Self-adjoint extensions and stochastic completeness of the Laplace–Beltrami operator on conic and anticonic surfaces

Abstract

We study the evolution of the heat and of a free quantum particle (described by the Schrödinger equation) on two-dimensional manifolds endowed with the degenerate Riemannian metric ds 2 = dx 2 + |x| −2α dθ 2 , where x ∈ R, θ ∈ T and the parameter α ∈ R. For α ≤ −1 this metric describes cone-like manifolds (for α = −1 it is a flat cone). For α = 0 it is a cylinder. For α ≥ 1 it is a Grushin-like metric. We show that the Laplace-Beltrami operator ∆ is essentially self-adjoint if and only if α / ∈ (−3, 1). In this case the only self-adjoint extension is the Friedrichs extension ∆ F , that does not allow communication through the singular set {x = 0} both for the heat and for a quantum particle. For α ∈ (−3, −1] we show that for the Schrödinger equation only the average on θ of the wave function can cross the singular set, while the solutions of the only Markovian extension of the heat equation (which indeed is ∆ F ) cannot. For α ∈ (−1, 1) we prove that there exists a canonical self-adjoint extension ∆ B , called bridging extension, which is Markovian and allows the complete communication through the singularity (both of the heat and of a quantum particle). Also, we study the stochastic completeness (i.e., conservation of the L 1 norm for the heat equation) of the Markovian extensions ∆ F and ∆ B , proving that ∆ F is stochastically complete at the singularity if and only if α ≤ −1, while ∆ B is always stochastically complete at the singularity.

Extensions of Brownian motion to a family of Grushin-type singularities

“…For this extension, we see that only the average over T of a function flows through the singularity under the corresponding semigroup (note that in this case, the bridging extension does not correspond to a diffusion on M α ). Thus the same phenomenon observed in [4] for non-Markov self-adjoint extensions of ∆ for α ∈ (−3, −1] is replicated here for Markov processes that respect the topology of M α when α ∈ (−1, 0). This is carried out in Section 3.…”

“…From the perspective of functional analysis, this means considering self-adjoint extensions of the Laplacian on M . Indeed, in [4], the following two basic results were proven. An immediate consequence of Theorem 1 is that for α ∈ (−∞, −3] ∪ [1, ∞) the only self-adjoint extension of ∆ is the Friedrich extension ∆ F .…”

“…Thus, if we do not allow killing at Z (or anywhere else), M α is stochastically complete for all α ∈ R. (This contrasts slightly with [4], since they consider only self-adjoint extensions and thus kill the process at Z when it is an exit-only boundary, instead of letting it be absorbed. But the real point is that the process never explodes to infinity in finite time.)…”

“…where (x 1 , θ 1 ) ∼ (x 2 , θ 2 ) if and only if x 1 = x 2 = 0. When α ≥ 0, extending the vector fields X and Y to M cylinder , the natural control-theoretic notion of the length of a curve shows that there are paths of finite length between M + = {x > 0} × T and M − = {x < 0} × T (which are the Riemannian geodesics for x = 0 and which are tangent to X when x = 0), and, as explained in [4], this extended distance makes M cylinder into a metric space (and a length space) in a way that induces on M cylinder its original topology. Similarly, when α < 0 the distance induced by the metric (1) extend naturally to M cone .…”

“…But such an extension clearly cannot descend to a strong Markov process on M α . Thus, here we treat extensions that are carried by M α itself, so that our results differ from, and complement, those of [4]. We also do not restrict our attention to symmetric extensions.…”