Tunisian Journal of Mathematics

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

Mouez Dimassi

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Abstract

In the semiclassical regime (i.e., ϵ0), 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.

Article information

Source
Tunisian J. Math., Volume 2, Number 1 (2020), 197-215.

Dates
Received: 10 October 2018
Revised: 17 November 2018
Accepted: 2 December 2018
First available in Project Euclid: 2 April 2019

Permanent link to this document
https://projecteuclid.org/euclid.tunis/1554170465

Digital Object Identifier
doi:10.2140/tunis.2020.2.197

Mathematical Reviews number (MathSciNet)
MR3933395

Zentralblatt MATH identifier
07074074

Subjects
Primary: 35P20: Asymptotic distribution of eigenvalues and eigenfunctions 47A55: Perturbation theory [See also 47H14, 58J37, 70H09, 81Q15] 47N50: Applications in the physical sciences 81Q10: Selfadjoint operator theory in quantum theory, including spectral analysis 81Q15: Perturbation theories for operators and differential equations

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

Citation

Dimassi, Mouez. Semiclassical approximation of the magnetic Schrödinger operator on a strip: dynamics and spectrum. Tunisian J. Math. 2 (2020), no. 1, 197--215. doi:10.2140/tunis.2020.2.197. https://projecteuclid.org/euclid.tunis/1554170465


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