A note on eigenvalue bounds for non-compact manifolds. (English) Zbl 1523.58032
Summary: In this article we prove upper bounds for the Laplace eigenvalues \(\lambda_k\) below the essential spectrum for strictly negatively curved Cartan-Hadamard manifolds. Our bound is given in terms of \(k^2\) and specific geometric data of the manifold. This applies also to the particular case of non-compact manifolds whose sectional curvature tends to \(- \infty\), where no essential spectrum is present due to a theorem of Donnelly/Li. The result stands in clear contrast to Laplacians on graphs where such a bound fails to be true in general.
{© 2021 The Authors. Mathematische Nachrichten published by Wiley-VCH GmbH}
{© 2021 The Authors. Mathematische Nachrichten published by Wiley-VCH GmbH}
MSC:
| 58J50 | Spectral problems; spectral geometry; scattering theory on manifolds |
| 35P20 | Asymptotic distributions of eigenvalues in context of PDEs |
References:
| [1] | S. G.Bobkov and C.Houdré, Isoperimetric constants for product probability measures, Ann. Probab.25 (1997), no. 1, 184-205. · Zbl 0878.60013 |
| [2] | M.Bonnefont, S.Golénia, and M.Keller, Eigenvalue asymptotics for Schrödinger operators on sparse graphs, Ann. Inst. Fourier (Grenoble)65 (2015), no. 5, 1969-1998. · Zbl 1336.47034 |
| [3] | I.Chavel, Eigenvalues in Riemannian geometry, Pure Appl. Math., vol. 115, Academic Press, Inc., Orlando, FL, 1984. · Zbl 0551.53001 |
| [4] | S. Y.Cheng, Eigenvalue comparison theorems and its geometric applications, Math. Z.143 (1975), no. 3, 289-297. · Zbl 0329.53035 |
| [5] | H.Donnelly and P.Li, Pure point spectrum and negative curvature for noncompact manifolds, Duke Math. J.46 (1979), no. 3, 497-503. · Zbl 0416.58025 |
| [6] | T.Kura, A Laplacian comparison theorem and its applications, Proc. Japan Acad. Ser. A Math. Sci.78 (2002), no. 1, 7-9. · Zbl 1082.53038 |
| [7] | Tsz ChiuKwok, Lap ChiLau, Yin TatLee, Sh. OveisGharan, and L.Trevisan, Improved Cheeger’s inequality: analysis of spectral partitioning algorithms through higher order spectral gap, STOC’13, Proceedings of the 2013 ACM Symposium on Theory of Computing, ACM, New York, 2013, pp. 11-20. · Zbl 1293.05301 |
| [8] | S.Liu, An optimal dimension‐free upper bound for eigenvalue ratios, (preprint), arXiv:1405.2213, 2014. |
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