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On the spectrum of the Schrödinger operator perturbed by a rapidly oscillating potential. (Russian, English) Zbl 1134.34337

J. Math. Sci., New York 139, No. 1, 6243-6322 (2006); translation from Probl. Mat. Anal. 33, 13-78 (2006).
Summary: We study the spectrum of the one-dimensional Schrödinger operator perturbed by a rapidly oscillating potential. The oscillation period is a small parameter. We find explicitly the essential spectrum and study the existence of the discrete spectrum. Complete asymptotic expansions of the eigenvalues and corresponding eigenfunctions are constructed.

MSC:

34L20 Asymptotic distribution of eigenvalues, asymptotic theory of eigenfunctions for ordinary differential operators
34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
47E05 General theory of ordinary differential operators
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
81Q15 Perturbation theories for operators and differential equations in quantum theory
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