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

The factorization method for Schrodinger operators with magnetic fields on a two-dimensional surface M-2 with nontrivial metric is investigated. This leads to the new integrable examples of such operators and brings a new loo k at some classical problems such as the Dirac magnetic monopole and the La ndau 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 th e class of Schrodinger operators with two integrable levels introduced in t he flat case by S. P. Novikov and one of the authors. (C) 2001 American Ins titute of Physics.