# Exactly Solvable Time-Dependent Oscillator-Like Potentials Generated by Darboux Transformations

## TL;DR

This paper introduces a method to generate exactly solvable, time-dependent oscillator-like potentials by applying a Darboux transformation that incorporates time as a parameter, extending traditional spatial-only approaches.

## Contribution

It extends the Darboux transformation technique to include time dependence, enabling the construction of new exactly solvable time-dependent potentials.

## Key findings

- Constructed new time-dependent oscillator-like potentials.
- Demonstrated the method's applicability to the Schrödinger equation.
- Extended the Darboux transformation framework to time-variable scenarios.

## Abstract

The stationary Schroedinger equation of the harmonic oscillator is deformed by a Darboux transformation to construct time-dependent potentials with the oscillator profile. The Darboux (supersymmetric or factorization) method is usually developed in the spatial variables of the Schroedinger equation. Here we follow a variation introduced by Bagrov, Samsonov and Shekoyan to include the time-variable as a parameter of the transformation.