1288:
228:), constructed examples of isospectral, non-isometric closed hyperbolic 2-manifolds and 3-manifolds as quotients of hyperbolic 2-space and 3-space by arithmetic subgroups, constructed using quaternion algebras associated with quadratic extensions of the rationals by
176:
was behind the conservation laws that were responsible for keeping solitons from dissipating. That is, the preservation of spectrum was an interpretation of the conservation mechanism. The identification of so-called
323:
are isospectral but not necessarily isometric. Although this does not recapture the arithmetic examples of Milnor and Vignéras, Sunada's method yields many known examples of isospectral manifolds. It led C. Gordon,
247:
technique, which, either in its original or certain generalized versions, came to be known as the Sunada method or Sunada construction. Like the previous methods it is based on the trace formula, via the
85:. Two such domains are called isospectral if their Laplacians are isospectral. The problem of inferring the geometrical properties of a domain from the spectrum of its Laplacian is often known as
343:
Sunada's idea also stimulated the attempt to find isospectral examples which could not be obtained by his technique. Among many examples, the most striking one is a simply connected example of
201:(Laplacians), counted multiplicities, coincide. One of fundamental problems in spectral geometry is to ask to what extent the eigenvalues determine the geometry of a given manifold.
54:
The theory of isospectral operators is markedly different depending on whether the space is finite or infinite dimensional. In finite-dimensions, one essentially deals with square
212:. After this example, many isospectral pairs in dimension two and higher were constructed (for instance, by M. F. Vignéras, A. Ikeda, H. Urakawa, C. Gordon). In particular
1177:
1013:
693:
Selberg, Atle (1956), "Harmonic analysis and discontinuous groups in weakly symmetric
Riemannian spaces with applications to Dirichlet series",
840:
449:
236:, the set of lengths of closed geodesics in each free homotopy class, along with the twist along the geodesic in the 3-dimensional case.
1003:
538:
497:
1130:
985:
961:
763:
853:
942:
833:
1212:
857:
204:
There are many examples of isospectral manifolds which are not isometric. The first example was given in 1964 by
589:
61:
In infinite dimensions, the spectrum need not consist solely of isolated eigenvalues. However, the case of a
1008:
198:
1291:
1064:
998:
826:
1028:
360:
333:
86:
1273:
1227:
1151:
1033:
325:
48:
232:. In this case Selberg's trace formula shows that the spectrum of the Laplacian fully determines the
1268:
1084:
348:
73:) is still tractable, since the eigenvalues are at most countable with at most a single limit point
1312:
1120:
1018:
921:
106:
1217:
993:
208:. He constructed a pair of flat tori of 16 dimension, using arithmetic lattices first studied by
144:
in a complicated way. This is an evolution of a matrix that happens inside one similarity class.
36:
647:
Schueth, D. (1999), "Continuous families of isospectral metrics on simply connected manifolds",
533:
1248:
1192:
1156:
253:
955:
249:
217:
102:
951:
77: = 0. The most studied isospectral problem in infinite dimensions is that of the
1231:
734:
Vignéras, Marie-France (1980), "Variétés riemanniennes isospectrales et non isométriques",
608:
109:. This doesn't however reduce completely the interest of the concept, since we can have an
55:
818:
8:
1197:
1135:
849:
560:
McKean, H. P. (1972), "Selberg's trace formula as applied to a compact
Riemann surface",
276:
194:
612:
525:
493:
395:
This amounts to knowing the conjugacy class of the corresponding group element in PSL(2,
1222:
1089:
808:
751:
723:
682:
674:
656:
529:
482:
351:
proved that certain isospectral arithmetic hyperbolic manifolds in are commensurable.
229:
631:
1202:
636:
585:
365:
40:
791:
Wolpert, Scott (1979), "The length spectra as moduli for compact
Riemann surfaces",
783:
686:
1207:
1125:
1094:
1074:
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1049:
800:
778:
743:
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666:
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462:
426:
240:
78:
62:
886:
430:
1069:
1023:
971:
966:
937:
599:
Milnor, John (1964), "Eigenvalues of the
Laplace operator on certain manifolds",
415:"Isospectrality and commensurability of arithmetic hyperbolic 2- and 3-manifolds"
24:
896:
414:
1258:
1110:
911:
244:
98:
552:
1306:
1263:
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could be adapted to compact manifolds. His method relies on the fact that if
252:. Sunada noticed that the method of constructing number fields with the same
152:
66:
1253:
906:
876:
640:
573:
272:
70:
621:
1182:
1172:
1079:
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205:
20:
336:" An elementary treatment, based on Sunada's method, was later given in
1115:
947:
812:
755:
727:
486:
209:
44:
678:
661:
185:, showed how linear machinery could explain the non-linear behaviour.
