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 \ensuremath{\beta} 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 〈${\mathit{x}}^{2}$〉\ensuremath{\sim}${\mathit{t}}_{0}^{2\mathit{D}}$, where ${\mathit{D}}_{0}$ is the box-counting dimension of the spectrum, and that ${\mathit{D}}_{0}$=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.