  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.
