Integrable Schr\"odinger operators with magnetic fields: factorisation method on curved surfaces

The factorisation method for Schr"odinger operators with magnetic fields on a two-dimensional surface $M^2$ with non-trivial metric is investigated. This leads to the new integrable examples of such operators and brings a new look at some classical problems such as Dirac magnetic monopole and Landau problem. The global geometric aspects and related spectral properties of the operators from the factorisation chains are discussed in details. We also consider the Laplace transformations on a curved surface and extend the class of Schr"odinger operators with two integrable levels introduced in the flat case by S.P.Novikov and one of the authors.