum
1673:
1671:
1667:
1666:
1664:
1663:
1658:
1653:
1648:
1642:
1640:
1636:
1635:
1633:
1632:
1631:
1630:
1620:
1615:
1610:
1605:
1600:
1594:
1588:
1586:
1582:
1581:
1579:
1578:
1573:
1568:
1563:
1558:
1553:
1548:
1543:
1538:
1532:
1530:
1526:
1525:
1523:
1522:
1517:
1512:
1507:
1502:
1497:
1496:
1495:
1490:
1485:
1474:
1472:
1468:
1467:
1465:
1464:
1459:
1454:
1449:
1444:
1439:
1433:
1431:
1425:
1424:
1422:
1421:
1416:
1408:
1400:
1392:
1386:
1384:
1380:
1379:
1377:
1376:
1371:
1366:
1361:
1356:
1351:
1346:
1340:
1338:
1334:
1333:
1331:
1330:
1328:Operator space
1325:
1320:
1315:
1310:
1305:
1300:
1295:
1290:
1288:Banach algebra
1285:
1279:
1277:
1276:Basic concepts
1273:
1272:
1263:
1261:
1260:
1253:
1246:
1238:
1229:
1228:
1226:
1225:
1214:
1211:
1210:
1208:
1207:
1202:
1197:
1192:
1190:Choquet theory
1187:
1182:
1176:
1174:
1170:
1169:
1167:
1166:
1156:
1151:
1146:
1141:
1136:
1131:
1126:
1121:
1116:
1111:
1106:
1100:
1098:
1094:
1093:
1091:
1090:
1085:
1079:
1077:
1073:
1072:
1070:
1069:
1064:
1059:
1054:
1049:
1044:
1042:Banach algebra
1038:
1036:
1032:
1031:
1029:
1028:
1023:
1018:
1013:
1008:
1003:
998:
993:
988:
983:
977:
975:
971:
970:
968:
967:
965:Banach–Alaoglu
962:
957:
952:
947:
942:
937:
932:
927:
921:
919:
913:
912:
909:
908:
906:
905:
900:
895:
893:Locally convex
890:
876:
871:
865:
863:
859:
858:
856:
855:
850:
845:
840:
835:
830:
825:
820:
815:
810:
804:
798:
794:
793:
779:
777:
776:
769:
762:
754:
748:
747:
736:
725:
724:(1975), 43-69.
712:
709:
705:
704:
693:
689:
683:
679:
674:
670:
667:
664:
661:
658:
654:
648:
644:
639:
633:
629:
623:
620:
613:
610:
607:
603:
599:
596:
582:Casimir energy
577:
574:
561:
558:
555:
552:
532:
529:
526:
522:
518:
515:
512:
509:
506:
503:
483:
480:
477:
474:
471:
451:
431:
428:
425:
422:
419:
416:
413:
410:
407:
404:
401:
381:
378:
375:
372:
357:
356:
345:
342:
339:
336:
331:
327:
311:
308:
295:, such as the
276:
273:
270:
267:
264:
253:
252:
241:
237:
231:
227:
222:
218:
215:
212:
209:
206:
203:
198:
194:
190:
187:
184:
179:
175:
169:
166:
159:
156:
153:
149:
145:
142:
117:
113:
97:
94:
92:for fermions.
90:Casimir effect
58:Dirac operator
13:
10:
9:
6:
4:
3:
2:
1913:
1902:
1899:
1897:
1894:
1893:
1891:
1876:
1873:
1871:
1868:
1866:
1863:
1861:
1858:
1856:
1853:
1851:
1848:
1846:
1843:
1841:
1838:
1836:
1833:
1831:
1828:
1826:
1823:
1821:
1818:
1816:
1813:
1811:
1808:
1806:
1803:
1800:
1796:
1793:
1791:
1788:
1786:
1783:
1782:
1780:
1776:
1770:
1767:
1766:
1764:
1760:
1754:
1751:
1749:
1746:
1744:
1741:
1739:
1736:
1734:
1731:
1729:
1726:
1724:
1721:
1719:
1716:
1714:
1711:
1710:
1708:
1706:Miscellaneous
1704:
1697:
1693:
1690:
1688:
1685:
1683:
1680:
1678:
1675:
1674:
1672:
1668:
1662:
1659:
1657:
1654:
1652:
1649:
1647:
1644:
1643:
1641:
1637:
1629:
1626:
1625:
1624:
1621:
1619:
1616:
1614:
1611:
1609:
1606:
1604:
1601:
1599:
1595:
1593:
1590:
1589:
1587:
1583:
1577:
1574:
1572:
1569:
1567:
1564:
1562:
1559:
1557:
1554:
1552:
1549:
1547:
1544:
1542:
1539:
1537:
1534:
1533:
1531:
1527:
1521:
1518:
1516:
1513:
1511:
1508:
1506:
1503:
1501:
1498:
1494:
1491:
1489:
1486:
1484:
1481:
1480:
1479:
1476:
1475:
1473:
1471:Decomposition
1469:
1463:
1460:
1458:
1455:
1453:
1450:
1448:
1445:
1443:
1440:
1438:
1435:
1434:
1432:
1430:
1426:
1420:
1417:
1415:
1412:
1409:
1407:
1404:
1401:
1399:
1396:
1393:
1391:
1388:
1387:
1385:
1381:
1375:
1372:
1370:
1367:
1365:
1362:
1360:
1357:
1355:
1352:
1350:
1347:
1345:
1342:
1341:
1339:
1335:
1329:
1326:
1324:
1321:
1319:
1316:
1314:
1311:
1309:
1306:
1304:
1301:
1299:
1296:
1294:
1291:
1289:
1286:
1284:
1281:
1280:
1278:
1274:
1270:
1266:
1259:
1254:
1252:
1247:
1245:
1240:
1239:
1236:
1224:
1216:
1215:
1212:
1206:
1203:
1201:
1198:
1196:
1195:Weak topology
1193:
1191:
1188:
1186:
1183:
1181:
1178:
