  We study the well posedness of the nonlinear Schr\"odinger (NLS) equation
with a point interaction and power nonlinearity in dimension two and three.
Behind the autonomous interest of the problem, this is a model of the evolution
of so called singular solutions that are well known in the analysis of
semilinear elliptic equations. We show that the Cauchy problem for the NLS
considered enjoys local existence and uniqueness of strong (operator domain)
solutions, and that the solutions depend continuously from initial data. In
dimension two well posedness holds for any power nonlinearity and global
existence is proved for powers below the cubic. In dimension three local and
global well posedness are restricted to low powers.
