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Direct problems attempt to infer the behavior of the eigenvalues of a
Riemannian manifold from knowledge of the geometry. The solutions to direct problems are typified by the
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570:
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47:
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1007:
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110:
28:
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have also been examined. The field concerns itself with two kinds of questions: direct problems and inverse problems.
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105:, which can be used to establish spectral rigidity for a special class of manifolds. However as the example given by
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93:. A refinement of Weyl's asymptotic formula obtained by Pleijel and Minakshisundaram produces a series of local
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gave rise to a plethora of such examples which clarifies the phenomenon of isospectral manifolds.
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174:, Lecture Notes in Mathematics (in French), vol. 194, Berlin-New York: Springer-Verlag
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Inverse problems seek to identify features of the geometry from information about the
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309:
294:
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167:
62:
1069:
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864:
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299:
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125:
58:
136:). Many versions of the inequality have been established since Cheeger's work (by
695:
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274:
149:
114:
106:
20:
128:
inequality which gives a relation between the first positive eigenvalue and an
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54:
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730:
1331:
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90:
24:
109:
tells us, the information of eigenvalues is not enough to determine the
204:
57:
of the
Laplacian. One of the earliest results of this kind was due to
196:
699:
215:
23:
which concerns relationships between geometric structures of
183:(1985), "Riemannian coverings and isospectral manifolds",
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in 1911 to show that the volume of a bounded domain in
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1307:Spectral theory of ordinary differential equations
571:Spectral theory of ordinary differential equations
46:has been most intensively studied, although other
1205:Schröder–Bernstein theorems for operator algebras
1392:
711:
227:
8:
1399:
1385:
718:
704:
696:
262:
234:
220:
212:
117:). A general and systematic method due to
48:Laplace operators in differential geometry
170:; Gauduchon, Paul; Mazet, Edmond (1971),
85:. This question is usually expressed as "
524:Group algebra of a locally compact group
1038:Spectral theory of normal C*-algebras
836:Spectral theory of normal C*-algebras
172:Le spectre d'une variété riemannienne
7:
1353:
1351:
1033:Spectral theory of compact operators
1371:. You can help Knowledge (XXG) by
1185:Cohen–Hewitt factorization theorem
14:
1190:Extensions of symmetric operators
87:Can one hear the shape of a drum?
1355:
1008:Positive operator-valued measure
680:
679:
606:Topological quantum field theory
79:Dirichlet boundary value problem
1292:Rayleigh–Faber–Krahn inequality
1:
1200:Limiting absorption principle
402:Uniform boundedness principle
89:", the popular phrase due to
826:Singular value decomposition
140:and P. Buser for instance).
1257:Hearing the shape of a drum
940:Decomposition of a spectrum
155:Hearing the shape of a drum
77:of the eigenvalues for the
73:can be determined from the
1449:
1350:
845:Special Elements/Operators
545:Invariant subspace problem
99:covariant differentiations
1433:Riemannian geometry stubs
1317:Superstrong approximation
1180:Banach algebra cohomology
1013:Projection-valued measure
998:Borel functional calculus
770:Projection-valued measure
675:
265:
113:class of a manifold (see
37:Laplace–Beltrami operator
909:Spectrum of a C*-algebra
780:Spectrum of a C*-algebra
514:Spectrum of a C*-algebra
1337:Wiener–Khinchin theorem
1272:Kuznetsov trace formula
1247:Almost Mathieu operator
1065:Banach function algebra
1054:Amenable Banach algebra
811:Gelfand–Naimark theorem
765:Noncommutative topology
611:Noncommutative geometry
31:of canonically defined
1367:-related article is a
1312:Sturm–Liouville theory
1210:Sherman–Takeda theorem
1090:Tomita–Takesaki theory
865:Hermitian/Self-adjoint
816:Gelfand representation
667:Tomita–Takesaki theory
642:Approximation property
586:Calculus of variations
130:isoperimetric constant
33:differential operators
1418:Differential geometry
806:Gelfand–Mazur theorem
662:Banach–Mazur distance
625:Generalized functions
1282:Proto-value function
1261:Dirichlet eigenvalue
1175:Abstract index group
1060:Approximate identity
1023:Rigged Hilbert space
899:Krein–Rutman theorem
745:Involution/*-algebra
407:Kakutani fixed-point
392:Riesz representation
1428:Riemannian geometry
1365:Riemannian geometry
1085:Von Neumann algebra
821:Polar decomposition
591:Functional calculus
550:Mahler's conjecture
529:Von Neumann algebra
243:Functional analysis
95:spectral invariants
75:asymptotic behavior
44:Riemannian manifold
1215:Unbounded operator
1144:Essential spectrum
1123:Schur–Horn theorem
1113:Bauer–Fike theorem
1108:Alon–Boppana bound
1101:Finite-Dimensional
1075:Nuclear C*-algebra
919:Spectral asymmetry
616:Riemann hypothesis
