Inverse localization of eigenfunctions and applications

We will discuss some situations where the following phenomenon can be shown to hold: let f be any solution of the Helmholtz equation (\Delta f+f=0) on the unit ball of Euclidean space; if one examines the eigenfunctions of the Laplace-Beltrami operator on certain Riemannian manifolds at increasingly high eigenvalues K2 and at small balls of radius 1/K, one ends up finding an eigenfunction that in rescaled coordinates approximates, with a given arbitrarily small precision, the function f.