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We develop further the approach to upper and lower bounds in quantum dynamicsvia complex analysis methods which was introduced by us in a sequence ofearlier papers. Here we derive upper bounds for non-time averaged outsideprobabilities and moments of the position operator from lower bounds fortransfer matrices at complex energies. Moreover, for the time-averagedtransport exponents, we present improved lower bounds in the special case ofthe Fibonacci Hamiltonian. These bounds lead to an optimal description of thetime-averaged spreading rate of the fast part of the wavepacket in the largecoupling limit. This provides the first example which demonstrates that thetime-averaged spreading rates may exceed the upper box-counting dimension ofthe spectrum.

Quantum Dynamics via Complex Analysis Methods: General Upper Bounds  Without Time-Averaging and Tight Lower Bounds for the Strongly Coupled  Fibonacci Hamiltonian