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Quantum limits of sub-Laplacians via joint spectral calculus. (arXiv:2007.00910v3 [math.AP] UPDATED)

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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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