Quantum walks with an anisotropic coin I : spectral theory

Serge Richard (1) , A Suzuki (2) , R Tiedra de Aldecoa (3)

1 PSPM - Probabilités, statistique, physique mathématique
2 Shinshu University [Nagano]
3 Departamento de Matemáticas [Santiago de Chile]

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.

Mots clés: Quantum walks; Spectral theory; Commutator methods; Unitary operators

Domaines: Physique mathématique [math-ph]

Serge Richard, A Suzuki, R Tiedra de Aldecoa. Quantum walks with an anisotropic coin I : spectral theory. Letters in Mathematical Physics, 2018, 108, pp.331-357. ⟨hal-01588656⟩