Darboux transformation of the Schrodinger equation

The recent developments in the theory of the generation of potentials for which the Schrodinger equation has an exact solution are discussed, The generalization of the Darboux transformation to the nonstationary Schrodinger equation is studied in detail. The supersymmetric generalization of the nonstationary Schrodinger equation is formulated, Versions corresponding to exact and spontaneously broken supersymmetry are discussed. New, exactly solvable nonstationary potentials are obtained as examples. The stationary Darboux transformation is viewed as a special case of the new transformation. Families of isospectral potentials with the spectra of the harmonic oscillator and the hydrogen-like atom are obtained. The effectiveness of these methods for describing the coherent states of the transformed Hamiltonians is demonstrated. (C) 1997 American Institute of Physics.