Semi-classical models for the Schrödinger equation with periodic potentials and band crossings

Lihui Chai; Shi Jin; Qin Li

doi:10.3934/krm.2013.6.505

Article Contents

2013, Volume 6, Issue 3: 505-532. Doi: 10.3934/krm.2013.6.505

This issue Previous Article Global existence and large time behavior of solutions to the electric-magnetohydrodynamic equations Next Article On a regularization of the magnetic gas dynamics system of equations

Semi-classical models for the Schrödinger equation with periodic potentials and band crossings

1.
Department of Mathematical Science, Tsinghua University, Bejing 100084
2.
Department of Mathematics, Institute of Natural Sciences, and MOE Key Lab in Scientific and Engineering Computing, Shanghai Jiao Tong University, Shanghai 200240
3.
Department of Mathematics, University of Wisconsin-Madison, Madison, WI 53706

Received: January 31, 2013

Revised: February 28, 2013

Published: August 2013

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Abstract

The Bloch decomposition plays a fundamental role in the study of quantum mechanics and wave propagation in periodic media. Most of the homogenization theory developed for the study of high frequency or semi-classical limit for these problems assumes no crossing of the Bloch bands, resulting in a classical Liouville equation in the limit along each Bloch band.
In this article, we derive semi-classical models for the Schrödinger equation in periodic media that take into account band crossings, which is important to describe quantum transitions between Bloch bands. Our idea is still based on the Wigner transform (on the Bloch eigenfunctions), but in taking the semi-classical approximation, we retain the off-diagonal entries of the Wigner matrix, which cannot be ignored near the points of band crossings. This results in coupled inhomogeneous Liouville systems that can suitably describe quantum tunneling between bands that are not well-separated. We also develop a domain decomposition method that couples these semi-classical models with the classical Liouville equations (valid away from zones of band crossings) for a multiscale computation. Solutions of these models are numerically compared with those of the Schrödinger equation to justify the validity of these new models for band-crossings.

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Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

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