Trace formula for the magnetic Laplacian

Abstract

The Guillemin-Uribe trace formula is a semiclassical version of the Selberg trace formula and more general Duistermaat-Guillemin formula for elliptic operators on compact manifolds, which reflects the dynamics of magnetic geodesic flows in terms of eigenvalues of a natural differential operator (the magnetic Laplacian) associated with the magnetic field. In this paper, we give a survey of basic notions and results related with the Guillemin-Uribe trace formula and provide concrete examples of its computation for two-dimensional constant curvature surfaces with constant magnetic fields and for the Katok example.

37 pages, v2: minor corrections, references added

References in corpus (1)

• The multiplicity problem for periodic orbits of magnetic flows on the 2-sphere

Cited by in corpus (2)

• Trace formula for the magnetic Laplacian on a compact hyperbolic surface
• The spectral density function of the renormalized Bochner Laplacian on a symplectic manifold