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

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Tunisian Journal of Mathematics an international publication organized by the Tunisian Mathematical Society msp Semiclassical approximation of the magnetic Schrödinger operator on a strip: dynamics and spectrum Mouez Dimassi 2020 vol. 2 no. 1 msp TUNISIAN JOURNAL OF MATHEMATICS Vol. 2, No. 1, 2020 dx.doi.org/10.2140/tunis.2020.2.197 Semiclassical approximation of the magnetic Schrödinger operator on a strip: dynamics and spectrum Mouez Dimassi In the semiclassical regime (i.e., & 0), we study the effect of a slowly varying potential V(t,z) on the magnetic Schrödinger operator P = D2 x + (Dz + µx)2 on a strip [−a,a] × Rz. 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(t,z) for small enough. All our results depend on the eigenvalues corresponding to D2 x + (µx + k)2 on L2([−a,a]) with Dirichlet boundary condition. 1. Introduction The quantum dynamics of an electron in a strip subject to an uniform magnetic ﬁeld and an external slowly varying potential is governed by the Schrödinger operator H() := P +V(t,z) = D2 x +(Dz +µx)2+V(t,z), Dν = 1 i ∂ν, ,µ>0, where µ is proportional to the strength of the magnetic ﬁeld and is a small pa- rameter. The potential V is assumed to be smooth and real valued. The operator P = D2 x + (Dz + µx)2, is deﬁned on {u ∈ H2(Ca); u|∂Ca = 0}, where H2(Ca) denotes the second order Sobolev space on a strip Ca := {(x, z) ∈ R

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