Generalized Heisenberg Algebras and Fibonacci Series

TL;DR

This paper introduces a generalized Heisenberg algebra framework that models quantum systems with energy spectra following Fibonacci sequences, depending on functions f(x) and g(x), and analyzes their algebraic structure and stability.

Contribution

It constructs a new algebraic structure for quantum systems with Fibonacci-like spectra and classifies its representations based on linear functions and stability analysis.

Findings

Algebra describes systems with Fibonacci energy spectra.

Classification of algebra representations based on linear functions.

Analysis of stability of fixed points in the algebraic structure.

Abstract

We have constructed a Heisenberg-type algebra generated by the Hamiltonian, the step operators and an auxiliar 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 Fibonacci sequence. Moreover, the algebraic structure depends on 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.