Long-time behavior of spreading solutions of Schrödinger and diffusion equations

We investigate the asymptotic time behavior of the solutions of a large class of linear differential equations that generalize the free-particle Schrödinger and diffusion equations, containing the standard ones as particular cases. We find general scalings that depend only on characteristic features of both the arbitrary initial condition and the Green function associated with the evolution equation. Basically, the amplitude of a long-time solution can be expressed in terms of low order moments of the initial condition (if finite) and low order spatial derivatives of the Green function. These derivatives can also be of the fractional type, which naturally arise when moments are divergent. We apply our results to a large class of differential equations that includes the fractional Schrödinger and Lévy diffusion equations. In particular, we show that, except for threshold cases, the amplitude of a packet may follow the asymptotic law t-alpha, with arbitrary positive alpha.