Quantum confinement on non-complete Riemannian manifolds

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 $V\in L_{\mathrm{loc}}^2(M)$. 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

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 [9].