Schwartz very weak solutions for Schr\"odinger type equations with distributional coefficients

TL;DR

This paper develops a framework for solving Schr"odinger equations with highly singular, distributional coefficients, introducing Schwartz very weak solutions that extend classical solutions to more irregular cases.

Contribution

The authors introduce Schwartz very weak solutions for Schr"odinger equations with distributional coefficients, ensuring existence and uniqueness even when classical solutions are not possible.

Findings

Existence of Schwartz very weak solutions for the Cauchy problem

Uniqueness of solutions modulo negligible perturbations

Consistency with classical solutions for regular coefficients

Abstract

This paper continues the analysis of Schr\"odinger type equations with distributional coefficients initiated by the authors in [3]. Here we consider coefficients that are tempered distributions with respect to the space variable and are continuous in time. We prove that the corresponding Cauchy problem, which in general cannot even be stated in the standard distributional setting, admits a Schwartz very weak solution which is unique modulo negligible perturbations. Consistency with the classical theory is proved in the case of regular coefficients and Schwartz Cauchy data.