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

Abstract: In a 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.