#### 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 $V\phantom{\rule{0.3em}{0ex}}\left(ϵt,ϵz\right)$ on the magnetic Schrödinger operator $P={D}_{\phantom{\rule{0.3em}{0ex}}x}^{2}+{\left({D}_{z}+\mu x\right)}^{2}$ on a strip $\left[-a,a\right]×{ℝ}_{z}$. The potential $V\phantom{\rule{0.3em}{0ex}}\left(t,z\right)$ 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 $P+V\phantom{\rule{0.3em}{0ex}}\left(ϵt,ϵz\right)$ for $ϵ$ small enough. All our results depend on the eigenvalues corresponding to ${D}_{\phantom{\rule{0.3em}{0ex}}x}^{2}+{\left(\mu x+k\right)}^{2}$ on ${L}^{2}\left(\left[-a,a\right]\right)$ with Dirichlet boundary condition.

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