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

TL;DR

This paper investigates the self-adjoint extensions and stochastic completeness of the Laplace-Beltrami operator on conic and anticonic surfaces, revealing conditions for essential self-adjointness, communication through singularities, and conservation properties of heat and quantum evolution.

Contribution

It characterizes all self-adjoint extensions of the Laplace-Beltrami operator on these surfaces and introduces a canonical bridging extension allowing communication across singularities.

Findings

The Laplace-Beltrami operator is essentially self-adjoint iff α not in (-3,1).

The Friedrichs extension does not allow communication through the singularity.

The bridging extension allows complete communication and is always stochastically complete.

Abstract

We study the evolution of the heat and of a free quantum particle (described by the Schr\"odinger 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 $\alpha\ge 1$ 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 Schr\"odinger 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.