Abstract
If λ k is thek th eigenvalue for the Dirichlet boundary problem on a bounded domain in ℝn, H. Weyl's asymptotic formula asserts that\(\lambda _k \sim C_n \left( {\frac{k}{{V(D)}}} \right)^{2/n} \), hence\(\sum\limits_{i = 1}^k {\lambda _i \sim \frac{{nC_n }}{{n + 2}}k^{\frac{{n + 2}}{n}} V(D)^{ - 2/n} } \). We prove that for any domain and for all\(\sum\limits_{i = 1}^k {\lambda _i \geqq \frac{{nC_n }}{{n + 2}}k^{\frac{{n + 2}}{n}} V(D)^{ - 2/n} } \). A simple proof for the upper bound of the number of eigenvalues less than or equal to -α for the operator Δ−V(x) defined on ℝn (n≧3) in terms of\(\int\limits_{\mathbb{R}^n } {(V + \alpha )_ - ^{n/2} } \) is also provided.
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Aubin, T.: Problèmes isopérimétriques et espaces de Sobolev. J. Diff. Geom.11, 573–598 (1976)
Cheng, S.Y., Li, P.: Heat kernel estimates and lower bound of eigenvalues. Commun. Math. Helv. (56)3, 327–338 (1981)
Cwikel, W.: Weak type estimates for singular values and the number of bound states of Schrödinger operators. Ann. Math.106, 93–100 (1977)
Lieb, E.: The number of bound states of one-body Schrödinger operators and the Weyl problem. Proc. Sym. Pure Math.36, 241–252 (1980)
Lieb, E., Thirring, W.: Inequalities for the moments of the eigenvalues of the Schrödinger equation and their relation to Sobolev inequalities. Studies in Math. Phys.: Essay in Honor of Valentine Bargmann. Princeton, NJ: Princeton University Press 1976
Pólya, G.: On the eigenvalues of vibrating membranes. Proc. London Math. Soc. (3)11, 419–433 (1961)
Rosenbljum, G.V.: Distribution of the discrete spectrum of singular operator. Dokl. Akad. Nauk SSSR202, 1012–1015 (1972)
Simon, B.: Weak trace ideals and the number of bound states of Schrödinger operators. Trans. Am. Math. Soc.224, 367–380 (1976)
Lieb, E.: Bounds on the eigenvalues of the Laplace and Schroedinger operators. Bull. AMS82, 751–753 (1976)
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Communicated by B. Simon
Research partially supported by a Sloan Fellowship and NSF Grant No. 81-07911-A1
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Li, P., Yau, ST. On the Schrödinger equation and the eigenvalue problem. Commun.Math. Phys. 88, 309–318 (1983). https://doi.org/10.1007/BF01213210
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DOI: https://doi.org/10.1007/BF01213210