ID | 54721 |
FullText URL | |
Author |
Dimassi, Mouez
Universit´e Bordeaux I, Institut de Math´ematiques de Bordeaux
Anh Tuan Duong
Department of Mathematics, Hanoi National University of Education
|
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 h1/2. 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
|
Published Date | 2017-01
|
Publication Title |
Mathematical Journal of Okayama University
|
Volume | volume59
|
Issue | issue1
|
Publisher | Department of Mathematics, Faculty of Science, Okayama University
|
Start Page | 149
|
End Page | 174
|
ISSN | 0030-1566
|
NCID | AA00723502
|
Content Type |
Journal Article
|
Official Url | http://www.math.okayama-u.ac.jp/mjou/
|
language |
English
|
Copyright Holders | Copyright©2017 by the Editorial Board of Mathematical Journal of Okayama University
|
File Version | publisher
|
Refereed |
True
|
Submission Path | mjou/vol59/iss1/12
|
JaLCDOI |