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 $ds^2=dx^2+|x|^{−2\alpha}d\theta^2$, where $x\in\mathbb{R}$, $\theta\in\mathbb{T}$ and the parameter $\alpha\in\mathbb{R}$. For $\alpha\le−1$ this metric describes cone-like manifolds (for $\alpha=−1$ it is a flat cone). For $\alpha=0$ it is a cylinder. For $\alpha\ge1$ it is a Grushin-like metric. We show that the Laplace–Beltrami operator $\Delta$ is essentially self-adjoint if and only if $\alpha\notin(−3,1)$. In this case the only self-adjoint extension is the Friedrichs extension $\Delta_F$, that does not allow communication through the singular set $x=0$ both for the heat and for a quantum particle. For $\alpha\in(−3,−1]$ we show that for the Schrödinger equation only the average on $\theta$ of the wave function can cross the singular set, while the solutions of the only Markovian extension of the heat equation (which indeed is $\Delta_F$) cannot. For $\alpha\in(−1,1)$ we prove that there exists a canonical self-adjoint extension $\Delta_B$, 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 $\Delta_F$ and $\Delta_B$, proving that $\Delta_F$ is stochastically complete at the singularity if and only if $\alpha\le−1$, while $\Delta_B$ is always stochastically complete at the singularity.

Publication
Journal of Differential Equations