Quasi-Fibonacci oscillators

Abstract

We study the properties of sequences of the energy eigenvalues for some generalizations of q-deformed oscillators including the p,q-oscillator, the 3-, 4- and 5-parameter deformed oscillators given in the literature. It is shown that most of the considered models belong to the class of so-called Fibonacci oscillators for which any three consequtive energy levels satisfy the relation E_{n+1}=\lambda E_n+\rho E_{n-1} with real constants \lambda, \rho. On the other hand, for certain \mu-oscillator known from 1993 we prove the fact of its non-Fibonacci nature. Possible generalizations of the three-term Fibonacci relation are discussed among which we choose, as most adequate for the \mu$-oscillator, the so-called quasi-Fibonacci (or local Fibonacci) property of the energy levels. The property is encoded in the three-term quasi-Fibonacci (QF) relation with non-constant, n-dependent coefficients \lambda and \rho. Various aspects of the QF relation are elaborated for the \mu-oscillator and some of its extensions.