On Geometric Quantum Confinement in Grushin-type Manifolds

TL;DR

This paper investigates geometric quantum confinement on Grushin-type manifolds by analyzing the self-adjointness of the Laplace-Beltrami operator using a fiber decomposition approach.

Contribution

It provides a complete characterization of conditions for essential self-adjointness on Grushin-type manifolds through a novel fiber analysis method.

Findings

Identifies regimes where quantum confinement occurs.

Characterizes the self-adjointness of the Laplace-Beltrami operator.

Develops a fiber decomposition approach for analysis.

Abstract

We study the problem of so-called geometric quantum confinement in a class of two-dimensional incomplete Riemannian manifold with metric of Grushin type. We employ a constant-fibre direct integral scheme, in combination with Weyl's analysis in each fibre, thus fully characterising the regimes of presence and absence of essential self-adjointness of the associated Laplace-Beltrami operator.