Coupled Susy, pseudo-bosons and a deformed $\mathfrak{su}(1,1)$ Lie algebra

TL;DR

This paper extends the analysis of non self-adjoint operators in quantum mechanics by exploring a set of four operators related to pseudo-bosons and a deformed $rak{su}(1,1)$ algebra, revealing their ladder operator properties and connections.

Contribution

It introduces a new framework involving four operators satisfying specific relations, expanding the understanding of pseudo-bosons and their algebraic structures in quantum mechanics.

Findings

Operators form ladder structures

Connection with biorthogonal vectors

Relation to $ ext{D}$-pseudo bosons

Abstract

In a recent paper a pair of operators $a$ and $b$ satisfying the equations $a^\dagger a=bb^\dagger+\gamma\1$ and $aa^\dagger=b^\dagger b+\delta\1$, has been considered, and their nature of ladder operators has been deduced and analysed. Here, motivated by the spreading interest in non self-adjoint operators in Quantum Mechanics, we extend this situation to a set of four operators, $c$, $d$, $r$ and $s$, satisfying $ dc=rs+\gamma\1$ and $cd=sr+\delta\1$, and we show that they are also ladder operators. We show their connection with biorthogonal families of vectors and with the so-called $\D$-pseudo bosons. Some examples are discussed.