The lower bound of the Ricci curvature that yields an infinite discrete spectrum of the Laplacian
Annales de l'Institut Fourier, Volume 61 (2011) no. 4, p. 1557-1572

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.

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.

DOI : https://doi.org/10.5802/aif.2651
Classification:  58J50,  53C21
Keywords: Laplace-Beltrami operator, discrete spectrum, Ricci curvature
@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},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {61},
     number = {4},
     year = {2011},
     pages = {1557-1572},
     doi = {10.5802/aif.2651},
     mrnumber = {2951504},
     zbl = {1252.58017},
     language = {en},
     url = {http://www.numdam.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, Volume 61 (2011) no. 4, pp. 1557-1572. doi : 10.5802/aif.2651. http://www.numdam.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, Academic Press Inc., Pure and Applied Mathematics, Tome 115 (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; Shiohama, Katsuhiro Applications of Laplacian and Hessian comparison theorems, Adv. Stud. Pure Math., 3, 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), Springer-Verlag, New York, Applied Mathematical Sciences (1996) | MR 1395148 | Zbl 0869.35003