Limits of regularizations for generalized function solutions to the Schr\"odinger equation with "square root of delta" initial value

TL;DR

This paper investigates the limits of regularization methods for generalized solutions of the Schrödinger equation with singular initial data, focusing on convergence properties of regularized solutions with 'square root of delta' initial values.

Contribution

It provides a detailed analysis of the convergence behavior of Colombeau-type generalized solutions for Schrödinger equations with non-smooth initial conditions, highlighting the limitations of regularization techniques.

Findings

Regularized solutions converge in certain dual spaces.

Limitations of regularizations are identified for specific singular initial data.

Compatibility with classical solutions is discussed.

Abstract

We briefly review results on Colombeau type generalized solutions to the Cauchy problem for linear Schr\"odinger-type equations with non-smooth principal part and their compatibility with classical and distributional solutions. In the main part, we study convergence properties of regularized solutions to the standard Schr\"odinger equation with initial values corresponding to "square roots" of Dirac measures in various duals of classical subspaces of the space of continuous functions.