Symplectic spectral geometry of semiclassical operators

TL;DR

This paper reviews recent advances in semiclassical spectral theory, highlighting the relationship between quantum spectral properties and classical symplectic geometry, especially in integrable systems.

Contribution

It synthesizes recent results connecting spectral theory of Berezin-Toeplitz and pseudodifferential operators with symplectic geometry of Hamiltonian systems.

Findings

Enhanced understanding of spectral properties in semiclassical regimes

Connections established between quantum spectra and classical symplectic invariants

Insights into the geometry of finite-dimensional integrable systems

Abstract

In the past decade there has been a flurry of activity at the intersection of spectral theory and symplectic geometry. In this paper we review recent results on semiclassical spectral theory for commuting Berezin-Toeplitz and h-pseudodifferential operators. The paper emphasizes the interplay between spectral theory of operators (quantum theory) and symplectic geometry of Hamiltonians (classical theory), with an eye towards recent developments on the geometry of finite dimensional integrable systems.