Heat equation with inverse-square potential of bridging type across two half-lines

Abstract

The heat equation with inverse square potential on both half-lines of $\mathbb{R}$ is discussed in the presence of \emph{bridging} boundary conditions at the origin. The problem is the lowest energy (zero-momentum) mode of the transmission of the heat flow across a Grushin-type cylinder, a generalisation of an almost Riemannian structure with compact singularity set. This and related models are reviewed, and the issue is posed of the analysis of the dispersive properties for the heat kernel generated by the underlying positive self-adjoint operator. Numerical integration is shown that provides a first insight and relevant qualitative features of the solution at later times.

References in corpus (9)

• Self-adjoint extensions and stochastic completeness of the Laplace-Beltrami operator on conic and anticonic surfaces
• Quantum confinement on non-complete Riemannian manifolds
• Spectral analysis and the Aharonov-Bohm~effect on certain almost-Riemannian manifolds
• On the essential self-adjointness of singular sub-Laplacians
• On the domains of Bessel operators
• On Geometric Quantum Confinement in Grushin-type Manifolds
• Intrinsic random walks in Riemannian and sub-Riemannian geometry via volume sampling
• Quantum Geometric Confinement and Dynamical Transmission in Grushin Cylinder
• Quantum particle across Grushin singularity