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Periodic Homogenization and Effective Mass Theorems for the Schrödinger Equation

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Book cover Quantum Transport

Part of the book series: Lecture Notes in Mathematics ((LNMCIME,volume 1946))

Abstract

The goal of this course is to give an introduction to periodic homogenization theory with an emphasis on applications to Schrödinger equation. We review the formal method of two-scale asymptotic expansions, then discuss the rigorous two-scale convergence method as well as the Bloch wave decomposition. Eventually these tools will be apply to the Schrödinger equation with a periodic potential perturbed by a small macroscopic potential. The notion of effective mass for the one electron model in solid state physics will be derived. Localization effects will also be considered

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Author information

Authors and Affiliations

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

    Grégoire Allaire

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  1. Grégoire Allaire
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Editors and Affiliations

  1. Institut de Mathématiques, Université de Toulouse, 118 route de Narbonne, 31062, Toulouse Cedex 9, France

    Naoufel Ben Abdallah

  2. Dipartimento di Matematica Applicata, Università di Firenze, Via S.Marta 3, 50139, Firenze, Italy

    Giovanni Frosali

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Allaire, G. (2008). Periodic Homogenization and Effective Mass Theorems for the Schrödinger Equation. In: Abdallah, N.B., Frosali, G. (eds) Quantum Transport. Lecture Notes in Mathematics, vol 1946. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-79574-2_1

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