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Demonstratio Mathematica

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Volume 47, Issue 3

Issues

One Can Hear the Area of a Torus by Hearing the Eigenvalues of the Polyharmonic Operators

Shunzi Guo
  • SCHOOL OF MATHEMATICS AND STATISTICS MINNAN NORMAL UNIVERSITY ZHANGZHOU, 363000, PEOPLE’S REPUBLIC OF CHINA / SCHOOL OF MATHEMATICS AND STATISTICS HUBEI UNIVERSITY WUHAN, 430062, PEOPLE’S REPUBLIC OF CHINA
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  • SCHOOL OF MATHEMATICS AND STATISTICS HUBEI UNIVERSITY WUHAN, 430062, PEOPLE’S REPUBLIC OF CHINA
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Published Online: 2014-09-02 | DOI: https://doi.org/10.2478/dema-2014-0048

Abstract

This paper considers the asymptotic properties for the spectrum of a positive integer power l of the Laplace-Beltrami operator acting on an n-dimensional torus T. If N(λ) is the number of eigenvalues counted with multiplicity, smaller than a real positive number, we establish a Weyl-type asymptotic formula for the spectral problem of the polyharmonic operators on T, that is, as λ → +∞

N (λ) ~ ωn (VolT)λn/21 /2(π)n

where ωn is the volume of the unit ball in ℝn and Vol T is the area of T, which gives the information of the area of the torus based on the spectrum of the polyharmonic operators.

Keywords: and phrases: asymptotic formula; counting function; polyharmonic operators; torus

References

  • [1] M. Ashbaugh, F. Gesztesy, M. Mitrea, G. Teschl, Spectral theory for perturbed Krein Laplacians in nonsmooth domains, Adv. Math. 223(4) (2010), 1372-1467.Web of ScienceGoogle Scholar

  • [2] M. Berger, P. Gauduchon, E. Mazet, Le Spectre d’une variété Riemannienes, Lecture Notes in Math. 194, Spinger, Berlin, New York, 1974.Google Scholar

  • [3] M. Sh. Birman, M. Z. Solomyak, Asymptotic behavior of the spectrum of differential equations, Itogi Nauki i Tekhniki Matemat. Anal. 14 (1977), 5-58, (in Russian); Engl. transl.: J. Soviet Math. 12(3) (1979), 247-283.Google Scholar

  • [4] R. Brooks, Constructing isospectral manifolds, Amer. Math. Monthly 95(9) (1988), 823-839.Google Scholar

  • [5] I. Chavel, Eigenvalues in Riemannian Geometry, Academic Press, New York, 1984.Google Scholar

  • [6] C. Gordon, D. Webb, Isospectral convex domains in Euclidean space, Math. Res. Lett. 5(1) (1994), 539-545.CrossrefGoogle Scholar

  • [7] C. Gordon, D. Webb, S. Wolpert, One cannot hear the shape of a drum, Bull. Amer. Math. Soc. 27(1) (1992), 134-138.CrossrefGoogle Scholar

  • [8] C. Gordon, D. Webb, S. Wolpert, Isospectral plane domains and surfaces via Riemannian orbifolds, Invent. Math. 110(1) (1992), 1-22.Google Scholar

  • [9] M. Kac, Can one hear the shape of a drum?, Amer. Math. Monthly (Slaught Mem. Papers, no. 11) 73(4) (1966), 1-23.Google Scholar

  • [10] G. Q. Liu, Some inequalities for eigenvalues on Riemannian manifold and asymptotic formulas, J. Math. Anal. Appl. 376(1) (2011), 349-364.Google Scholar

  • [11] G. Q. Liu, The Weyl-type asymptotic formula for biharmonic Steklov eigenvalues on Riemannian manifolds, Adv. Math. 228(4) (2011), 2162-2217.Web of ScienceGoogle Scholar

  • [12] H. P. McKean, I. M. Singer, Curvature and the eigenvalues of the Laplacian, J. Differential Geom. 1(1) (1967), 43-69.Google Scholar

  • [13] J. Milnor, Eigenvalues of the Laplace operator on certain manifolds, Proc. Nat. Acad. Sci. U.S.A. 51 (1964), 542.CrossrefGoogle Scholar

  • [14] Å. Pleijel, On the eigenvalues and eigenfunctions of elastic plates, Comm. Pure Appl. Math. 3(1) (1950), 1-10.Google Scholar

  • [15] T. Sunada, Riemannian coverings and isospectral manifolds, Ann. of Math. (2) 121 (1985), 169-186.Google Scholar

  • [16] H. Urakawa, Bounded domains which are isospectral but not congruent, Ann. Sci. École Norm. Sup. (4) 15(3) (1982), 441-456.Google Scholar

  • [17] H. Weyl, Über die asymptotische Verteilung der Eigenwerte, Nachr. Konigl. Ges. Wiss. Göttingen, (1911), 110-117.Google Scholar

  • [18] H. Weyl, Der asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen, Math. Ann. 71(4) (1912), 441-469. Google Scholar

About the article

Received: 2012-06-13

Revised: 2013-06-17

Published Online: 2014-09-02

Published in Print: 2014-07-01


Citation Information: Demonstratio Mathematica, Volume 47, Issue 3, Pages 607–614, ISSN (Online) 2391-4661, ISSN (Print) 0420-1213, DOI: https://doi.org/10.2478/dema-2014-0048.

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© by Shunzi Guo. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

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