Quantum confinement on non-complete Riemannian manifolds

TL;DR

This paper establishes intrinsic criteria for quantum confinement and self-adjointness of the Laplace-Beltrami operator on non-complete Riemannian manifolds, accounting for degeneracies, singularities, and curvature effects.

Contribution

It introduces an effective potential to formulate simple, curvature-based criteria for quantum confinement and self-adjointness in complex geometric and measure-theoretic settings.

Findings

Derived conditions for quantum confinement involving the effective potential.

Generalized classical theorems to Riemannian manifolds with singularities.

Proved essential self-adjointness of the Laplace-Beltrami operator in almost-Riemannian geometry.

Abstract

We consider the quantum completeness problem, i.e. the problem of confining quantum particles, on a non-complete Riemannian manifold $M$ equipped with a smooth measure $\omega$, possibly degenerate or singular near the metric boundary of $M$, and in presence of a real-valued potential $V\in L^2_{\mathrm{loc}}(M)$. The main merit of this paper is the identification of an intrinsic quantity, the effective potential $V_{\mathrm{eff}}$, which allows to formulate simple criteria for quantum confinement. Let $\delta$ be the distance from the possibly non-compact metric boundary of $M$. A simplified version of the main result guarantees quantum completeness if $V\ge -c\delta^2$ far from the metric boundary and \[ V_{\mathrm{eff}}+V\ge \frac3{4\delta^2}-\frac{\kappa}{\delta}, \qquad \text{close to the metric boundary}. \] These criteria allow us to: (i) obtain quantum confinement results for measures with degeneracies or singularities near the metric boundary of $M$; (ii) generalize the Kalf-Walter-Schmincke-Simon Theorem for strongly singular potentials to the Riemannian setting for any dimension of the singularity; (iii) give the first, to our knowledge, curvature-based criteria for self-adjointness of the Laplace-Beltrami operator; (iv) prove, under mild regularity assumptions, that the Laplace-Beltrami operator in almost-Riemannian geometry is essentially self-adjoint, partially settling a conjecture formulated in [Boscain, Laurent - Ann. Inst. Fourier, 2013] .