Calculus of Variations and Geometric Measure Theory
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D. Prandi - L. Rizzi - M. Seri

Quantum confinement on non-complete Riemannian manifolds

created by rizzi1 on 08 Sep 2016
modified on 28 Dec 2018


Published Paper

Inserted: 8 sep 2016
Last Updated: 28 dec 2018

Journal: Journal of Spectral Theory
Volume: 8
Number: 4
Pages: 1221--1280
Year: 2018
Doi: 10.4171/JST/226

ArXiv: 1609.01724 PDF
Links: arXiv preprint


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 .


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