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On form-preserving transformations for the time-dependent Schrödinger equation

Federico Finkel·Artemio Gonzalez-Lopez·Niky Kamran·Miguel A. Rodriguez
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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 Schrödinger 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 Schrödinger 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.

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Federico Finkel, Artemio Gonzalez-Lopez, Niky Kamran, Miguel A. Rodriguez. 1999-05-14. On form-preserving transformations for the time-dependent Schrödinger equation. https://doi.org/10.1063/1.532885

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