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 $d{s}^2=d{x}^2+|x{|}^{-2\alpha }d{\theta }^2$, 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 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.

29 pages, 2 figures, accepted version

References in corpus (3)

• Weyl-Titchmarsh Theory for Sturm-Liouville Operators with Distributional Potentials
• Spectral analysis and the Aharonov-Bohm~effect on certain almost-Riemannian manifolds
• Markovian Extensions of Symmetric Second Order Elliptic Differential Operators

Cited by in corpus (15)

• Spectral analysis and the Aharonov-Bohm~effect on certain almost-Riemannian manifolds
• On the essential self-adjointness of singular sub-Laplacians
• Self-adjoint extension schemes and modern applications to quantum Hamiltonians
• On Geometric Quantum Confinement in Grushin-type Manifolds
• Stochastic processes on surfaces in three-dimensional contact sub-Riemannian manifolds
• Quantum Geometric Confinement and Dynamical Transmission in Grushin Cylinder
• Quantum particle across Grushin singularity
• Direct sums of trace maps and self-adjoint extensions
• Extensions of Brownian motion to a family of Grushin-type singularities
• Uniform, localized asymptotics for sub-Riemannian heat kernels, their logarithmic derivatives, and associated diffusion bridges
• Heat equation with inverse-square potential of bridging type across two half-lines
• Drift-diffusion equations on domains in ${\mathbb{R}}^d$: essential self-adjointness and stochastic completeness
• Product formulas and convolutions for two-dimensional Laplace-Beltrami operators: beyond the trivial case
• Essential self-adjointness and the $L^2$-Liouville property
• Exact observability properties of subelliptic wave and Schr{ö}dinger equations