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Asymptotic Distribution of Eigenvalues for Schrödinger Operators with Magnetic Fields

Published online by Cambridge University Press:  22 January 2016

Hideo Tamura
Hideo Tamura*
Affiliation:
Department of Applied Physics, Nagoya University Chikusa-ku, Nagoya 464, Japan
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The asymptotic distribution of eigenvalues has been studied by many authors for the Schrõdinger operators —Δ+V with scalar potential growing unboundedly at infinity. Let N(λ) be the number of eigenvalues less than λ of —Δ + V on L2Rnx). Under suitable assumptions on V(x), N(λ) obeys the following asymptotic formula:

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 105 , March 1987 , pp. 49 - 69
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1987

References

[ 1 ] Avron, J. Herbst, I. and Simon, B., Schrödinger operators with magnetic fields, I, general interactions, Duke Math. J., 45 (1978), 847–883.CrossRefGoogle Scholar
[ 2 ] Birman, M. Sh. and Solomyak, M. Z., Asymptotic behavior of the spectrum of differential equations, J. Soviet Math., 12 (1979), 247–283.CrossRefGoogle Scholar
[ 3 ] Combes, J. M., Schrader, R. and Seiler, R., Classical bounds and limits for energy distributions of hamiltonian operators in electromagnetic fields, Ann. Physics, 111 (1978), 1–18.CrossRefGoogle Scholar
[ 4 ] Dufresnoy, A., Un exemple de champ magnetiques dan Rv , Duke Math. J., 50 (1983), 729–734.CrossRefGoogle Scholar
[ 5 ] Hörmander, L., Hyoelliptic second order differential equations, Acta. Math., 119 (1967), 147–171.CrossRefGoogle Scholar
[ 6 ] A. Iwatsuka, Magnetic Schrödinger operators with compact resolvent, To appear in J. Math. Kyoto Univ.Google Scholar
[ 7 ] Karamata, J., Neuer Beweis und Verallgemeinerung der Tauberschen Sätze, J. Reine angew. Math., 164 (1931), 27–39.CrossRefGoogle Scholar
[ 8 ] Metivier, G., Fonction spectrale et valeurs propres d’une class d’operateurs non elliptiques, Comm. Partial Differ. Eqs., 1 (1976), 467–519.CrossRefGoogle Scholar
[ 9 ] Reed, M. and Simon, B., Methods of Modern Mathematical Physics, Vol. IV Analysis of Operators, Academic Press, 1978.Google Scholar
[10] Rosenbljum, G. V., Asymptotics of the eigenvalues of the Schrödinger operator, Math. USSR Sb., 22 (1974), 349–371.CrossRefGoogle Scholar
[11] Schechter, M., Spectra of Partial Differential Operators, North-Holland, 1971.Google Scholar
[12] Simon, B., Functional Integration and Quantum Physics, Academic Press, 1977.Google Scholar
[13] Simon, B., Non-classical eigenvalue asymptotics, J. Funct. Anal., 53 (1983), 84–98.CrossRefGoogle Scholar
[14] Sjöstrand, J., On the eigenvalues of a class of hypoelliptic operators, IV, Ann. Inst. Fourier Grenoble, 30 (1980), 109–169.CrossRefGoogle Scholar