315:Topological vector
35:. The case of the
1380:
1379:
1345:
1344:
1322:Transfer operator
1297:Spectral geometry
982:Spectral abscissa
962:Approximate point
904:Normal eigenvalue
693:
692:
596:Integral operator
373:
372:
181:Sunada, Toshikazu
67:integral equation
17:Spectral geometry
1440:
1401:
1394:
1387:
1359:
1352:
1327:Transform theory
1047:Special algebras
1028:Spectral theorem
991:Spectral Theorem
831:Spectral theorem
720:
713:
706:
697:
683:
682:
601:Jones polynomial
519:Operator algebra
263:
236:
229:
222:
213:
207:
175:
134:Cheeger constant
119:Toshikazu Sunada
103:curvature tensor
83:Laplace operator
1448:
1447:
1443:
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1441:
1439:
1438:
1437:
1423:Spectral theory
1408:
1407:
1406:
1405:
1348:
1346:
1341:
1302:Spectral method
1287:Ramanujan graph
1235:
1219:
1195:Fredholm theory
1163:
1158:Shilov boundary
1154:Structure space
1132:Generalizations
1127:
1118:Numerical range
1096:
1080:Uniform algebra
1042:
1018:Riesz projector
1003:Min-max theorem
986:
972:Direct integral
928:
914:Spectral radius
885:
840:
794:
785:Spectral radius
733:
727:Spectral theory
724:
694:
689:
671:
635:Advanced topics
630:
554:
533:
492:
458:Hilbert–Schmidt
431:
422:Gelfand–Naimark
369:
319:
254:
240:
197:10.2307/1971195
179:
166:
163:
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71:Euclidean space
12:
11:
5:
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1269:
1264:
1254:
1252:Corona theorem
1249:
1243:
1241:
1237:
1236:
1234:
1233:
1231:Wiener algebra
1227:
1225:
1221:
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1202:
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1192:
1187:
1182:
1177:
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1161:
1151:
1149:Pseudospectrum
1146:
1141:
1139:Dirac spectrum
1135:
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1128:
1126:
1125:
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796:
795:
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790:Operator space
787:
782:
777:
772:
767:
762:
757:
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750:Banach algebra
747:
741:
739:
738:Basic concepts
735:
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725:
723:
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715:
708:
700:
691:
690:
688:
687:
676:
673:
672:
670:
669:
664:
659:
654:
652:Choquet theory
649:
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541:
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504:Banach algebra
500:
498:
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480:
475:
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445:
439:
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432:
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427:Banach–Alaoglu
424:
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368:
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355:Locally convex
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191:(1): 169–186,
177:
168:Berger, Marcel
162:
159:
158:
157:
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145:
142:
19:is a field in
13:
10:
9:
6:
4:
3:
2:
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1168:Miscellaneous
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933:Decomposition
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714:
709:
707:
702:
701:
698:
686:
678:
677:
674:
668:
665:
663:
660:
658:
657:Weak topology
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622:
619:
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614:
612:
609:
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604:
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597:
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592:
589:
587:
584:
582:
581:Index theorem
579:
577:
574:
572:
569:
567:
564:
563:
561:
557:
551:
548:
546:
543:
542:
540:
538:Open problems
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185:Ann. of Math.
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68:
65:'s theory of
64:
63:David Hilbert
60:
56:
51:
49:
45:
42:
38:
34:
30:
26:
22:
18:
1373:expanding it
1362:
1347:
1296:
1240:Applications
1070:Disk algebra
924:Spectral gap
799:Main results
647:Balanced set
621:Distribution
559:Applications
412:Krein–Milman
397:Closed graph
188:
184:
171:
123:
59:Hermann Weyl
52:
16:
15:
1267:Heat kernel
967:Compression
852:Isospectral
576:Heat kernel
566:Hardy space
473:Trace class
387:Hahn–Banach
349:Topological
150:Isospectral
115:isospectral
107:John Milnor
55:eigenvalues
21:mathematics
1412:Categories
945:Continuous
760:C*-algebra
755:B*-algebra
509:C*-algebra
324:Properties
161:References
97:involving
731:-algebras
483:Unbounded
478:Transpose
436:Operators
365:Separable
360:Reflexive
345:Algebraic
331:Barrelled
138:R. Brooks
61:who used
25:manifolds
1332:Weyl law
1277:Lax pair
1224:Examples
1058:With an
977:Discrete
955:Residual
891:Spectrum
876:operator
868:operator
860:operator
775:Spectrum
685:Category
497:Algebras
379:Theorems
336:Complete
305:Schwartz
251:glossary
144:See also
111:isometry
91:Mark Kac
873:Unitary
488:Unitary
468:Nuclear
453:Compact
448:Bounded
443:Adjoint
417:Min–max
310:Sobolev
295:Nuclear
285:Hilbert
280:Fréchet
245: (
205:1971195
126:Cheeger
101:of the
81:of the
29:spectra
857:Normal
463:Normal
300:Orlicz
290:Hölder
270:Banach
259:Spaces
247:topics
203:
41:closed
1363:This
950:Point
275:Besov
201:JSTOR
132:(the
39:on a
1369:stub
881:Unit
729:and
623:(or
341:Dual
27:and
193:doi
189:121
1414::
249:–
199:,
187:,
1400:e
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1386:v
1375:.
1263:)
1259:(
1160:)
1156:(
719:e
712:t
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351:)
347:/
343:(
253:)
235:e
228:t
221:v
208:.
195::
176:.
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