Supersymmetry of the Nonstationary Schrödinger Equation and Time-Dependent Exactly Solvable Quantum Models

An essential ingredient of the conventional supersymmetric quantum mechanics (for reviews see [1]) is the well known Darboux transformation [2] for the stationary Schrödinger equation. This transformation permits us to construct new exactly solvable stationary potentials from known ones. Similar constructions may be developed for the time-dependent Schrödinger equation [3]. Our approach to Darboux transformation is based on a general notion of transformation operator introduced by Delsart [4]. In terms of this notion Darboux [2] studied differential first order transformation operators for the Sturm-Liouville problem. This is the reason in our opinion to call every differential transformation operator Darboux transformation operator. Different approach to Darboux transformation is exposed in the book [5]. It is worthwhile mentioning that our approach in contrast to that of Ref. [5] leads to real potential differences. This property is crucial for constructing the supersymmetric extension of the nonstationary Schrödinger equation.