Fractal position spectrum for a class of oscillators

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.

9 pages, 2 figures

References in corpus (2)

• First experimental realization of the Dirac oscillator
• Playing relativistic billiards beyond graphene

Cited by in corpus (1)

• Dynamical symmetry in a minimal dimeric complex