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

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