 
Summary: LANDAU HAMILTONIANS ON SYMMETRIC
SPACES
J. E. AVRON AND A. PNUELI
Department of Physics, Technion, IIT, Haifa, 32000, ISRAEL
1 INTRODUCTION AND OVERVIEW
A classical particle on a manifold moves on geodesics: Straight lines on the plane,
great circles on the sphere and semicircles on the hyperpolicplane, represented as
the upperhalf plane with the usual metric1
.
The corresponding Schršodinger operators are (minus) the Laplacians on the manifolds
and their spectra are [0, ) for the plane, {0, 2, 6, ..., n(n+1), . . . , } for the sphere,
and [1
4
, ) for the hyperbolic plane [see e.g. McKean (1970) or Terras (1985)].
Magnetic fields are 2forms, and a constant magnetic field is a multiple, B, of the
area form, with B > 0, the magnetic field strength. Constant magnetic fields are
therefore natural for orientable two dimensional manifolds.
In contrast with magnetic fields, constant electric fields are not natural in general:
Electric fields are vector fields and on a curved manifold zero is the only constant
vector field; Constant electric field are defined only for flat spaces. For this reason
