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In this expository note, we discuss spatially inhomogeneous quantum walks inone dimension and describe a genre of mathematical methods that enables one totranslate information about the time-independent eigenvalue equation for theunitary generator into dynamical estimates for the corresponding quantum walk.To illustrate the general methods, we show how to apply them to a 1D coinedquantum walk whose coins are distributed according to an element of theThue--Morse subshift.

Resolvent Methods for Quantum Walks with an Application to a Thue-Morse Quantum Walk