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Homogenization of the Schrödinger Equation and Effective Mass Theorems

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Abstract

We study the homogenization of a Schrödinger equation with a large periodic potential: denoting by ∈ the period, the potential is scaled as ∈−2. We obtain a rigorous derivation of so-called effective mass theorems in solid state physics. More precisely, for well-prepared initial data concentrating on a Bloch eigenfunction we prove that the solution is approximately the product of a fast oscillating Bloch eigenfunction and of a slowly varying solution of an homogenized Schrödinger equation. The homogenized coefficients depend on the chosen Bloch eigenvalue, and the homogenized solution may experience a large drift. The homogenized limit may be a system of equations having dimension equal to the multiplicity of the Bloch eigenvalue. Our method is based on a combination of classical homogenization techniques (two-scale convergence and suitable oscillating test functions) and of Bloch waves decomposition.

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Authors and Affiliations

  1. Centre de Mathématiques Appliquées, Ecole Polytechnique, 91128, Palaiseau, France

    Grégoire Allaire

  2. Narvik Institute of Technology, HiN, P.O. Box 385, 8505, Narvik, Norway

    Andrey Piatnitski

  3. P.N.Lebedev Physical Institute RAS, Leninski prospect 53, Moscow, 117333, Russia

    Andrey Piatnitski

Authors
  1. Grégoire Allaire
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  2. Andrey Piatnitski
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Correspondence to Grégoire Allaire.

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Communicated by B. Simon

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Allaire, G., Piatnitski, A. Homogenization of the Schrödinger Equation and Effective Mass Theorems. Commun. Math. Phys. 258, 1–22 (2005). https://doi.org/10.1007/s00220-005-1329-2

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  • DOI: https://doi.org/10.1007/s00220-005-1329-2

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