# Klein-Gordon equations with homogeneous time-dependent electric fields

## TL;DR

This paper extends the analysis of Klein-Gordon equations to include homogeneous electric fields that vary over time, building on Veselić's earlier work on time-independent fields.

## Contribution

It generalizes previous results by establishing bounds for the time-evolution propagator with time-dependent electric fields.

## Key findings

- Boundaries of the propagator are established for time-dependent fields.
- Extension of Veselić's 1991 results to more general electric field conditions.
- Provides mathematical framework for analyzing Klein-Gordon equations with dynamic electric environments.

## Abstract

We consider a system associated to Klein-Gordon equations with homogeneous time-dependent electric fields. The upper and lower boundaries of a time-evolution propagator for this system were proven by Veseli\'c in 1991 for electric fields that are independent of time. We extend this result to time-dependent electric fields.