Abstract ladder operators and their applications

We consider a rather general version of ladder operator Z used by some authors in few recent papers, [H 0, Z] = λZ for some, H0=H0†, and we show that several interesting results can be deduced from this formula. Then we extend it in two ways: first we replace the original equality with formula [H 0, Z] = λZ[Z †, Z], and secondly we consider [H, Z] = λZ for some C, H ≠ H †. In both cases many applications are discussed. In particular we consider factorizable Hamiltonians and Hamiltonians written in terms of operators satisfying the generalized Heisenberg algebra or the D pseudo-bosonic commutation relations.