Fractal position spectrum for a class of oscillators

TL;DR

This paper demonstrates that the position operator for certain $f$-deformed oscillators exhibits a fractal spectrum similar to the Cantor set, using a connection to Harper's model and numerical diagonalization.

Contribution

It establishes a link between $f$-deformed oscillators and fractal spectra, revealing a new class of quantum systems with Cantor-set spectra.

Findings

Position spectrum is homeomorphic to the Cantor set.

The spectrum resembles Hofstadter's butterfly pattern.

Diagonalization confirms fractal spectral structure.

Abstract

We show that the position operator for a class of $f$-deformed oscillators has a fractal spectrum, homeomorphic to the Cantor set, via a unitary transformation to Harper's model. The class corresponds to a choice of ergodic operators for the deformation function. Hofstadter's butterfly is plotted by direct diagonalization of a position operator with an originally vanishing diagonal. This is equivalent to a one-dimensional hamiltonian without potential.