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January  2006, 6(1): 1-16. doi: 10.3934/dcdsb.2006.6.1

Homogenization of the Schrödinger equation with a time oscillating potential

Grégoire Allaire 1, and  M. Vanninathan 2,

1. 

Centre de Mathématiques Appliquées, Ecole Polytechnique, 91128 PALAISEAU Cedex, France
2. 

TIFR Center, P.O. Box 1234, Bangalore - 560012, India

Received  March 2005 Revised  August 2005 Published  October 2005

We study the homogenization of a Schrödinger equation in a periodic medium with a time dependent potential. This is a model for semiconductors excited by an external electromagnetic wave. We prove that, for a suitable choice of oscillating (both in time and space) potential, one can partially transfer electrons from one Bloch band to another. This justifies the famous "Fermi golden rule" for the transition probability between two such states which is at the basis of various optical properties of semiconductors. Our method is based on a combination of classical homogenization techniques (two-scale convergence and suitable oscillating test functions) and of Bloch waves theory.
Keywords: Bloch waves., Schrödinger equation, Homogenization.
Mathematics Subject Classification: Primary: 35B27; Secondary: 35Q4.
Citation: Grégoire Allaire, M. Vanninathan. Homogenization of the Schrödinger equation with a time oscillating potential. Discrete and Continuous Dynamical Systems - B, 2006, 6 (1) : 1-16. doi: 10.3934/dcdsb.2006.6.1
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