  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.

Mesoscale and Nanoscale Physics