Point interactions for 3D sub-Laplacians

Abstract: The aim of this seminar is to present some recent results on the essential self-adjointness of pointed sub-Laplacians in three dimensions. These are the natural (non-negative) hypoelliptic operators $H$ associated with a sub-Riemannian structure on a 3D manifold $M$, with domain $\operatorname{Dom}(H)=C^\infty_c(M\setminus \{p\})$, for $p\in M$.

If $M=\mathbb{R}^n$ and the geometry is Euclidean, $H$ is the standard Laplacian. It is then well-known that $H$ is essentially self-adjoint with $\operatorname{Dom}(H) =C^\infty_c(\mathbb{R}^n\setminus\{p\})$ if and only if $n\ge 4$. This follows, for instance, by the Euclidean Hardy inequality.

In this seminar we show that, unlike the Euclidean case, pointed sub-Laplacians (as-sociated with smooth measures) are essentially self-adjoint already for contact sub-Riemannian manifolds of (topological) dimension $3$. Although this is not surprising, since the Hausdorff dimension of these structures is $4$, we will sow that this result cannot be deduced via Hardy inequalities as in the Euclidean case but requires a much finer machinery. Indeed, our strategy of proof is based on a localicazion argument which allows to reduce to the study of the 3D Heisenberg pointed sub-Laplacian. The essential self-adjoitness of the latter is then obtained by exploiting non-commutative Fourier transform techniques.

This is a joint work with R. Adami (Politecnico di Torino, Italy), U. Boscain (CNRS &UPMC, Sorbonne Université, France), and V. Franceschi (Università di Padova, Italy).

analysis of PDEsdifferential geometrymetric geometryoptimization and controlspectral theory

Audience: researchers in the topic

( paper | video )

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Sub-Riemannian Seminars

Series comments: The "Sub-Riemannian seminars" are the union of the "Séminaire de géométrie et analyse sous-riemannienne" (held in Paris since 2011) and the "International Sub-Riemannian Seminars", which were born in spring 2020 as a reaction to the COVID-19 pandemic.

The new format will gather every 3 weeks on average, alternating between these types of sessions:

- physical session in Paris (Laboratoire Jacques-Louis Lions), also transmitted online on Zoom.

- fully online session on Zoom.

- special session hosted physically somewhere else, and transmitted online.

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