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

TL;DR

This paper studies the heat equation with inverse-square potential across two half-lines connected by bridging boundary conditions, analyzing dispersive properties and providing numerical insights into the solution's behavior over time.

Contribution

It introduces a novel analysis of the heat equation with inverse-square potential on a bridged domain, extending understanding of dispersive properties in singular geometric settings.

Findings

Numerical integration reveals qualitative features of heat flow over time.

Analysis indicates specific dispersive behaviors related to the inverse-square potential.

The model generalizes Riemannian structures with singularities, offering new insights into heat kernel properties.

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.