 
Summary: MSC 1991: 58G40, 81S10, 20C35, 35Q40
Magnetic Bloch Analysis and BochnerLaplacians
J. Asch, H. Over*
, R. Seiler
Technische Universität Berlin, MA 71, Straße des 17. Juni 136, D1000 Berlin 12
* Fritz Haber Institut, Faradayweg 46, D1000 Berlin 33
Abstract
Hamiltonians for a particle on a manifold in a magnetic field are constructed as
BochnerLaplacians. 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 QuantumHallEffect
[TKNN], [ASY], [AKPS]. 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 L2sections in a hermitian line bundle with con
