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Quantum walks with an anisotropic coin I: spectral theory

2017; Springer Science+Business Media; Volume: 108; Issue: 2 Linguagem: Inglês

10.1007/s11005-017-1008-1

1573-0530

Serge Richard, Akito Suzuki, Rafael Tiedra de Aldecoa,

Electronic and Structural Properties of Oxides

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.