Fractal position spectrum for a class of oscillators

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.