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In this paper we study the time dependent Schr\"odinger equation with allpossible self-adjoint singular interactions located at the origin, whichinclude the $\delta$ and $\delta'$-potentials as well as boundary conditions ofDirichlet, Neumann, and Robin type as particular cases.  We derive an explicit representation of the time dependent Green's functionand give a mathematical rigorous meaning to the corresponding integral forholomorphic initial conditions, using Fresnel integrals. Superoscillatoryfunctions appear in the context of weak measurements in quantum mechanics andare naturally treated as holomorphic entire functions. As an application of theGreen's function we study the stability and oscillatory properties of thesolution of the Schr\"odinger equation subject to a generalized pointinteraction when the initial datum is a superoscillatory function.

Green's Function for the Schr\"odinger Equation with a Generalized Point Interaction and Stability of Superoscillations