GENERALIZED HEISENBERG ALGEBRAS AND k - GENERALIZED

Curado and Rego-Monteiro introduced in [2] a new algebraic structure generalizing the Heisenberg algebra and containing also the q-deformed oscillator as a particular case. This algebra, called generalized Heisenberg algebra, depends on an analytical function f and the eigenvalues αn of the Hamiltonian are given by the one-step recurrence αn+1 = f(αn). This structure has been used in different physical situations, see the references given in the recent paper [1]. In the same paper [1] de Souza et al. introduced an extended two-step Heisenberg algebra having many interesting properties. In particular, they showed that in certain special cases the eigenvalues of the involved Hamiltonian are given by the well-known Fibonacci numbers, i.e., satisfy a two-step recurrence. It is the aim of the present note to show how one may introduce for arbitrary natural numbers k an extended k-step Heisenberg algebra which reduces for k = 2 to the one discussed in [1] (and for k = 1 to the one in [2]). In particular, the eigenvalues of the involved Hamiltonian are given in special cases by the k-generalized Fibonacci numbers [3]. For the convenience of the reader we now recall the structure of the extended twostep Heisenberg algebra, using the notations of [1]. It is generated by the set of operators {H,a†, a, J3} where H = H † is the Hamiltonian, a and a† with a = (a†)† are the usual step operators and J3 = J † 3 is an additional operator. These operators satisfy the following relations: