Time-dependent Parasupersymmetry in Quantum Mechanics

Parasupersymmetry of the one-dimensional time-dependent Schrödinger equation is established. It is intimately connected with a chain of the time-dependent Darboux transformations. As an example a parasupersymmetric model of nonrelativistic free particle with threefold degenerate discrete spectrum of an integral of motion is constructed. 1. Supersymmetric quantum mechanics originally introduced by Witten [1] for investigation of the supersymmetry breaking in quantum field theories attracts now a considerable attention from the different points of view (see a recent survey [2]). Different generalizations of the initial constructions are known. We can cite N -extended supersymmetric quantum mechanics [3], higher-derivative supersymmetry in quantum mechanics [4], and parasupersymmetric quantum mechanics [5]-[6]. It is worth stressing that all these constructions deal with a stationary Schrödinger equation and consequently can be referred to the stationary supersymmetric quantum mechanics. The nonstationary supersymmetric quantum mechanics needs to be developed. First steps in this direction have been made in Refs. [7] [9]. An essential ingredient of the stationary supersymmetric quantum mechanics constitutes the well-known Darboux transformation [11] of the stationary Schrödinger equation in mathematics (see Ref. [10]). Recently the time-dependent Darboux transformation has been proposed [7] and on its base the supersymmetry of the nonstationary Schrödinger equation has been established [8]. This transformation seems to be very fruitful for construction of the nondispersive wave packets (coherent states) for anharmonic oscillator Hamiltonians with equidistant and quasiequidistant spectra [9]. In this paper we establish the parasupersymmetry of the time-dependent Schrödinger equation which is intimately connected with a chain of the time dependent Darboux