  We consider momentum push-forwards of measures arising as quantum limits
(semi-classical measures) of eigenfunctions of a point scatterer on the
standard flat torus $\mathbb T^2 = \mathbb R^2/\mathbb Z^{2}$. Given any
probability measure arising by placing delta masses, with equal weights, on
$\mathbb Z^2$-lattice points on circles and projecting to the unit circle, we
show that the mass of certain subsequences of eigenfunctions, in momentum
space, completely localizes on that measure and is completely delocalized in
position (i.e., concentration on Lagrangian states.) We also show that the
mass, in momentum, can fully localize on more exotic measures, e.g. singular
continous ones with support on Cantor sets. Further, we can give examples of
quantum limits that are certain convex combinations of such measures, in
particular showing that the set of quantum limits is richer than the ones
arising only from weak limits of lattice points on circles. The proofs exploit
features of the half-dimensional sieve and behavior of multiplicative functions
in short intervals.
