Quantum limits of sub-Laplacians via joint spectral calculus

TL;DR

This paper investigates the quantum limits of sub-Laplacians, demonstrating spectral decomposition techniques under commutativity assumptions and analyzing specific cases on Heisenberg group quotients.

Contribution

It introduces a joint spectral calculus approach for sub-Laplacians, enabling spectral decomposition and analysis of quantum limits in degenerate spectral settings.

Findings

Quantum limits can be split into parts under commutativity assumptions.

Spectral decomposition reveals harmonic oscillators in sub-Laplacian spectra.

Results apply to sub-Laplacians on products of Heisenberg group quotients.

Abstract

We establish two results concerning the Quantum Limits (QLs) of some sub-Laplacians. First, under a commutativity assumption on the vector fields involved in the definition of the sub- Laplacian, we prove that it is possible to split any QL into several pieces which can be studied separately, and which come from well-characterized parts of the associated sequence of eigenfunctions. Secondly, building upon this result, we study in detail the QLs of a particular family of sub-Laplacians defined on products of compact quotients of Heisenberg groups. We express the QLs through a disintegration of measure result which follows from a natural spectral decomposition of the sub-Laplacian in which harmonic oscillators appear. Both results are based on the construction of an adequate elliptic operator commuting with the sub-Laplacian, and on the associated joint spectral calculus. They illustrate the fact that, because of the possible high degeneracies in the spectrum, the spectral theory of sub-Laplacians is very rich.