The lower bound of the Ricci curvature that yields an infinite discrete spectrum of the Laplacian
[Limite inférieure de la courbure de Ricci qui donne un nombre de spectre discret infini]
Annales de l'Institut Fourier, Tome 61 (2011) no. 4, pp. 1557-1572.

Ce document traite de la question si le spectre discret de l’opérateur de Laplace-Beltrami est infini ou fini. La ligne de démarcation du comportement des courbures de ce problème sera complètement déterminée.

This paper discusses the question whether the discrete spectrum of the Laplace-Beltrami operator is infinite or finite. The borderline-behavior of the curvatures for this problem will be completely determined.

DOI : 10.5802/aif.2651
Classification : 58J50, 53C21
Keywords: Laplace-Beltrami operator, discrete spectrum, Ricci curvature
Mot clés : opérateur de Laplace-Beltrami, spectre discret, courbure de Ricci
Kumura, Hironori 1

1 Shizuoka University Department of Mathematics Ohya, Shizuoka 422-8529 (Japan)
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Kumura, Hironori. The lower bound of the Ricci curvature that yields an infinite discrete spectrum of the Laplacian. Annales de l'Institut Fourier, Tome 61 (2011) no. 4, pp. 1557-1572. doi : 10.5802/aif.2651. http://archive.numdam.org/articles/10.5802/aif.2651/

[1] Akutagawa, Kazuo; Kumura, Hironori The uncertainty principle lemma under gravity and the discrete spectrum of Schrödinger operators (arXiv:0812.4663)

[2] Brooks, Robert A relation between growth and the spectrum of the Laplacian, Math. Z., Volume 178 (1981), pp. 501-508 | DOI | MR | Zbl

[3] Chavel, Isaac Eigenvalues in Riemannian Geometry, Pure and Applied Mathematics, 115, Academic Press Inc., 1984 | MR | Zbl

[4] Cheng, Shiu-Yuen Eigenvalue comparison theorems and its geometric application, Math. Z, Volume 143 (1982), pp. 289-297 | DOI | MR | Zbl

[5] Courant, Richard; Hilbert, David Methods of Mathematical Physics, Interscience Publishers, Inc.,(a division of John Wiley & Sons), New York-London, Vol. I ,1953; Vol. II, 1962 | Zbl

[6] Donnelly, Harold On the essential spectrum of a complete Riemannian manifold, Topology, Volume 20 (1981), pp. 1-14 | DOI | MR | Zbl

[7] Greene, Robert E.; Wu, Hung-Hsi Function Theory on Manifolds Which Possess a Pole, Lecture Notes in Math. 699, Springer-Verlag, Berlin, 1979 | MR | Zbl

[8] Kasue, Atsushi; Shiohama, Katsuhiro Applications of Laplacian and Hessian comparison theorems, Adv. Stud. Pure Math., 3, Elsevier Science Ltd, Tokyo, 1982, pp. 333-386 | MR | Zbl

[9] Kirsch, Werner; Simon, Barry Corrections to the classical behavior of the number of bound states of Schrödinger operators, Ann. Phys., Volume 183 (1988), pp. 122-130 | DOI | MR | Zbl

[10] Prüfer, Heinz Neue Herleitung der Sturm-Liouvilleschen Reihenentwicklung stetiger Funktionen, Math. Ann., Volume 95 (1926), pp. 499-518 | DOI | EuDML | JFM | MR

[11] Reed, Michael; Simon, Barry Methods of Modern Mathematical Physics, Vol. II, Academic Press, New York, 1972 | MR | Zbl

[12] Taylor, Michael E. Partial Differential Equations I, (Applied Math. Sci. 116), Applied Mathematical Sciences, Springer-Verlag, New York, 1996 | MR | Zbl

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