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∈L2loc(M). The main merit of this paper is the identification of an intrinsic quantity, the effective potential Veff, 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≥−cδ2 far from the metric boundary and Veff+V≥34δ2−κδ,closetothemetricboundary. 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].

Quantum confinement on non-complete Riemannian manifolds / Prandi, D.; Rizzi, L.; Seri, M.. - In: JOURNAL OF SPECTRAL THEORY. - ISSN 1664-039X. - 8:4(2018), pp. 1221-1280. [10.4171/jst/226]

Quantum confinement on non-complete Riemannian manifolds

Rizzi, L.;
2018-01-01

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∈L2loc(M). The main merit of this paper is the identification of an intrinsic quantity, the effective potential Veff, 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≥−cδ2 far from the metric boundary and Veff+V≥34δ2−κδ,closetothemetricboundary. 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].
2018
8
4
1221
1280
https://arxiv.org/abs/1609.01724
Prandi, D.; Rizzi, L.; Seri, M.
File in questo prodotto:
Non ci sono file associati a questo prodotto.

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/20.500.11767/128670
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 20
  • ???jsp.display-item.citation.isi??? 17
social impact