Physics Department, Tomsk State University,
36 Lenin Avenue, Tomsk, 634050 Russia
E-mail: samsonov@phys.tsu.ru

Time-dependent supersymmetry and parasupersymmetry in quantum mechanics

Abstract:
An essential ingredient of the conventional supersymmetric quantum mechanics is the well known Darboux transformation for the stationary Schroedinger equation. This transformation permits us to construct new exactly solvable stationary potentials from known ones. Similar constructions may be developed for the time-dependent Schroedinger equation. We obtain thus the time-dependent Darboux transformation which is a basis for the time dependent supersymmetry. A chain of these
transformations creates the time-dependent parasupersymmetry algebra.
Our approach to the problem in contrast to that known in soliton theory generates only real exactly solvable time-dependent potentials which is of crucial importance in quantum mechanics. The known time-independent supersymmetry and parasupersymmetry is a particular case of developed constructions.