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

Abstract

In the semiclassical regime (i.e., $ϵ\searrow 0$), we study the effect of a slowly varying potential $V\left(ϵt,ϵz\right)$ on the magnetic Schrödinger operator $P=D_x^2+{\left(D_z+\mu x\right)}^2$ on a strip $\left[-a,a\right]\times {ℝ}_z$. The potential $V\left(t,z\right)$ 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 $P+V\left(ϵt,ϵz\right)$ for $ϵ$ small enough. All our results depend on the eigenvalues corresponding to $D_x^2+{\left(\mu x+k\right)}^2$ on $L^2\left(\left[-a,a\right]\right)$ with Dirichlet boundary condition.