Solving the Fractional Schrödinger Equation with Singular Initial Data in the Extended Colombeau Algebra of Generalized Functions

Abstract

This manuscript aims to highlight the existence and uniqueness results for the following Schrödinger problem in the extended Colombeau algebra of generalized functions. $\left\{\begin{array}{c}1/ı\partial /\partial t\mathfrak{u}\left(t,x\right)-\mathfrak{△}\mathfrak{u}\left(t,x\right)+v\left(x\right)\mathfrak{u}\left(t,x\right)=0,t\in {\mathbb{R}}^{+},x\in {\mathbb{R}}^n\text{,}\\ v\left(x\right)=\delta \left(x\right)\text{,}\\ \mathfrak{u}\left(0,x\right)=\delta \left(x\right)\text{,}\end{array}\right.$ where $\delta$ is the Dirac distribution. The proofs of our main results are based on the Gronwall inequality and regularization method. We conclude our article by establishing the association concept of solutions.