Journal Article10.1007/S00220-005-1329-2
Homogenization of the Schrödinger Equation and Effective Mass Theorems
141
TL;DR: In this paper, the authors derived effective mass theorems in solid state physics for a Schrodinger equation with a large periodic potential, denoting by ∈ the period, the potential is scaled as ∈−2.
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Abstract: We study the homogenization of a Schrodinger 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 Schrodinger 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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References
•Book
Introduction to solid state physics
Charles Kittel
- 01 Jan 1953
TL;DR: In this paper, the Hartree-Fock Approximation of many-body techniques and the Electron Gas Polarons and Electron-phonon Interaction are discussed.
•Book
Perturbation theory for linear operators
Tosio Kato
- 01 Jan 1966
TL;DR: The monograph by T Kato as discussed by the authors is an excellent reference work in the theory of linear operators in Banach and Hilbert spaces and is a thoroughly worthwhile reference work both for graduate students in functional analysis as well as for researchers in perturbation, spectral, and scattering theory.
22K
•Book
Asymptotic analysis for periodic structures
Alain Bensoussan,Alain Bensoussan,Jacques-Louis Lions,George Papanicolaou +3 more
- 01 Jan 1978
TL;DR: In this article, the authors give a systematic introduction of multiple scale methods for partial differential equations, including their original use for rigorous mathematical analysis in elliptic, parabolic, and hyperbolic problems, and with the use of probabilistic methods when appropriate.
5.8K
•Book
Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert
Haim Brezis
- 01 Jan 1973
TL;DR: In this article, Operateurs Maximaux Monotones: Et Semi-Groupes De Contractions Dans Les Espaces De Hllbert are described and discussed. But the focus is not on the performance of the operators.
3.6K