Quantum Geometric Confinement and Dynamical Transmission in Grushin Cylinder

Abstract

We classify the self-adjoint realisations of the Laplace-Beltrami operator minimally defined on an infinite cylinder equipped with an incomplete Riemannian metric of Grushin type, in the non-trivial class of metrics yielding an infinite deficiency index. Such realisations are naturally interpreted as Hamiltonians governing the geometric confinement of a Schrödinger quantum particle away from the singularity, or the dynamical transmission across the singularity. In particular, we characterise all physically meaningful extensions qualified by explicit local boundary conditions at the singularity. Within our general classification we retrieve those distinguished extensions previously identified in the recent literature, namely the most confining and the most transmitting one. Contents 1. Introduction, setting, main results 1.1. Grushin structures and geometric quantum confinement 1.2. Scheme of our analysis. Main results 2. Preparatory materials 2.1. Unitary equivalence to a constant-fiber orthogonal sum structure 2.2. Orthogonal sum operators 2.3. Momentum-fibred extensions. Local and non-local extensions. 3. Extensions of the differential operator on each half-fibre 3.1. Homogeneous differential problem: kernel of A α (k) * 3.2. Non-homogeneous inverse differential problem 3.3. Operator closure A α (k) 3.4. Distinguished extension and induced classification 3.5. Proof of the classification theorem on fibre 4. Continuation: the mode k = 0 5. Bilateral-fibre extensions 6. General extensions of H α 7. Uniformly fibred extensions of H α 7.1. Generalities and classification theorem 7.2. General strategy 7.3. Integrability and Sobolev regularity of g 0 and g 1 7.4. Decomposition of the adjoint into singular terms 7.5. Detecting short-scale asymptotics and regularity 7.6. Control of ϕ 7.7. Control of ϑ 7.8. Proof of the classification theorem 8. Putting all together

Quantum particle across Grushin singularity

“…This work ideally completes a cycle we recently started in [13] and continued in [14], both works in collaboration with E. Pozzoli, and is part of the fast growing subject of geometric quantum confinement away from the metric's singularity, and transmission across it, for quantum particles on degenerate Riemannian manifolds. Such theme is particularly active with reference to Grushin structures on cylinder, cone, and plane [16,3,6,17,5,10,13,4], as well as, more generally, on two-dimensional orientable compact manifolds [3], and d-dimensional incomplete Riemannian manifolds [17].…”

“…i.e., if and only if γ 1 + γ 4 > 0 and γ 1 γ 4 γ 2 2 + γ 2 3 . In practice, the positivity of each realisation is determined by the positivity of the corresponding extension parameter (γ or Γ), a characterisation that takes such simple form thanks to the efficient choice of the labelling for each extension, made in [14] by exploiting the Kreȋn-Višik-Birman extension scheme [12].…”

“…As a matter of fact, the self-adjoint extension problem for H α in L 2 (M, dµ α ) is more conveniently dealt with in a suitable unitarily equivalent re-formulation that exploits the natural fibered structure of the Hilbert space once the Fourier transform is taken in the compact variable y. In this Section we collect the relevant properties of this construction required for our subsequent analysis, referring to our previous works [13,14] for further details and proofs.…”

“…In particular, the self-adjoint extension problem was solved in [14] based on the crucial circumstance that the adjoint of H α is reduced, with respect to the Hilbert space orthogonal decomposition (3.7), as…”

“…In the work [14], together with Pozzoli, we established an extensive and fairly explicit classification of the "physical" self-adjoint realisations of the Laplace-Beltrami operator on the manifold M α in the significant regime α ∈ (0, 1). The result is summarised as follows.…”