Integrable Schrödinger operators with magnetic fields: Factorization method on curved surfaces

Abstract

The factorization method for Schrödinger operators with magnetic fields on a two-dimensional surface M2 with nontrivial metric is investigated. This leads to the new integrable examples of such operators and brings a new look at some classical problems such as the Dirac magnetic monopole and the Landau problem. The global geometric aspects and related spectral properties of the operators from the factorization chains are discussed in detail. We also consider the Laplace transformations on a curved surface and extend the class of Schrödinger operators with two integrable levels introduced in the flat case by S. P. Novikov and one of the authors.