  Quantum systems whose classical counterpart have ergodic dynamics are quantum
ergodic in the sense that almost all eigenstates are uniformly distributed in
phase space. In contrast, when the classical dynamics is integrable, there is
concentration of eigenfunctions on invariant structures in phase space. In this
paper we study eigenfunction statistics for the Laplacian perturbed by a
delta-potential (also known as a point scatterer) on a flat torus, a popular
model used to study the transition between integrability and chaos in quantum
mechanics. The eigenfunctions of this operator consist of eigenfunctions of the
Laplacian which vanish at the scatterer, and new, or perturbed, eigenfunctions.
We show that almost all of the perturbed eigenfunctions are uniformly
distributed in configuration space.
