Annales Henri Poincaré

, Volume 7, Issue 2, pp 335–363

Energetic and Dynamic Properties of a Quantum Particle in a Spatially Random Magnetic Field with Constant Correlations along one Direction

Original Paper

DOI: 10.1007/s00023-005-0251-7

Cite this article as:
Leschke, H., Warzel, S. & Weichlein, A. Ann. Henri Poincaré (2006) 7: 335. doi:10.1007/s00023-005-0251-7

Abstract.

We consider an electrically charged particle on the Euclidean plane subjected to a perpendicular magnetic field which depends only on one of the two Cartesian co-ordinates. For such a “unidirectionally constant” magnetic field (UMF), which otherwise may be random or not, we prove certain spectral and transport properties associated with the corresponding one-particle Schrödinger operator (without scalar potential) by analysing its “energy-band structure”. In particular, for an ergodic random UMF we provide conditions which ensure that the operator’s entire spectrum is almost surely absolutely continuous. This implies that, along the direction in which the random UMF is constant, the quantum-mechanical motion is almost surely ballistic, while in the perpendicular direction in the plane one has dynamical localization. The conditions are verified, for example, for Gaussian and Poissonian random UMF’s with non-zero mean-values. These results may be viewed as “random analogues” of results first obtained by A. Iwatsuka [Publ. RIMS, Kyoto Univ. 21 (1985) 385] and (non-rigorously) by J.E. Müller [Phys. Rev. Lett. 68 (1992) 385].

In memoriam

Heinz BAUER (31 January 1928 - 15 August 2002)

former Professor of Mathematics at the University

of Erlangen-Nürnberg

Communicated by Frank den Hollander

Copyright information

© Birkhäuser Verlag, Basel 2006

Authors and Affiliations

  • Hajo Leschke
    • 1
  • Simone Warzel
    • 1
    • 2
  • Alexandra Weichlein
    • 1
  1. 1.Institut für Theoretische PhysikUniversität Erlangen-NürnbergErlangenGermany
  2. 2.Princeton UniversityPrincetonUSA