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

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

1998-09-15

arxiv.org/abs/math-ph/9809013v2

J. Math. Phys. 40 (1999) 3268-3274

Physics

Mathematical Physics

LaTeX2e (with amsmath, amssymb, amscd, cite packages), 11 pages

Scientific paper

10.1063/1.532885

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.

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