TY - JOUR

T1 - Semiclassical limit for the Schrödinger-Poisson equation in a crystal

AU - Bechouche, Philippe

AU - Mauser, Norbert

AU - Poupaud, Frederic

N1 - DOI: 10.1002/cpa.3004
Affiliations: Universität Wien, Institut für Mathematik, Strudlhofgasse 4, A-1090 Wien, Austria; Univ. de Nice Sophia-Antipolis, Laboratoire J. A. DieudonneŽ, UMR CNRS No. 6621, Parc Valrose, F-06108 Nice Cedex 2, France
Adressen: Bechouche, P.; Universität Wien; Institut für Mathematik; Strudlhofgasse 4 A-1090 Wien, Austria; email: [email protected]
Source-File: 506Scopus.csv
Import aus Scopus: 2-s2.0-0035593772
Importdatum: 24.01.2007 11:27:47
22.10.2007: Datenanforderung 1920 (Import Sachbearbeiter)
04.01.2008: Datenanforderung 2054 (Import Sachbearbeiter)

PY - 2001

Y1 - 2001

N2 - We give a mathematically rigorous theory for the limit from a weakly nonlinear Schrošdinger equation with both periodic and nonperiodic potential to the semiclassical version of the Vlasov equation. To this end we perform simultaneously a classical limit (vanishing Planck constant) and a homogenization limit of the periodic structure (vanishing lattice length taken proportional to the Planck constant). We introduce a new variant of Wigner transforms, namely the "Wigner-Bloch series," as an adaptation of the Wigner series for density matrices related to two different "energy bands." Another essential tool is estimates on the commutators of the projectors into the Floquet subspaces ("band subspaces") and the multiplicative potential operator that destroy the invariance of these band subspaces under the periodic Hamiltonian. We assume the initial data to be concentrated in isolated bands but allow for band-crossing of the other bands, which is the generic situation in more than one space dimension. The nonperiodic potential is obtained from a coupling to the Poisson equation; i.e., we take into account the self-consistent Coulomb interaction. Our results also hold for the easier linear case where this potential is given. We hence give the first rigorous derivation of the (nonlinear) "semiclassical equations" of solid state physics widely used to describe the dynamics of electrons in semiconductors. Œ 2001 John Wiley & Sons, Inc.

AB - We give a mathematically rigorous theory for the limit from a weakly nonlinear Schrošdinger equation with both periodic and nonperiodic potential to the semiclassical version of the Vlasov equation. To this end we perform simultaneously a classical limit (vanishing Planck constant) and a homogenization limit of the periodic structure (vanishing lattice length taken proportional to the Planck constant). We introduce a new variant of Wigner transforms, namely the "Wigner-Bloch series," as an adaptation of the Wigner series for density matrices related to two different "energy bands." Another essential tool is estimates on the commutators of the projectors into the Floquet subspaces ("band subspaces") and the multiplicative potential operator that destroy the invariance of these band subspaces under the periodic Hamiltonian. We assume the initial data to be concentrated in isolated bands but allow for band-crossing of the other bands, which is the generic situation in more than one space dimension. The nonperiodic potential is obtained from a coupling to the Poisson equation; i.e., we take into account the self-consistent Coulomb interaction. Our results also hold for the easier linear case where this potential is given. We hence give the first rigorous derivation of the (nonlinear) "semiclassical equations" of solid state physics widely used to describe the dynamics of electrons in semiconductors. Œ 2001 John Wiley & Sons, Inc.

U2 - 10.1002/cpa.3004

DO - 10.1002/cpa.3004

M3 - Article

VL - 54

SP - 851

EP - 890

JO - Communications on Pure and Applied Mathematics

JF - Communications on Pure and Applied Mathematics

SN - 0010-3640

IS - 7

ER -