Spectrum for a non-unitary one-dimensional two-state quantum walk with one defect

TL;DR

This work provides a complete spectral characterization of a non-unitary, one-defect, 1D two-state quantum walk with chiral symmetry. Using a transfer-matrix framework, it derives four explicit eigenvalues $\lambda_1 = R_{-}(\omega) + iR_{+}(\omega)$, $\lambda_2 = -\overline{\lambda_1}$, $\lambda_3 = -\lambda_1$, and $\lambda_4 = \overline{\lambda_1}$, with one-dimensional eigenvectors for each, and analyzes their existence via a detailed transfer-matrix analysis. The essential spectrum remains the homogeneous spectrum $\Sigma$, while the non-unitary perturbation introduces eigenvalues outside the unit circle, yielding a spiderweb-like spectral geometry. The results extend unitary quantum-walk spectral insights to non-Hermitian settings and illuminate localization phenomena under a single defect. The paper also provides explicit eigenvector constructions and discusses a concrete example at $\omega = -1$.

Abstract

Existence of the eigenvalues of the discrete-time quantum walks is deeply related to localization. Also, for the study of open quantum systems, non-Hermitian systems have attracted much attention. As mathematical models for such systems, non-unitary quantum walks with the chiral symmetry are essential for the study of the topological insulator. In this paper, we give the whole picture of the eigenvalues of a non-unitary one-dimensional two-state quantum walks with one defect and the chiral symmetry.

Figures (1)

• Figure 1: The eigenvalues and essential spectrum of $U_{\omega}$. ($\mathrm{i}$) Dashed line: $\omega > 0$ case; ($\mathrm{ii}$) Dotted line: $\omega < 0$ case; ($\mathrm{iii}$) Black line: the essential spectrum; ($\mathrm{iv}$) Gray line: the unit circle.

Theorems & Definitions (18)

• Theorem 3.1
• Lemma 4.1
• proof
• Lemma 4.2
• proof
• Remark 1
• Lemma 4.3
• Lemma 4.4
• proof
• Lemma 4.5