Spatial and Spectral Multifractality of the Local Density of States at the Mobility Edge

We performed numerical calculations of the local density of states (LDOS) at disorder induced localization-delocalization transitions. The LDOS defines a spatial measure for fixed energy and a spectral measure for fixed position. At the mobility edge both measures are multifractal and their generalized dimensions $D(q)$ and $\tilde{D}(q)$ are found to be proportional: $D(q)=d\tilde{D}(q)$, where $d$ is the dimension of the system. This observation is consistent with the identification of the frequency-dependent length scale $L_\omega \propto \omega^{-1/d}$ as an effective system size. The calculations are performed for two- and three-dimensional dynamical network models with local time evolution operators. The energy dependence of the LDOS is obtained from the time evolution of the local wavefunction amplitude of a wave packet, providing a numerically efficient way to obtain information about the multifractal exponents of the system.