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Spectra and pseudo-spectra of tridiagonal k-Toeplitz matrices and the topological origin of the non-Hermitian skin effect
(2024)SAM Research ReportWe 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 ...Report -
Scattering from time-modulated subwavelength resonators
(2024)SAM Research ReportWe consider wave scattering from a system of highly contrasting resonators with time-modulated material parameters. In this setting, the wave equation reduces to a system of coupled Helmholtz equations that models the scattering problem. We consider the one-dimensional setting. In order to understand the energy of the system, we prove a novel higher-order discrete, capacitance matrix approximation of the subwavelength resonant quasifrequencies. ...Report -
A Two-Scale Effective Model for Defect-Induced Localization Transitions in Non-Hermitian Systems
(2024)SAM Research ReportWe illuminate the fundamental mechanism responsible for the transition between the non-Hermitian skin effect and defect-induced localization in the bulk. We study a Hamiltonian with non-reciprocal couplings that exhibits the skin effect (the localization of all eigenvectors at one edge) and add an on-site defect in the center. Using a two-scale asymptotic method, we characterize the long-scale growth and decay of the eigenvectors and ...Report -
Tunable Localisation in Parity-Time-Symmetric Resonator Arrays with Imaginary Gauge Potentials
(2024)SAM Research ReportThe aim of this paper is to illustrate both analytically and numerically the interplay of two fundamentally distinct non-Hermitian mechanisms in a deep subwavelength regime. Considering a parity-time symmetric system of one-dimensional subwavelength resonators equipped with two kinds of non-Hermiticity — an imaginary gauge potential and on-site gain and loss — we prove that all but two eigenmodes of the system pass through exceptional ...Report -
Wave packets propagation in the subwavelength regime near the Dirac point
(2024)SAM Research ReportIn [Ammari et al., SIAM J Math Anal., 52 (2020), pp. 5441–5466], the first author with collaborators proved the existence of Dirac dispersion cones at subwavelength scales in bubbly honeycomb phononic crystals. In this paper, we study the time-evolution of wave packets, which are spectrally concentrated near such conical points. We prove that the wave packets dynamics is governed by a time-dependent effective Dirac system, which still ...Report -
Generalised Brillouin Zone for Non-Reciprocal Systems
(2024)SAM Research ReportRecently, it has been observed that the Floquet-Bloch transform with real quasiperiodicities fails to capture the spectral properties of non reciprocal systems. The aim of this paper is to introduce the notion of a generalised Brillouin zone by allowing the quasiperiodicities to be complex in order to rectify this. It is proved that this shift of the Brillouin zone into the complex plane accounts for the unidirectional spatial decay of ...Report -
Applications of Chebyshev polynomials and Toeplitz theory to topological metamaterials
(2024)SAM Research ReportWe survey the use of Chebyshev polynomials and Toeplitz theory for studying topological metamaterials. We consider both Hermitian and non-Hermitian systems of subwavelength resonators and provide a mathematical framework to explain some spectacular properties of metamaterials.Report -
Space-Time Wave Localisation in Systems of Subwavelength Resonators
(2024)SAM Research ReportIn this paper we study the dynamics of metamaterials composed of high-contrast subwavelength resonators and show the existence of localised modes in such a setting. A crucial assumption in this paper is time-modulated material parameters. We prove a so-called capacitance matrix approximation of the wave equation in the form of an ordinary differential equation. These formulas set the ground for the derivation of a first-principles ...Report -
Wave scattering with time-periodic coefficients: Energy estimates and harmonic formulations
(2024)SAM Research ReportThis paper investigates acoustic wave scattering from materials with periodic time-modulated material parameters. We consider the basic case of a single connected domain where absorbing or Neumann boundary conditions are enforced. Energy estimates limit the exponential growth of solutions to the initial value problem, thereby confining Floquet exponents to a complex half-space under absorbing boundary conditions and to a strip near the ...Report -
Truncated Floquet-Bloch transform for computing the spectral properties of large finite systems of resonators
(2024)SAM Research ReportThis paper aims at providing for the first time the mathematical foundations of the truncated Floquet-Bloch transform, which can be used to characterise the spectral properties of finite periodic and aperiodic large systems of resonators.Report