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 $\ddot{I}‰$, 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 $\hat{I}´$ be the distance from the possibly non-compact metric boundary of $M$. A simplified version of the main result guarantees quantum completeness if $V\ge -c\hat{I}{´}^2$ far from the metric boundary and \[ V_{\mathrm{eff}}+V\ge \frac3{4δ^2}-\fracκδ, \qquad \text{close to the metric boundary}. \] 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] .

40 pages, 7 figures. (V2) corrected typos and updated references. (V3) corrected typos and updated references. Final version to appear on Journal of Spectral Theory