×

Bounds on the eigenvalues of the Laplace and Schrödinger operators. (English) Zbl 0329.35018


MSC:

35J10 Schrödinger operator, Schrödinger equation
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35P15 Estimates of eigenvalues in context of PDEs
35P20 Asymptotic distributions of eigenvalues in context of PDEs
58J99 Partial differential equations on manifolds; differential operators
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] H. Weyl, Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen, Math. Ann. 71 (1911), 441-469. · JFM 43.0436.01
[2] Mark Kac, Can one hear the shape of a drum?, Amer. Math. Monthly 73 (1966), no. 4, 1 – 23. · Zbl 0139.05603 · doi:10.2307/2313748
[3] S. Minakshisundaram and A. Pleijel, Some properties of the eigenfunctions of the Laplace-operator on Riemannian manifolds, Canad. J. Math 1 (1949), 242-256. MR 11, 108. · Zbl 0041.42701
[4] Vojislav G. Avakumović, Über die Eigenfunktionen auf geschlossenen Riemannschen Mannigfaltigkeiten, Math. Z. 65 (1956), 327 – 344 (German). · Zbl 0070.32601 · doi:10.1007/BF01473886
[5] Lars Hörmander, The spectral function of an elliptic operator, Acta Math. 121 (1968), 193 – 218. · Zbl 0164.13201 · doi:10.1007/BF02391913
[6] A. Martin, Bound states in the strong coupling limit, Helv. Phys. Acta 45 (1972), 140-148.
[7] Hideo Tamura, The asymptotic eigenvalue distribution for non-smooth elliptic operators, Proc. Japan Acad. 50 (1974), 19 – 22. · Zbl 0312.35058
[8] M. Š. Birman and V. V. Borzov, On the asymptotic formula for the discrete spectrum of certain singular differential operators, Problemy Mat. Fiz., vyp. 5, Izdat. Leningrad. Gos Univ. Leningrad, 1971, pp. 24-38 = Topics in Math. Phys., no. 5, Plenum Press, New York, 1972, pp. 19-30; Consultants Bureau Transl., pp. 1-18. MR 46 #726.
[9] Michael Cwikel, Weak type estimates for singular values and the number of bound states of Schrödinger operators, Ann. of Math. (2) 106 (1977), no. 1, 93 – 100. · Zbl 0362.47006 · doi:10.2307/1971160
[10] Barry Simon, Analysis with weak trace ideals and the number of bound states of Schrödinger operators, Trans. Amer. Math. Soc. 224 (1976), no. 2, 367 – 380. · Zbl 0348.47017
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.