Darboux transformations for the matrix Schr\"odinger equation
The intertwining operator technique is applied to the matrix Schr\"odinger equation. The first- and second-order matrix Darboux transformations, factorization, supersymmetry, chains of transformations are studied. A relation between the matrix first-order Darboux transformation and supersymmetry is considered. The main differences between the matrix supersymmetry and the standard scalar supersymmetry for one Schr\"odinger equation and for the coupled systems of equations are
discussed. An interrelation is established between the differential and integral transformation operators. It is shown that in a particular case the second-order integral transformations turn into expressions for matrix solutions and potentials obtained by the inverse problem with degenerate integral kernels. Using unitary time-dependent transformations, we construct exactly soluble time-dependent generalizations of exactly soluble time-independent equations. The approach opens new opportunities for modelling the quantum dynamic systems with desired properties, for instance, quantum wells with the properties of dynamic localization.