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 andclose 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].

On geometric quantum confinement in Grushin-type manifolds

“…As is going to emerge in the course of the proofs, our approach has a two-fold feature. On the one hand it is relatively 'rigid', for it does not have an immediate generalisation in application to generic almost-Riemannian structures, for which the more versatile, typically perturbative analyses of [3,12,7] appear as more efficient and informative. On the other hand, it is particularly 'robust', whenever the problem can be boiled down to a constant-fiber direct integral scheme and to the study of self-adjointness along each fibre, and this allows us to cover a larger class of Grushin planes than that considered so far.…”

“…Recently, quantum confinement within incomplete Riemannian manifolds has attracted considerable attention especially when the measure has degeneracies or singularities near the metric boundary [3,12,7]. Such setting is intimately related with that of manifolds equipped with so-called almost-Riemannian structure [2], a notion that informally speaking refers to a smooth d-dimensional manifold M equipped with a family of smooth vector fields X 1 , ..., X d satisfying the Lie bracket generating condition: if Z ⊂ M is the embedded hyper-surface of points where the X j 's are not linearly independent, on M \ Z the fields X 1 , .…”

“…For Schrödinger operators on incomplete Riemannian manifolds with singular measure near the metric boundary, sufficient conditions of (essential) self-adjointness, including curvature-based criteria, have been recently established in [3,12] -we are going to comment further on such results in due time. In this context, a special focus is given to the case V ≡ 0, that is, a quantum particle not subject to external interaction: in this case quantum confinement, when it occurs, is then purely geometric.…”

“…• the novelty, and conciseness, of the approach, based on an analysis of constant-fibre direct integral on Hilbert space in combination with Weyl's limit-point limit-circle argument for each fibre; • the formulation of necessary and sufficient conditions for the essential selfadjointness of the considered Laplace-Beltrami operators -the previous works [3,12] were somewhat perturbative in nature, as we shall comment later, and only provided sufficient conditions for quantum confinement, without characterising the absence of it; • the possibility of treating the 'non-compact' case, as in the above example (M, g) = (R + × R, dx 2 + x −2 dy 2 ), thus generalising the analysis of [3, Sect. 4] made for the counterpart case M = R +…”

“…x × T y , where the y-variable was compactified; • some degree of 'robustness' of our approach, as we can immediately export it to a fairly general class of Grushin-type manifolds, say, the half-plane M = R + x × R y with metric g = dx 2 + F (x)dy 2 for suitable functions F , thus simplifying the effective potential approach of [12] based on Agmondtype estimates.…”