Multifractal Energy Spectra and Their Dynamical Implications

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Abstract
We present a method for constructing lattice tridiagonal Hamiltonians having a preassigned multifractal measure as local spectrum. Using this construction we investigate how the fractal structure of the spectrum affects the motion of wave packets. We find that the quantum evolution is intermittent: The moments of particle's position on the lattice are characterized by a nontrivial scaling function, even when the spectrum is a one-scale, balanced Cantor set. Numerical data show that the minimum scaling exponent is always larger than the information dimension of the spectral measure, and qualitatively follows the behavior of this quantity, as the spectral measure is varied.