R. Killip, A. Kiselev and Y. Last
Dynamical Upper Bounds on Wavepacket Spreading
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ABSTRACT.  We derive a general upper bound on the spreading 
rate of wavepackets in the framework of Schr\"odinger 
time evolution. Our result consists of showing that 
a portion of the wavepacket cannot escape outside a 
ball whose size grows dynamically in time, where 
the rate of this growth is determined by 
properties of the spectral measure and by 
spatial properties of solutions 
of an associated time independent Schr\"odinger 
equation. We also derive a new lower bound on 
the spreading rate, which is strongly connected 
with our upper bound. We apply these new bounds to 
the Fibonacci Hamiltonian---the most studied 
one-dimensional model of quasicrystals. As a result, we 
obtain for this model upper and lower dynamical bounds 
establishing wavepacket spreading rates which are 
intermediate between ballistic transport and localization. 
The bounds have the same qualitative behavior in the 
limit of large coupling.