Greens Function for the Schrodinger Equation with a Generalized Point Interaction and Stability of Superoscillations

In this paper we study the time dependent Schrodinger 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 Greens 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 Greens function we study the stability and oscillatory properties of the solution of the Schrodinger equation subject to a generalized point interaction when the initial datum is a superoscillatory function.