Journal of Spectral Theory

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Volume 8, Issue 4, 2018, pp. 1221–1280
DOI: 10.4171/JST/226

Published online: 2018-07-25

Quantum confinement on non-complete Riemannian manifolds

Dario Prandi[1], Luca Rizzi[2] and Marcello Seri[3]

(1) CentraleSupélec, Gif-sur-Yvette, France
(2) Universite Grenoble Alpes, France
(3) University of Groningen, The Netherlands

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 $$V_\mathrm {eff}+V\ge \frac3{4\delta^2}-\frac{\kappa}{\delta}, \quad \mathrm {close \:to \:the \:metric \:boundary}.$$ 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].

Keywords: Quantum completeness, almost-Riemannian geometry, Schrödinger operators

Prandi Dario, Rizzi Luca, Seri Marcello: Quantum confinement on non-complete Riemannian manifolds. J. Spectr. Theory 8 (2018), 1221-1280. doi: 10.4171/JST/226