Abstract ladder operators and their applications

Abstract

We consider a rather general version of ladder operator $Z$ used by some authors in few recent papers, $[H_0,Z]=\lambda Z$ for some $\lambda \in \mathbb{R}$, $H_0=H_0^{†}$, 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 $[H_0,Z]=\lambda Z[Z^{†},Z]$, and secondly we consider $[H,Z]=\lambda Z$ for some $\lambda \in \mathbb{C}$, $H\ne H^{†}$. 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.

arXiv commentaccepted in Journal of Physics A

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