We estimate the spreading of the solution of the Schrödinger equation asymptotically in time, in term of the fractal properties of the associated spectral measures. For this, we exhibit a lower bound for the moments of order $p$ at time $T$ for the state $\psi$ defined by $[\frac{1}{T}{\int }_0^T\parallel |X{|}^{p/2}{\mathrm{e}}^{-itH}\psi {\parallel }^2\mathrm{d}t]$. We show that this lower bound can be expressed in term of the generalized Rényi dimension of the spectral measure ${\mu }_{\psi }$ associated to the hamiltonian $H$ and the state $\psi$. We especially concentrate on continuous models.

We study the dispersion relations and spectra of invariant Schrödinger operators on a graphyne structure (lithographite). In particular, description of different parts of the spectrum, band-gap structure, and Dirac points are provided.