1177:
1175:
1171:
1164:
1160:
1157:
1155:
1152:
1150:
1147:
1145:
1142:
1140:
1137:
1135:
1132:
1130:
1127:
1125:
1122:
1120:
1119:Index theorem
1117:
1115:
1112:
1110:
1107:
1105:
1102:
1101:
1099:
1095:
1089:
1086:
1084:
1081:
1080:
1078:
1076:Open problems
1074:
1068:
1065:
1063:
1060:
1058:
1055:
1053:
1050:
1048:
1045:
1043:
1040:
1039:
1037:
1033:
1027:
1024:
1022:
1019:
1017:
1014:
1012:
1009:
1007:
1004:
1002:
999:
997:
994:
992:
989:
987:
984:
982:
979:
978:
976:
972:
966:
963:
961:
958:
956:
953:
951:
948:
946:
943:
941:
938:
936:
933:
931:
928:
926:
923:
922:
920:
918:
914:
904:
901:
899:
896:
894:
891:
888:
884:
880:
877:
875:
872:
870:
867:
866:
864:
860:
854:
851:
849:
846:
844:
841:
839:
836:
834:
831:
829:
826:
824:
821:
819:
816:
814:
811:
809:
806:
805:
802:
799:
795:
790:
786:
782:
775:
770:
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1778:Applications
1608:Disk algebra
1462:Spectral gap
1456:
1337:Main results
1185:Balanced set
1159:Distribution
1097:Applications
950:Krein–Milman
935:Closed graph
743:
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732:
728:
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360:
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99:
86:Witten index
25:
15:
1805:Heat kernel
1505:Compression
1390:Isospectral
1114:Heat kernel
1104:Hardy space
1011:Trace class
925:Hahn–Banach
887:Topological
102:eigenvalues
34:eigenvalues
18:mathematics
1890:Categories
1483:Continuous
1298:C*-algebra
1293:B*-algebra
1047:C*-algebra
862:Properties
711:References
576:Discussion
293:regulators
96:Definition
1901:Asymmetry
1269:-algebras
1021:Unbounded
1016:Transpose
974:Operators
903:Separable
898:Reflexive
883:Algebraic
869:Barrelled
678:ω
666:−
660:
643:ω
628:∑
609:→
557:α
531:α
528:−
508:α
476:α
421:α
406:α
377:α
344:α
326:ω
291:. Other
266:
226:ω
214:−
208:
193:ω
186:
174:∑
155:→
112:ω
1870:Weyl law
1815:Lax pair
1762:Examples
1596:With an
1515:Discrete
1493:Residual
1429:Spectrum
1414:operator
1406:operator
1398:operator
1313:Spectrum
1223:Category
1035:Algebras
917:Theorems
874:Complete
843:Schwartz
789:glossary
494:we have
82:fermions
38:operator
30:spectrum
1411:Unitary
1026:Unitary
1006:Nuclear
991:Compact
986:Bounded
981:Adjoint
955:Min–max
848:Sobolev
833:Nuclear
823:Hilbert
818:Fréchet
783: (
310:Example
287:is the
64:of the
22:physics
1395:Normal
1001:Normal
838:Orlicz
828:Hölder
808:Banach
797:Spaces
785:topics
392:obeys
359:where
255:where
80:. For
54:charge
36:of an
24:, the
1488:Point
813:Besov
74:quark
1419:Unit
1267:and
1161:(or
879:Dual
733:B140
479:<
473:<
20:and
744:187
657:exp
602:lim
263:sgn
205:exp
183:sgn
148:lim
44:on
32:of
16:In
1892::
787:–
722:77
306:.
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1797:(
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1694:(
1257:e
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766:t
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688:|
682:n
673:|
669:t
663:(
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647:n
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632:n
622:2
619:1
612:0
606:t
598:=
595:E
560:)
554:(
551:B
525:2
521:/
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514:=
511:)
505:(
502:B
482:1
470:0
450:m
430:)
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418:(
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412:=
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403:(
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374:(
371:B
361:n
341:+
338:n
335:=
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269:(
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230:n
221:|
217:t
211:(
202:)
197:n
189:(
178:n
168:2
165:1
158:0
152:t
144:=
141:B
116:n
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