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Spectra of manifolds less a small domain. (English) Zbl 0645.58042
Let M be a compact connected \(C^{\infty}\) Riemannian manifold with Laplacian \(\Delta\). Then \(\Delta\) has pure point spectrum consisting of eigenvalues \(\lambda_ i\), \(i\geq 0\). Suppose M *\(\subset M\) is a compact submanifold, of codimension at least two, with tubular neighborhood \(B_{\epsilon}\) of radius \(\epsilon >0\). The associated Laplacian acts on L \(2(M-B_{\epsilon})\), where Dirichlet boundary conditions are imposed. Let \(\lambda_{i,\epsilon}\), \(i\geq 1\), denote the eigenvalues of \(\Delta_{\epsilon}\). It is well known that \(\lambda_{j,\epsilon}\downarrow \lambda_{j-1}\), as \(\epsilon\downarrow 0\). The authors compute the first correction term \(\beta (\epsilon)=\lambda_{j,\epsilon}-\lambda_{j-1}\), in terms of the eigenfunction \(\phi_{j-1}\) associated to \(\lambda_{j-1}\).
Reviewer: H.Donelly

MSC:
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
53C20 Global Riemannian geometry, including pinching
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[1] P. Bérard and S. Gallot, Remarques sur quelques estimées géométriques explicites , C. R. Acad. Sci. Paris Sér. I Math. 297 (1983), no. 3, 185-188. · Zbl 0535.53036
[2] G. Besson, Comportement asymptotique des valeurs propres du laplacien dans un domaine avec un trou , Bull. Soc. Math. France 113 (1985), no. 2, 211-230. · Zbl 0577.58033 · numdam:BSMF_1985__113__211_0 · eudml:87481
[3] I. Chavel, Eigenvalues in Riemannian geometry , Pure and Applied Mathematics, vol. 115, Academic Press Inc., Orlando, FL, 1984. · Zbl 0551.53001
[4] I. Chavel and E. A. Feldman, Spectra of domains in compact manifolds , J. Funct. Anal. 30 (1978), no. 2, 198-222. · Zbl 0392.58016 · doi:10.1016/0022-1236(78)90070-8
[5] I. Chavel and E. A. Feldman, The Lenz shift and Wiener sausage in Riemannian manifolds , Compositio Math. 60 (1986), no. 1, 65-84. · Zbl 0607.60066 · numdam:CM_1986__60_1_65_0 · eudml:89798
[6] I. Chavel and E. A. Feldman, The Lenz shift and Wiener sausage in insulated domains , From local times to global geometry, control and physics (Coventry, 1984/85) ed. D. Elwarthy, Pitman Res. Notes Math. Ser., vol. 150, Longman Sci. Tech., Harlow, 1986, pp. 47-67. · Zbl 0616.60076
[7] G. Courtois, Comportement du spectre d’une variété riemannienne compacte sous perturbation topologique par excision d’une domaine , dissertation, Inst. Fourier, Grenoble, 1987.
[8] G. Del Grosso and F. Marchetti, Asymptotic estimates for the principal eigenvalue of the Laplacian in a geodesic ball , Appl. Math. Optim. 10 (1983), no. 1, 37-50. · Zbl 0546.35049 · doi:10.1007/BF01448378
[9] M. Kac, Probabilistic methods in some problems of scattering theory , Rocky Mountain J. Math. 4 (1974), 511-537. · Zbl 0314.47006 · doi:10.1216/RMJ-1974-4-3-511
[10] H. M. MacDonald, Zeroes of the spherical harmonic \(P^^m_n(\mu)\) considered as a function of \(n\) , Proc. London Math. Soc. (1) 31 (1899), 264-278. · JFM 30.0412.01
[11] T. Matsuzawa and S. Tanno, Estimates of the first eigenvalue of a big cup domain of a \(2\)-sphere , Compositio Math. 47 (1982), no. 1, 95-100. · Zbl 0498.53033 · numdam:CM_1982__47_1_95_0 · eudml:89561
[12] V. G. Maz’ya, S. A. Nazarov, and B. A. Plamenevskii, Asymptotic expansions of the eigenvalues of boundary value problems for the Laplace operator in domains with small holes , Math. USSR-Izv. 24 (1985), 321-345, (Engl. trans). · Zbl 0566.35031 · doi:10.1070/IM1985v024n02ABEH001237
[13] M Pinsky, The first eigenvalue of a spherical cap , Appl. Math. Optim. 7 (1981), no. 2, 137-139. · Zbl 0456.92002 · doi:10.1007/BF01442111
[14] 1 S. Ozawa, Singular Hadamard’s variation of domains and eigenvalues of the Laplacian , Proc. Japan Acad. Ser. A Math. Sci. 56 (1980), no. 7, 306-310. · Zbl 0483.35063 · doi:10.3792/pjaa.56.306
[15] 2 S. Ozawa, Singular Hadamard’s variation of domains and eigenvalues of the Laplacian. II , Proc. Japan Acad. Ser. A Math. Sci. 57 (1981), no. 5, 242-246. · Zbl 0509.35060 · doi:10.3792/pjaa.57.242
[16] S. Ozawa, Singular variation of domains and eigenvalues of the Laplacian , Duke Math. J. 48 (1981), no. 4, 767-778. · Zbl 0483.35064 · doi:10.1215/S0012-7094-81-04842-0
[17] S. Ozawa, The first eigenvalue of the Laplacian on two-dimensional Riemannian manifolds , Tôhoku Math. J. (2) 34 (1982), no. 1, 7-14. · Zbl 0486.53035 · doi:10.2748/tmj/1178229304
[18] S. Ozawa, Electrostatic capacity and eigenvalues of the Laplacian , J. Fac. Sci. Univ. Tokyo Sect. IA Math. 30 (1983), no. 1, 53-62. · Zbl 0531.35061
[19] S. Ozawa, An asymptotic formula for the eigenvalues of the Laplacian in a three-dimensional domain with a small hole , J. Fac. Sci. Univ. Tokyo Sect. IA Math. 30 (1983), no. 2, 243-257. · Zbl 0541.35060
[20] J. Rauch and M. Taylor, Potential and scattering theory on wildly perturbed domains , J. Funct. Anal. 18 (1975), 27-59. · Zbl 0293.35056 · doi:10.1016/0022-1236(75)90028-2
[21] S. Sato, Barta’s inequalities and the first eigenvalue of a cap domain of a \(2\)-sphere , Math. Z. 181 (1982), no. 3, 313-318. · Zbl 0498.53034 · doi:10.1007/BF01161979 · eudml:173235
[22] B. Simon, Functional integration and quantum physics , Pure and Applied Mathematics, vol. 86, Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York, 1979. · Zbl 0434.28013
[23] C. A. Swanson, Asymptotic variational formulae for eigenvalues , Canad. Math. Bull. 6 (1963), 15-25. · Zbl 0111.29803 · doi:10.4153/CMB-1963-004-9
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