Form-preserving transformations for the time-dependent Schrödinger equation in \((n+1)\) dimensions (Q2467630)

The author defines a form-preserving transformation for the time-dependent Schrödinger equation (TDSE) in \((n+1)\) dimensions. This transformation changes the dependent and independent variables of a TDSE in such a way that the transformed equation maintains the form of a TDSE, but for a potential different from the original one. It is shown that the form-preserving transformation is invertible and preserves \(L^2\)-normalizability. The form-preserving transformation is the method most used for generating solvable cases of the TDSE. The author gives a class of time-dependent TDSEs that can be mapped onto stationary Schrödinger equations by form-preserving transformation. Several examples are included. In particular, a solvable, time-dependent potential of Coulombic ring-shaped type together with the corresponding exact solution of the TDSE in \((3+1)\) dimensions is generated. Moreover, author considers TDSEs with position-dependent (effective) masses and shows that there is no form-preserving transformation between them and the conventional TDSEs, if the spatial dimension of the system is higher than one.