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 (D) over tilde(q) are found to be proportional: D(q)=d (D) over tilde(q), where d is the dimension of the system. This observation is consistent with the identification of the frequency-dependent length scale L(omega)proportional to 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 wave-function amplitude of a wave packet, providing a numerically efficient way to obtain information about the multifractal exponents of the system.