arXiv subjects

Yannick De Bruijn

Publications and source records attributed to Yannick De Bruijn.

4 recordsLinked to original sources

arXiv2025

Complex Band Structure and localisation transition for tridiagonal non-Hermitian k-Toeplitz operators with defects

Using the Bloch-Floquet theory, we propose an innovative technique to obtain the eigenvectors of tridiagonal k-Toeplitz operators. This method offers a more extensive and quantitative basis for describing localised eigenvectors beyond the non-trivial winding zone, yielding sharp decay bounds. The validity of our results is confirmed numerically in one-dimensional resonator chains, showcasing non-Hermitian skin localisation, bulk localisation, and tunnelling effects. We conclude the paper by analysing non-Hermitian tight binding Hamiltonians, illustrating the broad applicability of the complex band structure.

math.AP↗

arXiv2025

Complex Brillouin Zone for Localised Modes in Hermitian and Non-Hermitian Problems

We develop a mathematical and numerical framework for studying evanescent waves in subwavelength band gap materials. By establishing a link between the complex Brillouin zone and various Hermitian and non-Hermitian phenomena, including defect localisation in band gap materials and the non-Hermitian skin effect, we provide a unified perspective on these systems. In two-dimensional structures, we develop analytical techniques and numerical methods to study singularities of the complex band structure. This way, we demonstrate that gap functions effectively predict the decay rates of defect states. Furthermore, we present an analysis of the Floquet transform with respect to complex quasimomenta. Based on this, we show that evanescent waves may undergo a phase transition, where local oscillations drastically depend on the location of corresponding frequency inside the band gap.

math.AP↗

arXiv2024

Complex Band Structure for Subwavelength Evanescent Waves

We present the mathematical and numerical theory for evanescent waves in subwavelength band gap materials. We begin in the one-dimensional case, whereby fully explicit formulas for the complex band structure, in terms of the capacitance matrix, are available. As an example, we show that the gap functions can be used to accurately predict the decay rate of the interface mode of a photonic analogue of the SSH-model. In two dimensions, we derive the band gap Green's function and characterise the subwavelength gap functions via layer potential techniques. By generalising existing lattice-summation techniques, we illustrate our results numerically by computing the complex band structure in a variety of settings.

math.AP↗

arXiv2024

Spectra and pseudo-spectra of tridiagonal $k$-Toeplitz matrices and the topological origin of the non-Hermitian skin effect

We establish new results on the spectra and pseudo-spectra of tridiagonal $k$-Toeplitz operators and matrices. In particular, we prove the connection between the winding number of the eigenvalues of the symbol function and the exponential decay of the associated eigenvectors (or pseudo-eigenvectors). Our results elucidate the topological origin of the non-Hermitian skin effect in general one-dimensional polymer systems of subwavelength resonators with imaginary gauge potentials, proving the observation and conjecture in arXiv:2307.13551. We also numerically verify our theory for these systems.

math-ph↗