Irreducible Darboux Transformations

Detailed analysis of second order Darboux transformations for a growing potential of the Schrödinger equation is given. It is shown, that except for the known annihilation of two levels from a discrete spectrum other possibilities can be released as well: a) the transformed spectrum remains identical to the initial one; b) one level is deleted from a spectrum; c) one additional level is created; d) two additional levels are created.