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Trace formula for the magnetic Laplacian. (English. Russian original) Zbl 1433.58019

Russ. Math. Surv. 74, No. 2, 325-361 (2019); translation from Usp. Mat. Nauk 74, No. 2, 149-186 (2019).
The paper under review is devoted to a survey of some basic results related to the Guillemin-Uribe trace formula, which reflects the dynamics of magnetic geodesic flows in terms of eigenvalues of the magnetic Laplacian operator. The authors also give some examples of its computation in the framework of two-dimensional constant curvature surfaces with constant magnetic fields, as well as for the Katok example.

MSC:

58J05 Elliptic equations on manifolds, general theory
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
58J37 Perturbations of PDEs on manifolds; asymptotics
81Q20 Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory
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