# Existence of Propagators for Time Dependent Coulomb-like Potentials

## TL;DR

This paper establishes the existence of propagators for a class of time-dependent Coulomb-like potentials in Schrödinger equations, providing explicit Fourier transforms and analyzing spectral properties within time-dependent density functional theory.

## Contribution

It introduces a new class of softened Coulomb potentials that are time-dependent, proving the existence of propagators and analyzing their spectral characteristics.

## Key findings

- Existence of propagators for the new potentials.
- Explicit Fourier transform of the potentials.
- Potentials are dilatation analytic, enabling spectral analysis.

## Abstract

We prove existence of propagators for a time dependent Schr\"odinger equation with a new class of softened Coulomb potentials, which we allow to be time dependent, in the context of time dependent density functional theory. We compute explicitly the Fourier transform of these new potentials, and provide an alternative proof for the Fourier transform of the Coulomb potential using distribution theory. Finally we show the new potentials are dilatation analytic, and so the spectrum of the corresponding Hamiltonian can be fully characterized.