706:
Sunada, T. (1985), "Riemannian coverings and isospectral manifolds",
517:
182:
138:
804:
747:
719:
478:
670:
329:
178:
97:
In the case of operators on finite-dimensional vector spaces, for
451:
Variétés riemanniennes isospectrales non isométriques, exposé 705
148:
764:"The eigenvalue spectrum as moduli for compact Riemann surfaces"
328:
and S. Wolpert to the discovery in 1991 of a counter example to
848:
101:
square matrices, the relation of being isospectral for two
524:
337:
197:
are said to be isospectral if the eigenvalues of their
39:. Roughly speaking, they are supposed to have the same
260:
is a finite covering of a compact
Riemannian manifold
473:(9), Mathematical Association of America: 823–839,
301:in the same number of elements, then the manifolds
1178:Spectral theory of ordinary differential equations
243:found a general method of construction based on a
1304:
579:
532:; Doyle, Peter; Semmler, Klaus-Dieter (1994),
465:(1988), "Constructing Isospectral Manifolds",
384:
834:
181:(P,L) giving rise to analogous equations, by
92:
447:
841:
827:
539:International Mathematics Research Notices
782:
660:
630:
620:
551:
516:
1131:Group algebra of a locally compact group
733:
582:The Arithmetic of Hyperbolic 3-manifolds
213:
188:
790:
761:
692:
646:
344:
16:Linear operators with a common spectrum
1305:
705:
598:
580:Maclachlan, C.; Reid, Alan W. (2003),
559:
461:
822:
492:
412:
13:
457:, SĂ©minaire Bourbaki, vol. 31
155:analogue of that equation, namely
14:
1324:
534:"Some planar isospectral domains"
334:Can one hear the shape of a drum?
1287:
1286:
1213:Topological quantum field theory
297:meeting each conjugacy class of
784:10.1090/S0002-9904-1977-14425-X
584:, Springer, pp. 383–394,
498:"Isospectral Riemann surfaces"
406:
389:
378:
47:, when those are counted with
1:
1009:Uniform boundedness principle
505:Annales de l'Institut Fourier
467:American Mathematical Monthly
441:
431:10.1215/S0012-7094-92-06508-2
448:Bérard, Pierre (1988–1989),
7:
361:Hearing the shape of a drum
354:
87:hearing the shape of a drum
10:
1329:
1152:Invariant subspace problem
601:Proc. Natl. Acad. Sci. USA
385:Maclachlan & Reid 2003
1282:
1241:
1165:
1144:
1103:
1042:
984:
930:
872:
865:
553:10.1155/S1073792894000437
419:Duke Mathematical Journal
199:Laplace–Beltrami operator
147:A fundamental insight in
93:Finite dimensional spaces
1121:Spectrum of a C*-algebra
371:
1218:Noncommutative geometry
762:Wolpert, Scott (1977),
103:diagonalizable matrices
1274:Tomita–Takesaki theory
1249:Approximation property
1193:Calculus of variations
771:Bull. Amer. Math. Soc.
574:10.1002/cpa.3160250302
562:Comm. Pure Appl. Math.
413:Reid, Alan W. (1992).
254:Dedekind zeta function
35:if they have the same
1269:Banach–Mazur distance
1232:Generalized functions
793:Annals of Mathematics
736:Annals of Mathematics
708:Annals of Mathematics
649:Annals of Mathematics
622:10.1073/pnas.51.4.542
347:. On the other hand,
250:Selberg zeta function
218:Selberg trace formula
189:Isospectral manifolds
113:of matrices of shape
1014:Kakutani fixed-point
999:Riesz representation
695:J. Indian Math. Soc.
