Discrete and Continuous Dynamical Systems - Series B (DCDS-B)

Homogenization of the Schrödinger equation with a time oscillating potential

Pages: 1 - 16, Volume 6, Issue 1, January 2006      doi:10.3934/dcdsb.2006.6.1

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Grégoire Allaire - Centre de Mathématiques Appliquées, Ecole Polytechnique, 91128 PALAISEAU Cedex, France (email)
M. Vanninathan - TIFR Center, P.O. Box 1234, Bangalore - 560012, India (email)

Abstract: 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:  Homogenization, Schrödinger equation, Bloch waves.
Mathematics Subject Classification:  Primary: 35B27; Secondary: 35Q40.

Received: March 2005;      Revised: August 2005;      Available Online: October 2005.