We consider the quantum completeness problem, i.e. the problem of confining quantum particles, on a non-complete Riemannian manifold equipped with a smooth measure , possibly degenerate or singular near the metric boundary of , and in presence of a real-valued potential . The main merit of this paper is the identification of an intrinsic quantity, the effective potential , which allows to formulate simple criteria for quantum confinement. Let be the distance from the possibly non-compact metric boundary of . A simplified version of the main result guarantees quantum completeness if 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 ; (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 .
Cite this article
Dario Prandi, Luca Rizzi, Marcello Seri, Quantum confinement on non-complete Riemannian manifolds. J. Spectr. Theory 8 (2018), no. 4, pp. 1221–1280DOI 10.4171/JST/226