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MSC 1991: 58G40, 81S10, 20C35, 35Q40 Magnetic Bloch Analysis and Bochner-Laplacians
 

Summary: MSC 1991: 58G40, 81S10, 20C35, 35Q40
Magnetic Bloch Analysis and Bochner-Laplacians
J. Asch, H. Over*
, R. Seiler
Technische Universität Berlin, MA 7-1, Straße des 17. Juni 136, D-1000 Berlin 12
* Fritz Haber Institut, Faradayweg 4-6, D-1000 Berlin 33
Abstract
Hamiltonians for a particle on a manifold in a magnetic field are constructed as
Bochner-Laplacians. We show for the case of a torus and a given magnetic
field that they are in one to one correspondence with the constituents in the
Bloch decomposition of the unique Hamiltonian on the universal covering.
Introduction
We consider a Schrödinger Hamiltonian on a Riemannian manifold with
magnetic field and its relation to the corresponding Hamiltonian on the
universal covering manifold. This problem is motivated by and our results may
be useful for some questions around models for the Quantum-Hall-Effect
[TKNN], [A-S-Y], [A-K-P-S]. We review first the well known geometrical
construction of the Hilbert space of quantum mechanical states and the Hamil-
tonian of the system for magnetic fields with integral flux. In this setting the
Hilbert space consists of L2-sections in a hermitian line bundle with con-

  

Source: Asch, Joachim - Centre De Physique Theorique, Campus de Luminy, Case 907

 

Collections: Mathematics