  In this paper we study the time dependent Schr\"odinger equation with all
possible self-adjoint singular interactions located at the origin, which
include the $\delta$ and $\delta'$-potentials as well as boundary conditions of
Dirichlet, Neumann, and Robin type as particular cases.
  We derive an explicit representation of the time dependent Green's function
and give a mathematical rigorous meaning to the corresponding integral for
holomorphic initial conditions, using Fresnel integrals. Superoscillatory
functions appear in the context of weak measurements in quantum mechanics and
are naturally treated as holomorphic entire functions. As an application of the
Green's function we study the stability and oscillatory properties of the
solution of the Schr\"odinger equation subject to a generalized point
interaction when the initial datum is a superoscillatory function.
