Isospectrality for quantum toric integrable systems

TL;DR

This paper proves that the semiclassical joint spectrum uniquely determines the classical symplectic structure of quantum toric integrable systems, providing a comprehensive spectral theory description.

Contribution

It establishes the spectral uniqueness for quantum toric integrable systems and fully characterizes their semiclassical spectral properties.

Findings

Semiclassical joint spectrum determines the classical system up to symplectomorphism.

Complete description of the spectral theory for quantum toric integrable systems.

Addresses classical spectral theory questions with quantum and semiclassical insights.

Abstract

We settle affirmatively the isospectral problem for quantum toric integrable systems: the semiclassical joint spectrum of such a system, given by a sequence of commuting Toeplitz operators on a sequence of Hilbert spaces, determines the classical integrable system given by the symplectic manifold and Poisson commuting functions, up to symplectomorphisms. We also give a full description of the semiclassical spectral theory of quantum toric integrable systems. This type of problem belongs to the realm of classical questions in spectral theory going back to pioneer works of Colin de Verdiere, Guillemin, Sternberg and others in the 1970s and 1980s.