Observability Inequality for a Multi-Dimensional Degenerate Schrödinger Equation with an Interior Degeneracy Point

Abstract

We study the observability inequality for a multi-dimensional degenerate Schrödinger equation possessing an interior degeneracy point, where the principal coefficient vanishes at the origin inside the domain and breaks ellipticity. Within a weighted Sobolev space, we establish the spectral properties, well-posedness, and regularity estimates for the degenerate operator. To tackle the failure of classical controllability tools, we develop a regularization approximation scheme. We construct a sequence of non-degenerate approximating problems. We then establish uniform a-priori estimates and solution convergence. Furthermore, we derive an observability inequality for the regularized problems and pass to the limit. This yields a non-standard observability inequality for the original degenerate equation for sufficiently large observation time, which quantifies the influence of the degeneracy strength on the observation time. Crucially, we establish the precise parameter restriction $3α+ N > 4$ to ensure that every test function in $C_0^\infty(Ω)$ belongs to $D(\mathcal A^{\frac 32})$.