277:deck transformations
195:Riemannian manifolds
151:theory was that the
1198:Functional calculus
1157:Mahler's conjecture
1136:Von Neumann algebra
850:Functional analysis
613:1964PNAS...51..542M
338:Buser et al. (1994)
1223:Riemann hypothesis
922:Topological vector
230:class field theory
111:isospectral family
1300:
1299:
1203:Integral operator
980:
979:
366:Spectral geometry
293:are subgroups of
137:) depending on a
1320:
1290:
1289:
1208:Jones polynomial
1126:Operator algebra
870:
869:
843:
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777:(6): 1306–1308,
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518:10.5802/aif.1054
502:
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410:
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387:
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241:Toshikazu Sunada
164:
79:Laplace operator
63:compact operator
25:linear operators
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1322:
1321:
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1313:Spectral theory
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1242:Advanced topics
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1140:
1099:
1065:Hilbert–Schmidt
1038:
1029:Gelfand–Naimark
976:
926:
861:
847:
805:10.2307/1971114
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748:10.2307/1971319
720:10.2307/1971195
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479:10.2307/2322897
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234:length spectrum
216:, based on the
214:Vignéras (1980)
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81:on a domain in
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12:
11:
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1259:Choquet theory
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1111:Banach algebra
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1034:Banach–Alaoglu
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1001:
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962:Locally convex
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799:(2): 323–351,
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714:(1): 169–186,
703:
690:
671:10.2307/121026
655:(1): 287–308,
644:
596:
590:
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568:(3): 225–246,
557:
546:(9): 391–400,
522:
511:(2): 167–192,
490:
463:Brooks, Robert
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345:Schueth (1999)
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245:covering space
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1264:Weak topology
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1188:Index theorem
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1145:Open problems
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662:dg-ga/9711010
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153:infinitesimal
150:
145:
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136:
132:
128:
124:
120:
116:
112:
108:
104:
100:
90:
88:
84:
80:
76:
72:
68:
67:Hilbert space
64:
59:
57:
52:
50:
46:
42:
38:
34:
30:
26:
22:
1254:Balanced set
1228:Distribution
1166:Applications
1019:Krein–Milman
1004:Closed graph
796:
792:
774:
770:
742:(1): 21–32,
739:
735:
711:
707:
698:
694:
652:
648:
604:
600:
581:
565:
561:
543:
537:
530:Conway, John
526:Buser, Peter
508:
504:
494:Buser, Peter
470:
466:
450:
422:
418:
408:
400:
396:
391:
380:
342:
332:'s problem "
320:
313:
309:
302:
298:
294:
287:
280:
273:finite group
268:
261:
257:
238:
233:
225:
224:) and PSL(2,
221:
203:
192:
175:
170:
166:
159:
146:
141:
134:
130:
126:
122:
118:
114:
110:
96:
82:
74:
71:Banach space
60:
53:
49:multiplicity
32:
28:
18:
1183:Heat kernel
1173:Hardy space
1080:Trace class
994:Hahn–Banach
956:Topological
399:) or PSL(2,
206:John Milnor
193:Two closed
45:eigenvalues
29:isospectral
27:are called
21:mathematics
1116:C*-algebra
931:Properties
607:(4): 542,
591:0387983864
442:References
220:for PSL(2,
210:Ernst Witt
107:similarity
33:cospectral
1090:Unbounded
1085:Transpose
1043:Operators
972:Separable
967:Reflexive
952:Algebraic
938:Barrelled
349:Alan Reid
183:Peter Lax
179:Lax pairs
139:parameter
1307:Category
1292:Category
1104:Algebras
986:Theorems
943:Complete
912:Schwartz
858:glossary
687:10898684
641:16591156
496:(1986),
355:See also
330:Mark Kac
239:In 1985
169:−
105:is just
56:matrices
37:spectrum
1095:Unitary
1075:Nuclear
1060:Compact
1055:Bounded
1050:Adjoint
1024:Min–max
917:Sobolev
902:Nuclear
892:Hilbert
887:Fréchet
852: (
813:1971114
756:1971319
728:1971195
701:: 47–87
609:Bibcode
487:2322897
326:D. Webb
149:soliton
99:complex
1070:Normal
907:Orlicz
897:Hölder
877:Banach
866:Spaces
854:topics
811:
754:
726:
685:
679:121026
677:
639:
632:300113
629:
588:
485:
23:, two
882:Besov
809:JSTOR
767:(PDF)
752:JSTOR
724:JSTOR
683:S2CID
675:JSTOR
657:arXiv
501:(PDF)
483:JSTOR
455:(PDF)
425:(2).
372:Notes
267:with
165:= =
65:on a
1230:(or
948:Dual
637:PMID
586:ISBN
544:1994
312:and
279:and
271:the
121:) =
69:(or
41:sets
801:doi
797:109
779:doi
744:doi
740:112
716:doi
712:121
667:doi
653:149
627:PMC
617:doi
570:doi
548:doi
513:doi
475:doi
427:doi
275:of
43:of
31:or
19:In
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807:,
795:,
775:83
773:,
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625:,
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