  We perform the spectral analysis of the evolution operator U of quantum walks
with an anisotropic coin, which include one-defect models, two-phase quantum
walks, and topological phase quantum walks as special cases. In particular, we
determine the essential spectrum of U, we show the existence of locally
U-smooth operators, we prove the discreteness of the eigenvalues of U outside
the thresholds, and we prove the absence of singular continuous spectrum for U.
Our analysis is based on new commutator methods for unitary operators in a
two-Hilbert spaces setting, which are of independent interest.
