Spectra and pseudospectra of non-Hermitian Toeplitz operators: Eigenvector decay transitions in banded and dense matrices

TL;DR

This work extends Floquet-Bloch-type analysis to non-Hermitian Toeplitz operators with algebraic off-diagonal decay, establishing sharp eigenvector decay estimates and a convergent pseudoeigenvector framework that connects semi-infinite theory to large finite dense matrices. It reveals how the complex band structure governs localization, showing a transition from exponential skin localization to algebraic bulk localization as coupling range increases, and it applies these insights to three-dimensional subwavelength resonator systems with defects. The results unify spectral theory, pseudospectra, and defect analysis to explain non-Hermitian localization transitions in banded and dense Toeplitz matrices, with practical implications for metamaterials and photonics. Numerical demonstrations illustrate the limits of band truncations versus dense coupling and quantify defect-induced localization transitions via the gauge capacitance matrix.

Abstract

Using a generalised Floquet-Bloch theory, we present a mathematical method to construct eigenvectors for non-Hermitian Toeplitz operators. We extend the method to both banded Toeplitz operators and those with algebraically decaying, fully dense off-diagonal structure. We present sharp decay estimates for the amplitude of bulk eigenmodes as well as eigenmodes associated with defect eigenfrequencies inside the spectral band gap. The validity of those results is illustrated numerically and we show that banded approximations give poor reconstructions of the dense operators, due to the slow algebraic decay. We apply the insights gained to model the non-Hermitian skin effect in a three-dimensional system of subwavelength resonators, where the corresponding operator exhibits only algebraic decay of off-diagonal entries. We use our approach to demonstrate the fundamental mechanism responsible for the transition between the non-Hermitian skin effect and defect-induced localisation in the bulk.

Figures (11)

• Figure 2.1: We plot the complex band structure of two different $m$-banded, pseudo-Hermitian Toeplitz operators. Here, the real part $\alpha$ (black dashed lines) describes the local oscillations, while the imaginary part $\beta$ (red solid lines) describes the exponential decay of the eigenmodes. The decay length of the eigenmodes of a finite truncation of the operator (blue crosses) agree remarkably well with the complex band structure as asserted by Theorem \ref{['thm: eigvec m banded construction']}. Unlike the tridiagonal case, the decay length of the eigenmodes is frequency dependent.
• Figure 2.2: The asymptotics of $\beta(m)$ are correctly predicted by \ref{['eq: asymptotic beta']} and $\beta(m) \to 0$ as $m\to \infty$.
• Figure 2.3: The algebraic bound established by Jaffard correctly bounds the first entries of the eigenvector. As discussed previously, both Demko's exponential bound Theorem \ref{['thm: demko off diagonal']} and the bound by the complex band structure are valid. However, the exponential bound achieved by the complex band structure is much tighter.
• Figure 2.4: The red lines denote the complex band structure for a $50$-banded non-Hermitian Toeplitz matrix with algebraic off-diagonal decay. The red region denotes the open spectrum $\sigma_{\text{open}}$ defined in \ref{['def: open spectrum']}. Eigenvectors for eigenvalues in the open spectrum are exponentially skin localised, and the exponential decay rate is predicted by the complex band structure. The green shaded region, $\sigma_{\mathrm{wind}}^\mathsf{c}$ denotes the eigenvalues for which Jaffard's theorem is applicable. The blue region, which fully contains the spectrum, denotes the winding region where Jaffard's estimate is not applicable. At the transition between $\sigma_{\mathrm{wind}}$ and $\sigma_{\mathrm{wind}}^\mathsf{c}$, the eigenmodes are constant leading up to the defect. Consequently the qualitative decay behaviour is accurately captured by the regions $\sigma_{\mathrm{wind}}$, $\sigma_{\mathrm{wind}}^\mathsf{c}$ and $\sigma_{\text{open}}$
• Figure 2.5: We numerically verify the bound established in Proposition \ref{['prop: Pseudoeigenvector construction']} for the convergence of the $\varepsilon_N$-pseudospectrum for dense non-Hermitian Toeplitz matrices. The pseudospectrum is algebraically convergent, which is to be expected as the decay rate of the eigenvalues decreases as $\mathcal{O}\bigl(\log(N)/N\bigr)$.

Theorems & Definitions (22)

• Theorem 2.1: Gohberg
• Theorem 2.2: Schmidt-Spitzer
• Definition 2.3
• Theorem 2.4
• proof
• Theorem 2.5
• proof
• Corollary 2.6
• proof
• Theorem 2.7