Spectral dimension and dynamics for Harper’s equation

The spectrum of Harper's equation (a model for Bloch electrons in a magnetic field) is a fractal Cantor set if the ratio {β} of the area of a unit cell to that of a flux quantum is not a rational number. It has been conjectured that the second moment of an initially localized wave packet has a power-law growth of the form 〈x²〉{~}t₀2D, where D₀ is the box-counting dimension of the spectrum, and that D₀=1/2. We present numerical results on the dimension of the spectrum and the spread of a wave packet indicating that these relationships are at best approximate. We also present heuristic arguments suggesting that there should be no general relationships between the dimension and the spread of a wave packet.