In this paper we study the time dependent Schrödinger equation with all possible self-adjoint singular interactions located at the origin, which include the δ and ${\delta }^{\prime }$-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 Schrödinger equation subject to a generalized point interaction when the initial datum is a superoscillatory function.

在本文中，我们研究了与时间相关的Schrödinger方程，其中所有可能的自伴随奇异相互作用都位于原点，包括δ和${\delta }^{\prime }$势以及Dirichlet，Neumann和Robin类型的边界条件。我们使用菲涅尔积分，导出了与时间相关的格林函数的显式表示，并为全纯初始条件的相应积分提供了严格的数学含义。超级振荡功能出现在量子力学中的弱测量范围内，自然被视为全纯全功能。作为格林函数的一种应用，我们研究了当初始基准为超振荡函数时，服从广义点相互作用的薛定ding方程解的稳定性和振动性质。