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

In this paper we point out a close connection between the Darboux transform ation and the group of point transformations which preserve the form of the time-dependent Schroumldinger equation (TDSE). In our main result, we prov e that any pair of time-dependent real potentials related by a Darboux tran sformation for the TDSE may be transformed by a suitable point transformati on into a pair of time-independent potentials related by a usual Darboux tr ansformation for the stationary Schroumldinger equation. Thus, any (real) p otential solvable via a time-dependent Darboux transformation can alternati vely be solved by applying an appropriate form-preserving point transformat ion of the TDSE to a time-independent potential. The pre-eminent role of th e latter type of transformations in the solution of the TDSE is illustrated with a family of quasi-exactly solvable time-dependent anharmonic potentia ls. (C) 1999 American Institute of Physics. [S0022-2488(99)00207-8].