Quantum limits of sub-Laplacians via joint spectral calculus

Cyril Letrouit

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arXiv · 2007.00910

Quantum limits of sub-Laplacians via joint spectral calculus

Cyril Letrouit

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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.

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Cyril Letrouit. 2020-07-02. Quantum limits of sub-Laplacians via joint spectral calculus. https://arxiv.org/abs/2007.00910

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