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Semi-classical asymptotics in solid state physics. (English) Zbl 0672.35014
Summary: 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 “semi-classical model.” For the time- independent equation the authors obtain asymptotic eigenfunctions (or “quasimodes”) for the energy levels predicted by Onsager’s relation.

MSC:
35J10 Schrödinger operator, Schrödinger equation
35Q99 Partial differential equations of mathematical physics and other areas of application
35P20 Asymptotic distributions of eigenvalues in context of PDEs
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