Localization of quantum systems at Liouville tori

Abstract

We consider a collection of pairwise commuting quantum observables in the setting of Berezin--Toeplitz quantization of a closed K\"{a}hler manifold and assume that the Arnold--Liouville theorem applies to their principal symbols. We use joint eigensections of these observables to define isometric embeddings of the quantum spaces into $L^2(\Lambda_{a_0})$, where $\Lambda_{a_0}$ is a fixed Liouville torus. These embeddings allow a broad class of quantum observables, including some defined by discontinuous functions, to be realized as sequences of operators on $L^2(\Lambda_{a_0})$ that converge strongly to multiplication operators. We discuss the spectral implications of this convergence and give applications to contractions of Lie algebra representations and to pairs of spectral projections of quantum observables.