Spectral limits of semiclassical commuting self-adjoint operators

TL;DR

This paper proves that the joint spectrum of commuting semiclassical self-adjoint operators converges to the classical spectrum, extending previous results and including various quantization methods, with implications for quantum integrable systems.

Contribution

It establishes a general convergence result for the joint spectrum of commuting operators under an abstract semiclassical quantization framework, extending prior work.

Findings

Joint spectrum converges to classical spectrum in semiclassical limit

Includes Berezin-Toeplitz and certain pseudodifferential quantizations

Provides insights into inverse problems for quantum integrable systems

Abstract

Using an abstract notion of semiclassical quantization for self-adjoint operators, we prove that the joint spectrum of a collection of commuting semiclassical self-adjoint operators converges to the classical spectrum given by the joint image of the principal symbols, in the semiclassical limit. This includes Berezin-Toeplitz quantization and certain cases of $\hbar$-pseudodifferential quantization, for instance when the symbols are uniformly bounded, and extends a result by L. Polterovich and the authors. In the last part of the paper we review the recent solution to the inverse problem for quantum integrable systems with periodic Hamiltonians, and explain how it also follows from the main result in this paper.