Quantum walks with an anisotropic coin I: spectral theory

Abstract

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.Comment: 26 page

A weak limit theorem for a class of long-range-type quantum walks in 1d

“…This proof is a slight modification of [12,Lemma 4.13]. In the proof of Proposition 4.5 of [12], we see that U 0 ∈ C 2 (A 0 ). Since J is unitary, it follows thatŨ 0 ∈ C 2 (A) ⊂ C 1+ (A).…”

“…Now we check two conditions in Theorem 4.3. We note that A 0 has a following form on H fin : For more details, see the proof of [12,Lemma 4.10]. On H fin , it follows that…”

“…In this section, we recall some definitions and notations related to commutator theory. We mainly refer [2,12]. We denote the set of bounded linear operators from a Hilbert space Here we introduce three regularity conditions which are stronger than T ∈ C 1 (A).…”

“…X is essentially self-adjoint [12,Lemma 4.3] and we denote the closure of X by the same symbol. Moreover we introduce the following operator:…”

“…on H fin can be extended to a bounded operator on H. We denote it by the same symbol. According to [2, p.325-328] or [12,Lemma 4.13], following estimate holds:…”