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 ω, possibly degenerate or singular near the metric boundary of M , and in presence of a real-valued potential V ∈ L 2 loc (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 δ be the distance from the possibly non-compact metric boundary of M. A simplified version of the main result guarantees quantum completeness if V ≥ −cδ 2 far from the metric boundary and V eff + V ≥ 3 4δ 2 − κ δ , close to the metric boundary. These criteria allow us to: (i) obtain sharp 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].
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Contributeur : Luca Rizzi <>
Soumis le : jeudi 8 septembre 2016 - 09:19:41
Dernière modification le : jeudi 11 janvier 2018 - 06:27:34
Document(s) archivé(s) le : vendredi 9 décembre 2016 - 12:24:57


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  • HAL Id : hal-01362030, version 1
  • ARXIV : 1609.01724


Dario Prandi, Luca Rizzi, Marcello Seri. Quantum confinement on non-complete Riemannian manifolds. IF_PREPUB. 40 pages, 7 figures. 2016. 〈hal-01362030v1〉



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