Quantum confinement in $\alpha$-Grushin planes

TL;DR

This paper investigates quantum confinement phenomena in Grushin-type geometries, characterizing conditions for self-adjointness of associated Laplace-Beltrami operators on degenerate half-planes.

Contribution

It introduces a fiber direct integral approach to rigorously analyze self-adjointness in singular Grushin geometries, extending understanding of quantum confinement.

Findings

Established criteria for self-adjointness of Grushin Laplacians.

Compared confinement in Grushin planes with half-cylinder cases.

Provided a rigorous mathematical framework for degenerate geometries.

Abstract

We consider here a family of singular Laplace-Beltrami operators, focussing our attention on the problem of so-called quantum confinement on the half-plane equipped with Riemannian metrics of Grushin type degenerate at the boundary. By introducing a costant-fiber direct integral scheme we are able to rigorously characterize the presence or absence of self-adjointness of these operators. We also compare our technique and results with the already studied problem of quantum confinement on the half-cylinder.