[Limite inférieure de la courbure de Ricci qui donne un nombre de spectre discret infini]
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.
Keywords: Laplace-Beltrami operator, discrete spectrum, Ricci curvature
Mot 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{\textquoteright}institut Fourier},
volume = {61},
number = {4},
year = {2011},
doi = {10.5802/aif.2651},
zbl = {1252.58017},
mrnumber = {2951504},
language = {en},
url = {http://www.numdam.org/articles/10.5802/aif.2651/}
}
TY - JOUR AU - Kumura, Hironori TI - The lower bound of the Ricci curvature that yields an infinite discrete spectrum of the Laplacian JO - Annales de l'Institut Fourier PY - 2011 SP - 1557 EP - 1572 VL - 61 IS - 4 PB - Association des Annales de l’institut Fourier UR - http://www.numdam.org/articles/10.5802/aif.2651/ DO - 10.5802/aif.2651 LA - en ID - AIF_2011__61_4_1557_0 ER -
%0 Journal Article %A Kumura, Hironori %T The lower bound of the Ricci curvature that yields an infinite discrete spectrum of the Laplacian %J Annales de l'Institut Fourier %D 2011 %P 1557-1572 %V 61 %N 4 %I Association des Annales de l’institut Fourier %U http://www.numdam.org/articles/10.5802/aif.2651/ %R 10.5802/aif.2651 %G en %F 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. http://www.numdam.org/articles/10.5802/aif.2651/
[1] The uncertainty principle lemma under gravity and the discrete spectrum of Schrödinger operators (arXiv:0812.4663)
[2] A relation between growth and the spectrum of the Laplacian, Math. Z., Volume 178 (1981), pp. 501-508 | DOI | MR | Zbl
[3] Eigenvalues in Riemannian Geometry, Pure and Applied Mathematics, 115, Academic Press Inc., 1984 | MR | Zbl
[4] Eigenvalue comparison theorems and its geometric application, Math. Z, Volume 143 (1982), pp. 289-297 | DOI | MR | Zbl
[5] Methods of Mathematical Physics, Interscience Publishers, Inc.,(a division of John Wiley & Sons), New York-London, Vol. I ,1953; Vol. II, 1962 | Zbl
[6] On the essential spectrum of a complete Riemannian manifold, Topology, Volume 20 (1981), pp. 1-14 | DOI | MR | Zbl
[7] Function Theory on Manifolds Which Possess a Pole, Lecture Notes in Math. 699, Springer-Verlag, Berlin, 1979 | MR | Zbl
[8] Applications of Laplacian and Hessian comparison theorems, Adv. Stud. Pure Math., 3, Elsevier Science Ltd, Tokyo, 1982, pp. 333-386 | MR | Zbl
[9] 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] Neue Herleitung der Sturm-Liouvilleschen Reihenentwicklung stetiger Funktionen, Math. Ann., Volume 95 (1926), pp. 499-518 | DOI | EuDML | JFM | MR
[11] Methods of Modern Mathematical Physics, Vol. II, Academic Press, New York, 1972 | MR | Zbl
[12] Partial Differential Equations I, (Applied Math. Sci. 116), Applied Mathematical Sciences, Springer-Verlag, New York, 1996 | MR | Zbl
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