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Integrable Schrödinger operators with magnetic fields: Factorization method on curved surfaces. (English) Zbl 1013.81007
Summary: The factorization method for Schrödinger operators with magnetic fields on a one-dimensional surface M 2 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 A. P. Veselov [Transl., Ser. 2, Am. Math. Soc. 179(33), 109-132 (1997; Zbl 0922.35141)].
MSC:
81Q05Closed and approximate solutions to quantum-mechanical equations
81U15Exactly and quasi-solvable systems (quantum theory)
35Q40PDEs in connection with quantum mechanics
81T20Quantum field theory on curved space backgrounds
37K10Completely integrable systems, integrability tests, bi-Hamiltonian structures, hierarchies
37K25Relations of infinite-dimensional systems with differential geometry
58J90Applications of PDE on manifolds