Quantum limits of sub-Laplacians via joint spectral calculus

  • Cyril Letrouit

    Massachusetts Institute of Technology, Cambridge, USA; École normale supérieure and CNRS, Université Paris Sciences et Lettres, France
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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, Quantum limits of sub-Laplacians via joint spectral calculus. Doc. Math. 28 (2023), no. 1, pp. 55–104

DOI 10.4171/DM/908