Closure of the Laplace-Beltrami operator on 2D almost-Riemannian manifolds and semi-Fredholm properties of differential operators on Lie manifolds

Abstract

The problem of determining the domain of the closure of the Laplace-Beltrami operator on a 2D almost-Riemannian manifold is considered. Using tools from theory of Lie groupoids natural domains of perturbations of the Laplace-Beltrami operator are found. The main novelty is that the presented method allows us to treat geometries with tangency points. This kind of singularity is difficult to treat since those points do not have a tubular neighbourhood compatible with the almost-Riemannian metric.

Self-adjoint extension schemes and modern applications to quantum Hamiltonians

“…The study of a quantum particle on degenerate Riemannian manifolds, and the problem of the purely geometric confinement away from the singularity locus of the metric, as opposite to the dynamical transmission across the singularity, has recently attracted a considerable amount of attention in relation to Grushin structures (defined, e.g., in [54], as well as in Section 5.1) and to the induced confining effective potentials on cylinder, cone, and plane (as in the works by Nenciu and Nenciu [193], Boscain and Laurent [44], Boscain, Prandi and Seri [47], Prandi, Rizzi and Seri [201], Boscain and Prandi [46], Franceschi, Prandi and Rizzi [104], Gallone, Michelangeli and Pozzoli [114,115,116], Boscain and Neel [45], Pozzoli [199], Beschastnnyi, Boscain and Pozzoli [37], Gallone and Michelangeli [112]), as well as, more generally, on two-step two-dimensional almost-Riemannian structures (Boscain and Laurent [44], Beschastnnyi, Boscain and Pozzoli [37], Beschastnnyi [36]), or also generalisations to almost-Riemannian structures in any dimension and of any step, and even to sub-Riemannian geometries, provided that certain geometrical assumptions on the singular set are taken (Franceschi, Prandi, and Rizzi [104], Prandi, Rizzi, and Seri [201]). On a related note, a satisfactory interpretation of the heat-confinement in the Grushin cylinder is known in terms of Brownian motions (Boscain and Neel [45]) and random walks (Agrachev, Boscain, Neel, and Rizzi [3]).…”

“…Virtually nothing is known on the heat or the quantum confinement on such singular structures, including the simplest example (5.92) (see [103] for further remarks). First preliminary results in this respect were recently obtained in [36], where the interpretation of almost-Riemannian structures as special Lie manifolds permits to study some closure properties of singular perturbations of the Laplace-Beltrami operator even in the presence of tangency points. This opens new perspectives of treating several types of different singularities in sub-Riemannian geometry within the same unifying theory.…”

Heat equation with inverse-square potential of bridging type across two half-lines

“…The subject of geometric quantum confinement away from the metric's singularity, and transmission across it, for quantum particles or for the heat flow on degenerate Riemannian manifolds is experiencing a fast growth in the recent years. Such themes are particularly active with reference to Grushin structures on cylinder, cone, and plane [5,8,7,17,20,6,4,3], as well as, more generally, on two-dimensional orientable compact almost-Riemannian manifolds of step two [5], 𝑑-dimensional regular almost-Riemannian and sub-Riemannian manifolds [22,14]. Of significant relevance is the counterpart model to the Grushin-type cylinder, but in the lack of compact variable.…”

“…Virtually nothing in known on the heat or the quantum confinement on such singular structures, including the simplest example (46) (see [13] for further remarks). First preliminary results in this respect were recently obtained in [3], where the interpretation of almost-Riemannian structures as special Lie manifolds permits to study some closure properties of singular perturbations of the Laplace-Beltrami operator even in the presence of tangency points. This opens new perspectives of treating several types of different singularities in sub-Riemannian geometry within the same unifying theory.…”

“…The first one, 𝑔 + 0 (𝑡) = 𝑔 − 0 (𝑡) can be interpreted as the continuity of the function, up to the weight |𝑥| 𝛼 2 that allows 𝑢 to have some degree of singularity at the origin; analogously, the condition 𝑔 + 1 (𝑡) = −𝑔 − 1 (𝑡) links the right and left derivative at zero, up to certain weights, and taken directionally from each side. In the regime 𝐶 𝛼 > 3 4 such conditions are obviously redundant, but when 𝐶 𝛼 3 4 an ad hoc analysis is needed to recognise that the prescribed behaviour at the origin expresses another condition of self-adjointness and positivity. Such an analysis has been carried out in several recent works [7,17,18,20,15] and is concisely reviewed in Section 2.…”