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 $x\in \mathbb R$, $θ\in\mathbb T$ and the parameter $α\in\mathbb R$. For $α\le-1$ this metric describes cone-like manifolds (for $α=-1$ it is a flat cone). For $α=0$ it is a cylinder. For $α\ge 1$ it is a Grushin-like metric. We show that the Laplace-Beltrami operator $Δ$ is essentially self-adjoint if and only if $α\notin(-3,1)$. In this case the only self-adjoint extension is the Friedrichs extension $Δ_F$, that does not allow communication through the singular set $\{x=0\}$ both for the heat and for a quantum particle. For $α\in(-3,-1]$ 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 $Δ_F$) cannot. For $α\in(-1,1)$ we prove that there exists a canonical self-adjoint extension $Δ_B$, 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 $Δ_F$ and $Δ_B$, proving that $Δ_F$ is stochastically complete at the singularity if and only if $α\le -1$, while $Δ_B$ is always stochastically complete at the singularity.