Quantum Confinement for the Curvature Laplacian −Δ + cK on 2D-Almost-Riemannian Manifolds

Abstract

Two-dimension almost-Riemannian structures of step 2 are natural generalizations of the Grushin plane. They are generalized Riemannian structures for which the vectors of a local orthonormal frame can become parallel. Under the 2-step assumption the singular set , where the structure is not Riemannian, is a 1D embedded submanifold. While approaching the singular set, all Riemannian quantities diverge. A remarkable property of these structures is that the geodesics can cross the singular set without singularities, but the heat and the solution of the Schrödinger equation (with the Laplace-Beltrami operator Δ) cannot. This is due to the fact that (under a natural compactness hypothesis), the Laplace-Beltrami operator is essentially self-adjoint on a connected component of the manifold without the singular set. In the literature such phenomenon is called quantum confinement.In this paper we study the self-adjointness of the curvature Laplacian, namely −Δ + , for ∈ (0, 1/2) (here is the Gaussian curvature), which originates in coordinate-free quantization procedures (as for instance in path-integral or covariant Weyl quantization). We prove that there is no quantum confinement for this type of operators.

Quantum confinement detection using a coupled Schrödinger system

“…The diffusion coefficient g in these equations represents the coefficient of diffusion, while curv u and curv v denote the Gaussian curvature (GC) and mean curvature (MC) for the variables u and v, respectively. For a more comprehensive understanding of the self-adjointness of the operator −0.5∆ s + cg(curv), please refer to the relevant details provided in [14].…”

“…The theoretical explanation for this phenomenon, under the compactness hypothesis, is that the LBO is essentially self-adjoint on a connected component of the manifold without the singular set. This phenomenon is referred to as quantum confinement [12,[14][15][16].…”

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]).…”

“…for suitable c > 0. In the recent work [37] it was indeed shown, for generic twostep two-dimensional almost-Riemannian manifolds with compact singular set, that irrespective of c ∈ (0, 1 2 ) the above correction washes essential self-adjointness out, yielding a quantum picture where the Schrödinger particle does reach the singularity much as the classical particle does. (At the expenses of some further technicalities, the whole regime c > 0 can be covered as well.…”