Journal Article Scattering and semi-classical asymptotics for periodic Schrödinger operators with oscillating decaying potential

Dimassi, Mouez  ,  Anh Tuan Duong,

59 ( 1 )  , pp.149 - 174 , 2017-01 , Department of Mathematics, Faculty of Science, Okayama University
In the semi-classical regime (i.e., <i>h</i> &searr; 0), we study the effect of an oscillating decaying potential <i>V</i> (<i>hy, y</i>) on the periodic Schr&ouml;dinger operator <i>H</i>. The potential <i>V</i> (<i>x, y</i>) is assumed to be smooth, periodic with respect to <i>y</i> and tends to zero as |<i>x</i>| &rarr; &infin;. We prove the existence of <i>O</i>(<i>h<sup>−n</sup></i>) eigenvalues in each gap of the operator <i>H</i> + <i>V</i> (<i>hy, y</i>). 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 <i>h</i> of the spectral shift function corresponding to the pair (<i>H</i> + <i>V</i> (<i>hy, y</i>),<i>H</i>). 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 <i>h<sup>1/2</sup></i>. All our results depend on the Floquet eigenvalues corresponding to the periodic Schr&ouml;dinger operator <i>H</i> +<i>V</i> (<i>x, y</i>), (here <i>x</i> is a parameter).

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