On form-preserving transformations for the time-dependent Schr\"odinger equation
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.
Cite
@article{arxiv.math-ph/9809013,
title = {On form-preserving transformations for the time-dependent Schr\"odinger equation},
author = {Federico Finkel and Artemio Gonzalez-Lopez and Niky Kamran and Miguel A. Rodriguez},
journal= {arXiv preprint arXiv:math-ph/9809013},
year = {2009}
}
Comments
LaTeX2e (with amsmath, amssymb, amscd, cite packages), 11 pages