Discrete Spectrum of the Periodic Schrödinger Operator with a Variable Metric Perturbed by a Nonnegative Potential

M. Sh. Birman¹ ; V. A. Sloushch¹

¹ Department of Physics, St. Petersburg State University, Ul’yanovskaya 3, Petrodvorets, St. Petersburg, 198504, RUSSIA

Mathematical modelling of natural phenomena, Tome 5 (2010) no. 4, pp. 32-53.

Résumé

We study discrete spectrum in spectral gaps of an elliptic periodic second order differential operator in L2(ℝd) perturbed by a decaying potential. It is assumed that a perturbation is nonnegative and has a power-like behavior at infinity. We find asymptotics in the large coupling constant limit for the number of eigenvalues of the perturbed operator that have crossed a given point inside the gap or the edge of the gap. The corresponding asymptotics is power-like and depends on the observation point.

Détail

DOI : 10.1051/mmnp/20105402

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