Quantum confinement on non-complete Riemannian manifolds

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 w, possibly degenerate or singular near the metric boundary of M, and in presence of a real-valued potential V is an element of L-loc(2) (M). The main merit of this paper is the identification of an intrinsic quantity, the effective potential V-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 >= -c delta(2) far from the metric boundary and

V-eff + V >= 3/4 delta(2)-k/delta, 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 [9].