Backlund transformations for the Schrodinger equation with a spectral dependence in the potential

The Schrodinger equation on the line with a potential depending linearly on the spectral parameter is considered. It is shown that this spectral problem admits one-parameter (called elementary) Backlund transformations and two-parameter (called full) Backlund transformations. The transformations of the spectral data under all these Backlund transformations (BTS) are explicitly computed. Different nonlinear superposition formulae for the different BTS considered are given and used to obtain soliton solutions of the evolution equations associated with the spectral problem.