On counterexamples to unique continuation for critically singular wave equations

Abstract: We consider wave equations with a critically singular potential $\xi \cdot \sigma^{-2}$ diverging as an inverse square at a hypersurface $\sigma = 0$. Our aim is to construct counterexamples to unique continuation from $\sigma = 0$ for this equation, provided there exists a family of null geodesics trapped near $\sigma = 0$. This extends the classical geometric optics construction of Alinhac-Baouendi (i) to linear differential operators with singular coefficients, and (ii) over non-small portions of $\sigma = 0$ - by showing that such counterexamples can be further continued as long as this null geodesic family remains trapped and regular. As an application to relativity and holography, we construct counterexamples to unique continuation from the conformal boundaries of asymptotically Anti-de Sitter spacetimes for some Klein-Gordon equations; this complements the unique continuation results of the second author with Chatzikaleas, Holzegel, and McGill and suggests a potential mechanism for counterexamples to the AdS/CFT correspondence.