Semiclassical approximation of the magnetic Schrödinger operator on a strip : dynamics and spectrum

In the semiclassical regime (i.e., $ϵ ↘ 0$ ), we study the effect of a slowly varying potential $V (ϵ t, ϵ z)$ on the magnetic Schrödinger operator $P = D_{x}^{2} + {(D_{z} + μ x)}^{2}$ on a strip $[- a, a] \times ℝ_{z}$ . The potential $V (t, z)$ is assumed to be smooth. We derive the semiclassical dynamics and we describe the asymptotic structure of the spectrum and the resonances of the operator $P + V (ϵ t, ϵ z)$ for $ϵ$ small enough. All our results depend on the eigenvalues corresponding to $D_{x}^{2} + {(μ x + k)}^{2}$ on $L^{2} ([- a, a])$ with Dirichlet boundary condition.

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Keywords

semiclassical analysis, periodic Schrödinger operator, Bohr–Sommerfeld quantization, spectral shift function, asymptotic expansions, limiting absorption theorem

Mathematical Subject Classification 2010

Primary: 35P20, 47A55, 47N50, 81Q10, 81Q15

Milestones

Received: 10 October 2018

Revised: 17 November 2018

Accepted: 2 December 2018

Published: 22 March 2019

Authors

Mouez Dimassi
IMB Université Bordeaux I 33405 Talence France

Vol. 2, No. 1, 2020

Mouez Dimassi

Abstract

PDF Access Denied

Keywords

Mathematical Subject Classification 2010

Milestones

Authors