Quantum confinement for the curvature Laplacian $-\Delta+cK$ on 2D-almost-Riemannian manifolds

TL;DR

This paper investigates the self-adjointness of the curvature Laplacian on 2D almost-Riemannian manifolds, showing that unlike the standard Laplace-Beltrami operator, it does not exhibit quantum confinement.

Contribution

It demonstrates that the curvature-dependent Laplacian $- abla + cK$ lacks quantum confinement on 2D almost-Riemannian manifolds for $c eq 0$, extending understanding of quantum behavior near singularities.

Findings

The curvature Laplacian $- abla + cK$ is not essentially self-adjoint on the manifold with singular set.

Quantum confinement phenomena do not occur for the curvature Laplacian with $c eq 0$.

The result applies to operators arising from coordinate-free quantization procedures.

Abstract

Two-dimension almost-Riemannian structures of step 2 are natural generalizations of the Grushin plane. They are generalized Riemannian structures for which the vectors of a local orthonormal frame can become parallel. Under the 2-step assumption the singular set $Z$, where the structure is not Riemannian, is a 1D embedded submanifold. While approaching the singular set, all Riemannian quantities diverge. A remarkable property of these structures is that the geodesics can cross the singular set without singularities, but the heat and the solution of the Schr\"{o}dinger equation (with the Laplace-Beltrami operator $\Delta$) cannot. This is due to the fact that (under a natural compactness hypothesis), the Laplace-Beltrami operator is essentially self-adjoint on a connected component of the manifold without the singular set. In the literature such phenomenon is called quantum confinement. In this paper we study the self-adjointness of the curvature Laplacian, namely $-\Delta+cK$, for $c\in(0,1/2)$ (here $K$ is the Gaussian curvature), which originates in coordinate-free quantization procedures (as for instance in path-integral or covariant Weyl quantization). We prove that there is no quantum confinement for this type of operators.