A semiclassical Birkhoff normal form for symplectic magnetic wells

TL;DR

This paper develops a semiclassical Birkhoff normal form for magnetic Schrödinger operators with non-degenerate magnetic wells, enabling precise eigenvalue expansions and Weyl asymptotics in a geometric setting.

Contribution

It introduces a novel normal form for semiclassical magnetic Schrödinger operators with non-degenerate wells on Riemannian manifolds, facilitating spectral analysis.

Findings

Derived eigenvalue expansions in powers of h^{1/2}

Established semiclassical Weyl asymptotics for the operator

Provided a new analytical framework for magnetic wells on manifolds

Abstract

In this paper we construct a Birkhoff normal form for a semiclassical magnetic Schr{\"o}dinger operator with non-degenerate magnetic field, and discrete magnetic well, defined on an even dimensional riemannian manifold M. We use this normal form to get an expansion of the first eigenvalues in powers of h^{1/2}, and semiclassical Weyl asymptotics for this operator.