Extensions beyond an isolated radially symmetric singularity

Develop extensions of the analysis for the singular heat equation with an inverse-square potential from an isolated radially symmetric singularity to equations with multiple singularities or more general singular potentials.

Background

The results in the paper concern a single singular point at $x=0$ and the radially symmetric inverse-square potential $\mu/|x|^2$. The authors identify extending this framework to multiple singularities and to more general singular potentials as an unresolved direction. Such extensions would test whether the measurable-set observability, null-controllability, uniqueness, and bang-bang conclusions remain valid in more general singular geometries and potential classes.