Abstract ladder operators for non self-adjoint Hamiltonians, with applications

TL;DR

This paper extends the concept of ladder operators to non-self-adjoint Hamiltonians, providing existence criteria for coherent states and applying the framework to pseudo-quons and deformed Heisenberg algebras in quantum mechanics.

Contribution

It introduces a detailed analysis of ladder operators for non-self-adjoint Hamiltonians, including existence criteria for coherent states and applications to specific quantum systems.

Findings

Pseudo-quons can diagonalize non-self-adjoint oscillator Hamiltonians.

Existence criteria for coherent states as eigenstates of lowering operators.

Application of the framework to deformed quantum algebras.

Abstract

Ladder operators are useful, if not essential, in the analysis of some given physical system since they can be used to find easily eigenvalues and eigenvectors of its Hamiltonian. In this paper we extend our previous results on abstract ladder operators considering in many details what happens if the Hamiltonian of the system is not self-adjoint. Among other results, we give an existence criterion for coherent states constructed as eigenstates of our lowering operators. In the second part of the paper we discuss two different examples of our framework: pseudo-quons and a deformed generalized Heisenberg algebra. Incidentally, and interestingly enough, we show that pseudo-quons can be used to diagonalize an oscillator-like Hamiltonian written in terms of (non self-adjoint) position and momentum operators which obey a deformed commutation rule of the kind often considered in minimal length quantum mechanics.