We consider the Schrödinger operator with electric potential V, which decays at infinity, and magnetic potential A. We study the asymptotic behaviour for large values of the electric field coupling constant of the eigenvalues situated under the essential-spectrum lower bound. We concentrate on the cases of rapidly decaying V (e.g. V ∈ L m/2(ℝm) for m ≥ 3) and arbitrary A, or slowly decaying V (i.e. V(x ∼ |x|−α, α ∈ (0,2), as |x| → ∞) and magnetic potentials A corresponding to constant magnetic fields B = curl A.
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AvronJ., HerbstI., and SimonB., Duke Math. J. 45, 847 (1978).
CyconH. L., FroeseR. G., KirschW., and SimonB., Schrödinger Operators, Springer, Berlin, 1987.
RaikovG. D., C.R. Acad. Sci. Paris 309, 559 (1989).
IwatsukaA., J. Math. Kyoto Univ. 26, 357 (1986).
BirmanM. Sh. and SolomjakM. Z., Quantitative Analysis in Sobolev Imbedding Theorems and Applications to Spectral Theory. Amer. Math. Soc. Transl. Ser. 2 114, AMS, Providence, R.I., 1980.
ReedM. and SimonB. Methods of Modern Mathematical Physics. IV. Analysis of Operators. Academic Press, New York, 1978.
TamuraH., J. Math. Soc. Jpn. 36, 355 (1984).
LiebE. H., Bull. Amer. Math. Soc. 82, 751 (1976).
Colin de VerdièreY., Comm. Math. Phys. 305, 327 (1986).
RozenbljumG. V., Math. Notes 21, 222 (1977).
TamuraH., Osaka J. Math. 25, 633 (1988).
Partially supported by the Ministry of Culture, Science and Education under Grant No. 52.
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Raikov, G.D. Strong electric field eigenvalue asymptotics for the Schrödinger operator with electromagnetic potential. Letters in Mathematical Physics 21, 41–49 (1991). https://doi.org/10.1007/BF00414634
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