Annales Henri Poincaré

, Volume 7, Issue 2, pp 335–363 | Cite as

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

  • Hajo LeschkeEmail author
  • Simone Warzel
  • Alexandra Weichlein
Original Paper


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


Magnetic Field Field Theory Elementary Particle Quantum Field Theory Transport Property 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Birkhäuser Verlag, Basel 2006

Authors and Affiliations

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

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