Trace formula for the magnetic Laplacian at zero energy level

TL;DR

This paper develops a trace formula for the magnetic Laplacian at zero energy, generalizing the Gutzwiller trace formula and focusing on critical energy levels on compact manifolds.

Contribution

It introduces a new trace formula at zero energy for the magnetic Laplacian, extending previous formulas and providing concrete examples and approaches for its proof.

Findings

Generalizes the Gutzwiller trace formula to zero energy

Provides concrete examples of the trace formula computation

Reviews approaches to trace formula proof at critical energy levels

Abstract

The paper is devoted to the trace formula for the magnetic Laplacian associated with a magnetic system on a compact manifold. This formula is a natural generalization of the semiclassical Gutzwiller trace formula and reduces to it in the case when the magnetic field form is exact. It differs somewhat from the Guillemin-Uribe trace formula studied in the author's previous work with I.A. Taimanov. Moreover, in contrast to that work, the focus is on the trace formula at the zero energy level, which is a critical energy level. The paper gives an overview of the main notions and results related to the trace formula at the zero energy level, describes various approaches to its proof, and gives concrete examples of its computation. In addition, a brief review of the Gutzwiller trace formula for regular and critical energy levels is given.