An Effective Mass Theorem for the Bidimensional Electron Gas in a Strong Magnetic Field

• Published in: Communications in mathematical physics Vol. 292; no. 3; pp. 829 - 870
• Main Authors: Delebecque, Fanny, Méhats, Florian
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer-Verlag 2009; Springer Verlag
• Subjects: Analysis of PDEs; Classical and Quantum Gravitation; Complex Systems; Condensed Matter; Mathematical and Computational Physics; Mathematical Physics; Mathematics; Other; Physics; Physics and Astronomy; Quantum Physics; Relativity Theory; Theoretical; Initial System; Maximal Solution; Limit System; Gronwall Lemma; Unique Global Solution
• ISSN: 0010-3616; 1432-0916
• DOI: 10.1007/s00220-009-0868-3

We study the limiting behavior of a singularly perturbed Schrödinger-Poisson system describing a 3-dimensional electron gas strongly confined in the vicinity of a plane (
x
,
y
) and subject to a strong uniform magnetic field in the plane of the gas. The coupled effects of the confinement and of the magnetic field induce fast oscillations in time that need to be averaged out. We obtain at the limit a system of 2-dimensional Schrödinger equations in the plane (
x
,
y
), coupled through an effective selfconsistent electrical potential. In the direction perpendicular to the magnetic field, the electron mass is modified by the field, as the result of an averaging of the cyclotron motion. The main tools of the analysis are the adaptation of the second order long-time averaging theory of ODEs to our PDEs context, and the use of a Sobolev scale adapted to the confinement operator.