Abstract ladder operators and their applications

TL;DR

This paper generalizes ladder operator frameworks for quantum Hamiltonians, exploring their algebraic properties and applications to factorizable Hamiltonians, generalized Heisenberg algebras, and pseudo-bosonic systems.

Contribution

It introduces extended ladder operator relations and applies them to various Hamiltonian classes, broadening the algebraic tools available in quantum mechanics.

Findings

Derived new algebraic relations for ladder operators.

Applied the generalized framework to factorizable Hamiltonians.

Explored connections with pseudo-bosonic and generalized Heisenberg algebra systems.

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 $\lambda\in\mathbb{R}$, $H_0=H_0^\dagger$, 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 $[H_0,Z]=\lambda Z[Z^\dagger, Z]$, and secondly we consider $[H,Z]=\lambda Z$ for some $\lambda\in\mathbb{C}$, $H\neq H^\dagger$. 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.