  We construct a specific family of eigenfunctions for a Laplace operator with
coefficients having a jump across an interface. These eigenfunctions have an
exponential concentration arbitrarily close to the interface, and therefore
could be considered as whispering gallery modes. The proof is based on an
appropriate Agmon estimate. We deduce as a corollary that the quantitative
unique continuation result for waves propagating in singular media proved by
the author in [Fil22] is optimal.
