Vol. 2, No. 1, 2020

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

Mouez Dimassi

Vol. 2 (2020), No. 1, 197–215
Abstract

In the semiclassical regime (i.e., ϵ 0), we study the effect of a slowly varying potential V (ϵt,ϵz) on the magnetic Schrödinger operator P = Dx2 + (Dz + μx)2 on a strip [a,a] × 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 (ϵt,ϵz) for ϵ small enough. All our results depend on the eigenvalues corresponding to Dx2 + (μx + k)2 on L2([a,a]) with Dirichlet boundary condition.

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