Lebeau–Robbiano spectral inequality for the singular operator

Establish the analogue of the Lebeau–Robbiano spectral inequality for the singular operator $A_\mu=-\Delta-\mu|x|^{-2}$ with Dirichlet boundary conditions in dimensions $n\geq 3$.

Background

The paper establishes observability for the heat equation with an inverse-square potential by combining Carleman estimates, real-analyticity away from the singularity, propagation of smallness, and a telescoping-series argument. The authors explicitly state that this strategy avoids relying on the corresponding Lebeau–Robbiano spectral inequality because that inequality is not currently known for the operator AμA_\mu in dimensions n≥3n\geq3. Establishing such an inequality would provide an alternative spectral route to observability and could broaden the analytical tools available for singular parabolic equations.

References

Unlike , our starting point is the classical open set observability inequality in , since the analogue of the Lebeau-Robbiano inequality for $A_\mu$ with $n\geq3$ remains unknown.

— Time optimal control for the heat equation with inverse-square potentials  (2609.29059 - Lu et al., 24 Sep 2026) in Introduction, paragraph beginning “In this paper, we extend the observability inequality…”