zbMATH — the first resource for mathematics

Transplantation and isospectrality. I. (Transplantation et isospectralité. I.) (French) Zbl 0735.58008
This paper gives new combinatorial geometric proofs of Sunada’s theorem concerning isospectral manifolds (both for the eigenvalue spectrum and for the length spectrum). The proofs also apply to non-smooth metrics and to natural operators. The paper includes examples of non-isometric complete non-compact manifolds that are isospectral in the following sense: they have the same discrete part of the spectrum and their scattering matrices are unitarily conjugate (in this sense, one can say that these manifolds have the same continuous spectra).
Reviewer: P.Bérard

58C40 Spectral theory; eigenvalue problems on manifolds
Full Text: DOI EuDML
[1] Aubin, Th.: Nonlinear analysis on manifolds, Monge-Amp?re equations. Berlin Heidelberg New York: Springer 1982 · Zbl 0512.53044
[2] Baider, A.: Noncompact Riemannian manifolds with discrete spectra. J. Diff. Geom.14, 41-57 (1979) · Zbl 0411.58022
[3] B?rard, P.: Vari?t?s riemanniennes isospectrales non isom?triques. Ast?risque177-178, 127-154 (1989)
[4] B?rard, P.: Transplantation et isospectralit?. II. (Pr?tirage Institut Fourier 1991)
[5] B?rard, P.: Transplantation et isospectralit?. I. In: S?minaire Th?orie Spectrale et G?om?trie 90-91. Grenoble: Institut Fourier 1991
[6] Brooks, R.: Constructing isospectral manifolds. Am. Math. Mon.95, 823-839 (1988) · Zbl 0673.58046 · doi:10.2307/2322897
[7] Brooks, R., Tse, R.: Isospectral surfaces of small genus. Nagoya Math. J.107, 13-24 (1987) · Zbl 0605.58041
[8] Buser, P.: Isospectral Riemann surfaces. Ann. Inst. Fourier36, 167-192 (1986) · Zbl 0579.53036
[9] Buser, P.: Cayley graphs and planar isospectral domains. In: Sunada, T. (ed.) Proc. Taniguchi Symp. ?Geometry and Analysis on manifolds?. (Lect. Notes Math., vol. 1339, pp. 64-77, Berlin Heidelberg New York: Springer 1988 · Zbl 0647.53034
[10] Buser, P.: Geometry and spectrum of compact Riemann surfaces (livre en pr?paration)
[11] Colin de Verdi?re, Y.: Pseudo-laplaciens. II. Ann. Inst. Fourier33, 87-113 (1983)
[12] DeTurck, D., Gordon, S.: Isospectral deformations. I. Riemannian structures on two-step nilspaces. Commun. Pure Appl. Math.40, 367-387 (1987) · Zbl 0649.53025 · doi:10.1002/cpa.3160400306
[13] DeTurck, D., Gordon, S.: Isospectral deformations. II. Trace formulas, metrics, and potentials. Commun. Pure Appl. Math.42, 1067-1095 (1989) · Zbl 0709.53030 · doi:10.1002/cpa.3160420803
[14] Gordon, C.S.: The Laplace spectrum versus the length spectra of Riemannian manifolds. In: DeTurck, D.M. (ed.) Nonlinear problems in geometry. (Contemp. Math., vol. 51, pp. 63-80) Providence, RI: Am. Math. Soc. 1986
[15] Gordon, C.S., Webb, D., Wolpert, S.: One can’t hear the shape of a drum. Annonce de Recherche (1991) · Zbl 0756.58049
[16] M?ller, W.: Spectral geometry and scattering theory for certain complete surfaces of finite volume. (Pr?tirage 1991)
[17] Serre, J.-P.: Linear representations of finite groups. Berlin Heidelberg New York: Springer 1977
[18] Sunada, T.: Riemannian coverings and isospectral manifolds. Ann. Math.121, 169-186 (1985) · Zbl 0585.58047 · doi:10.2307/1971195
[19] Zelditch, S.: Isospectrality in the category of F.I.O. (Pr?tirage 1989)
[20] Zelditch, S.: Kuznecov sum formula on manifolds. (Pr?tirage 1989)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.