Abstract
This article studies the Schrödinger equation for an electron in a lattice of ions with an external magnetic field. In a suitable physical scaling the ionic potential becomes rapidly oscillating, and one can build asymptotic solutions for the limit of zero magnetic field by multiple scale methods from “homogenization.” For the time-dependent Schrödinger equation this construction yields wave packets which follow the trajectories of the “semiclassical model.” For the time-independent equation one gets asymptotic eigenfunctions (or “quasimodes”) for the energy levels predicted by Onsager's relation.
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Communicated by A. Jaffe
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Guillot, J.C., Ralston, J. & Trubowitz, E. Semi-classical asymptotics in solid state physics. Commun.Math. Phys. 116, 401–415 (1988). https://doi.org/10.1007/BF01229201
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DOI: https://doi.org/10.1007/BF01229201