# Point interactions for 3D sub-Laplacians

## TL;DR

This paper proves that 3D sub-Laplacians on certain manifolds do not admit point interactions and are essentially self-adjoint, contrasting with Riemannian cases, with applications to quantum models of thin molecules.

## Contribution

It establishes the non-existence of point interactions for 3D sub-Laplacians and demonstrates their essential self-adjointness on complete manifolds, with implications for quantum molecular dynamics.

## Key findings

- No point interaction exists for 3D sub-Laplacians on complete manifolds.
- Sub-Laplacians are essentially self-adjoint on $C_0^inity(M\setminus\{q_0\})$.
- Contrast with Riemannian Laplacians in dimensions ≤ 3.

## Abstract

In this paper we show that, for a sub-Laplacian $\Delta$ on a $3$-dimensional manifold $M$, no point interaction centered at a point $q_0\in M$ exists. When $M$ is complete w.r.t. the associated sub-Riemannian structure, this means that $\Delta$ acting on $C^\infty_0(M\setminus\{q_0\})$ is essentially self-adjoint. A particular example is the standard sub-Laplacian on the Heisenberg group. This is in stark contrast with what happens in a Riemannian manifold $N$, whose associated Laplace-Beltrami operator is never essentially self-adjoint on $C^\infty_0(N\setminus\{q_0\})$, if $\dim N\le 3$. We then apply this result to the Schr\"odinger evolution of a thin molecule, i.e., with a vanishing moment of inertia, rotating around its center of mass.