Conditionally exactly solvable potentials: A supersymmetric construction method

We present in this paper a rather general method for the construction of so-called conditionally exactly solvable potentials. This method is based on algebraic tools known from supersymmetric quantum mechanics. Various families of one-dimensional potentials are constructed whose corresponding Schr\"odinger eigenvalue problem can be solved exactly under certain conditions of the potential parameters. Examples of quantum systems on the real line, the half line as well as on some finite interval are studied in detail.