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

Abstract

In a recent paper a pair of operators a and b satisfying the equations a † a = bb † + γ1 1 and aa † = b † b + δ1 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 + γ1 1 and cd = sr + δ1 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.