Abstract
The author presents a method to construct potentials for Schrodinger equations with some prescribed features, for which all eigenfunctions and the time-dependent propagator can be explicitly calculated. The prescribed features can be formulated by choosing arbitrarily the N lowest eigenvalues. Alternatively one can prescribe some qualitative behaviour for the potential, like the number and relative depths of wells and barriers. The results can also be applied to the construction of Fokker-Planck models with prescribed properties and explicitly calculable transition probability density. The method is based on ideas of supersymmetric quantum mechanics and the theory of solitons, that can be traced back to the work of Darboux (1894) and Crum (1955).