The Fibonacci-Deformed Harmonic Oscillator

Abstract

Given an infinite countable, strictly increasing sequence of positive real numbers, it is possible to associate to it a triplet (Identity I, Creation a†, Annihilation a) of operators acting on some Hilbert space and an overcomplete set of coherent states labeled with complex numbers (“holomorphic map”). The triplet generates a quantum algebra that reduces to the q-oscillator algebra when the sequence is the set of q-deformed natural numbers.

We shall present here another type of quantum algebra. The latter is based on the quasiperiodic sequence of numbers (the half-infinite Fibonacci chain) that are the counterparts of the positive integers in “golden-mean” basis. As a byproduct, we shall give a survey on that Fibonacci zoology: new remarkable numbers and related special functions.