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On the eigenvalues for slowly varying perturbations of a periodic Schrödinger operator. (English) Zbl 1124.34063

One-dimensional periodic Schrödinger operators perturbed by a slowly decaying potential are considered. An asymptotic expansion of the eigenvalues in the gaps of the periodic operator is given. If one slides the perturbation along the periodic potential, these eigenvalues oscillate. An exponentially small amplitude of the oscillations has been computed.

MSC:

34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
34L25 Scattering theory, inverse scattering involving ordinary differential operators
34L15 Eigenvalues, estimation of eigenvalues, upper and lower bounds of ordinary differential operators
81Q15 Perturbation theories for operators and differential equations in quantum theory
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