Generalized Heisenberg algebras and Fibonacci series

We have constructed a Heisenberg-type algebra generated by the Hamiltonian, the step operators and an auxiliary operator. This algebra describes quantum systems having eigenvalues of the Hamiltonian depending on the eigenvalues of the two previous levels. This happens, for example, for systems having the energy spectrum given by a Fibonacci sequence. Moreover, the algebraic structure depends on the two functions f ( x ) and g ( x ). When these two functions are linear we classify, analysing the stability of the fixed points of the functions, the possible representations for this algebra.