On form-preserving transformations for the time-dependent Schr\"odinger equation

TL;DR

This paper reveals a connection between Darboux transformations and point transformations that preserve the form of the time-dependent Schrödinger equation, enabling solutions of time-dependent potentials via time-independent methods.

Contribution

It proves that time-dependent Darboux-related potentials can be transformed into time-independent ones, simplifying their solvability using known stationary Schrödinger techniques.

Findings

Time-dependent potentials can be mapped to time-independent ones via form-preserving transformations.

Any real potential solvable by time-dependent Darboux transformations can be solved through stationary Schrödinger methods.

Illustrated with quasi-exactly solvable anharmonic potentials.

Abstract

In this paper we point out a close connection between the Darboux transformation and the group of point transformations which preserve the form of the time-dependent Schr\"odinger equation (TDSE). In our main result, we prove that any pair of time-dependent real potentials related by a Darboux transformation for the TDSE may be transformed by a suitable point transformation into a pair of time-independent potentials related by a usual Darboux transformation for the stationary Schr\"odinger equation. Thus, any (real) potential solvable via a time-dependent Darboux transformation can alternatively be solved by applying an appropriate form-preserving transformation of the TDSE to a time-independent potential. The preeminent role of the latter type of transformations in the solution of the TDSE is illustrated with a family of quasi-exactly solvable time-dependent anharmonic potentials.