On the eigenvalues for slowly varying perturbations of a periodic Schrödinger operator

Article type: Research Article

Authors: Marx, Magali

Affiliations: Institut Fourier, 100, rue des Mathématiques, BP 74, 38402 Saint-Martin d'Hères, France E-mail: [email protected]; Tel.: +33 4 76 51 35 28; Fax: +33 4 76 51 44 78

Abstract: In this paper, I consider one-dimensional periodic Schrödinger operators perturbed by a slowly decaying potential. In the adiabatic limit, I give an asymptotic expansion of the eigenvalues in the gaps of the periodic operator. When one slides the perturbation along the periodic potential, these eigenvalues oscillate. I compute the exponentially small amplitude of the oscillations.

Keywords: eigenvalues, complex WKB method, scattering, adiabatic perturbations

Journal: Asymptotic Analysis, vol. 48, no. 4, pp. 295-357, 2006