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.

Reçu le : 2010-01-22
Accepté le : 2010-11-26
DOI : https://doi.org/10.5802/aif.2651
Classification : 58J50,  53C21
Mots clés: opérateur de Laplace-Beltrami, spectre discret, courbure de Ricci
@article{AIF_2011__61_4_1557_0,
     author = {Kumura, Hironori},
     title = {The lower bound of the Ricci curvature that yields an infinite discrete spectrum of the Laplacian},
     journal = {Annales de l'Institut Fourier},
     pages = {1557--1572},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {61},
     number = {4},
     year = {2011},
     doi = {10.5802/aif.2651},
     zbl = {1252.58017},
     mrnumber = {2951504},
     language = {en},
     url = {aif.centre-mersenne.org/item/AIF_2011__61_4_1557_0/}
}
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. https://aif.centre-mersenne.org/item/AIF_2011__61_4_1557_0/

[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., Tome 178 (1981), pp. 501-508 | Article | MR 638814 | Zbl 0458.58024

[3] Chavel, Isaac Eigenvalues in Riemannian Geometry, Pure and Applied Mathematics, Tome 115, Academic Press Inc., 1984 | MR 768584 | Zbl 0551.53001

[4] Cheng, Shiu-Yuen Eigenvalue comparison theorems and its geometric application, Math. Z, Tome 143 (1982), pp. 289-297 | Article | MR 378001 | Zbl 0329.53035

[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 0729.00007

[6] Donnelly, Harold On the essential spectrum of a complete Riemannian manifold, Topology, Tome 20 (1981), pp. 1-14 | Article | MR 592568 | Zbl 0463.53027

[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 521983 | Zbl 0414.53043

[8] Kasue, Atsushi Applications of Laplacian and Hessian comparison theorems, Adv. Stud. Pure Math., 3 (Shiohama, Katsuhiro, ed.), Elsevier Science Ltd, Tokyo, 1982, pp. 333-386 | MR 758660 | Zbl 0578.53029

[9] Kirsch, Werner; Simon, Barry Corrections to the classical behavior of the number of bound states of Schrödinger operators, Ann. Phys., Tome 183 (1988), pp. 122-130 | Article | MR 952875 | Zbl 0646.35019

[10] Prüfer, Heinz Neue Herleitung der Sturm-Liouvilleschen Reihenentwicklung stetiger Funktionen, Math. Ann., Tome 95 (1926), pp. 499-518 | Article | | JFM 52.0455.01 | MR 1512291

[11] Reed, Michael; Simon, Barry Methods of Modern Mathematical Physics, Vol. II, Academic Press, New York, 1972 | MR 493419 | Zbl 0242.46001

[12] Taylor, Michael E. Partial Differential Equations I, (Applied Math. Sci. 116), Applied Mathematical Sciences, Springer-Verlag, New York, 1996 | MR 1395148 | Zbl 0869.35003