Abstract
In this Letter we prove inequalities on the number of bound states of Schrödinger operators by improving the method of Li and Yau relying on heat kernel estimates by Sobolev inequalities. We extend the range of applications and obtain better estimates. In particular, our technique yields a simple proof of Bargmann's bound and very good numerical values.
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References
Lieb, E., The number of bound states of one-body Schrödinger operators and the Weyl problem, Proc. Symp. Pure Math. 36, 241–252 (1980).
Li, P. and Yau, S. T., On the Schrödinger equation and the eigenvalue problem, Commun. Math. Phys. 88, 309–318, (1983).
Lieb, E., Bounds of the eigenvalue of the Laplace and Schrödinger operators, Bull. AMS 82, 751–753 (1976).
Cwickel, M., Weak type estimats for singular values and the number of bound states of Schrödinger operators, Ann. Math. 106, 93–100 (1977).
Rosenbljum, G. V., Distribution of the discrete spectrum of singular operators, Dokl. Akad. Nauk. SSR, 202, 1012–1015 (1972).
Simon, B., On the growth of the number of bound states which increase in potential strength, J. Math. Phys. 10, 1123–1126 (1969).
Martin, A., A bound of the total number of bound states in a potential, CERN Preprint TH 2085, 17 October 1975, unpublished.
Lieb, E., Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. Math. 118, 349–374 (1983).
Reed, M. and Simon, B., Methods of Modern Mathematical Physics IV Analysis of Operators, Academic Press, New York, 1978.
Jost, R. and Pais, A., On the scattering of a particle by a static potential, Phys. Rev. 82, 840–850 (1951).
Glaser, V., Grosse, H., Martin, A., and Thirring, W., A family of optimal condition for the absence of bound states in a potential, in E. Lieb, B. Simon, and A.S. Wightman (eds.), Studies in Mathematical Physics, Princeton University Press, 1976, pp. 169–194.
Glaser, V., Grosse, H., and Martin, A., Bounds of the number of eigenvalues of the Schrödinger operator, Commun. Math. Phys. 59, 197–212 (1978).
Simon, B., Functional Integration and Quantum Physics, Academic Press, New York, 1979.
Reed, M. and Simon, B., Methods of Modern Mathematical Physics II, Fourier Analysis, Self-Adjointness, Academic Press, New York (1975).
Lieb, E. and Thirring, W., Inequalities for the moments of the Schrödinger Hamiltonians and their relation to obolev inequalities, in E. Lieb, B. Simon, and A. S. Wightman (eds.), Studies in Mathematical Physics, Princeton University Press, 1976.
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Blanchard, P., Stubbe, J. & Rezende, J. New estimates on the number of bound states of Schrödinger operators. Lett Math Phys 14, 215–225 (1987). https://doi.org/10.1007/BF00416851
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DOI: https://doi.org/10.1007/BF00416851