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

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American Institute of Physics
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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 Schroumldinger 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 Schroumldinger equation. Thus, any (real) potential solvable via a time-dependent Darboux transformation can alternatively be solved by applying an appropriate form-preserving point transformation of the TDSE to a time-independent potential. The pre-eminent 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.
© 1999 American Institute of Physics. F.F., A.G.-L., and M.A.R. would like to acknowledge the partial financial support of the DGICYT under Grant No. PB95-0401. N.K. was supported in part by NSERC Grant No. 0GP0105490.
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