Systems with higher-order shape invariance: spectral and algebraic properties

We study systems of two intertwining relations of first or second order for the same (up to a constant shift) partner Schrodinger operators, It is shown that the corresponding Hamiltonians possess a higher order shape invariance which is equivalent to the ladder equation, We analyze with particular attention irreducible second order Darboux transformations which together with the first order act as building blocks. For the third order shape-invariance irreducible Darboux transformations entail only one sequence of equidistant levels while for the reducible case the structure consists of up to three infinite sequences of equidistant levels and, in some cases, singlets or doublets of isolated levels. (C) 2000 Published by Elsevier Science B.V. All rights reserved.