  We prove an analogue of Shnirelman, Zelditch and Colin de Verdiere's Quantum
Ergodicity Theorems in a case where there is no underlying classical
ergodicity. The system we consider is the Laplacian with a delta potential on
the square torus. There are two types of wave functions: old eigenfunctions of
the Laplacian, which are not affected by the scatterer, and new eigenfunctions
which have a logarithmic singularity at the position of the scatterer. We prove
that a full density subsequence of the new eigenfunctions equidistribute in
phase space. Our estimates are uniform with respect to the coupling parameter,
in particular the equidistribution holds for both the weak and strong coupling
quantizations of the point scatterer.
