Mathematical Journal of Okayama University 59巻 1号

2017-01 発行

Dimassi, Mouez
Universit´e Bordeaux I, Institut de Math´ematiques de Bordeaux

Anh Tuan Duong
Department of Mathematics, Hanoi National University of Education

Publication Date

2017-01

Abstract

In the semi-classical regime (i.e., *h* ↘ 0), we study the effect of an oscillating decaying potential *V* (*hy, y*) on the periodic Schrödinger operator *H*. The potential *V* (*x, y*) is assumed to be smooth, periodic with respect to *y* and tends to zero as |*x*| → ∞. We prove the existence of *O*(*h*^{−n}) eigenvalues in each gap of the operator *H* + *V* (*hy, y*). We also establish a Weyl type asymptotics formula of the counting function of eigenvalues with optimal remainder estimate. We give a weak and pointwise asymptotic expansions in powers of *h* of the spectral shift function corresponding to the pair (*H* + *V* (*hy, y*),*H*). Finally, under some analytic assumption on the potential V we prove the existence of shape resonances, and we give their asymptotic expansions in powers of *h*^{1/2}. All our results depend on the Floquet eigenvalues corresponding to the periodic Schrödinger operator *H* +*V* (*x, y*), (here *x* is a parameter).

Keywords

Periodic Schrödinger operator

oscillating potential

spectral shift function

asymptotic expansions

resonances