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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 -
Exponentially localised interface eigenmodes in finite chains of resonators
(2024)SAM Research ReportThis paper studies wave localisation in chains of finitely many resonators. There is an extensive theory predicting the existence of localised modes induced by defects in infinitely periodic systems. This work extends these principles to finite-sized systems. We consider finite systems of subwavelength resonators arranged in dimers that have a geometric defect in the structure. This is a classical wave analogue of the Su-Schrieffer-Heeger ...Report -
The non-Hermitian skin effect with three-dimensional long-range coupling
(2023)SAM Research ReportWe study the non-Hermitian skin effect in a three-dimensional system of finitely many subwavelength resonators with an imaginary gauge potential. We introduce a discrete approximation of the eigenmodes and eigenfrequencies of the system in terms of the eigenvectors and eigenvalues of the so-called gauge capacitance matrix, which is a dense matrix due to long-range interactions in the system. Based on translational invariance of this matrix ...Report -
Stability of the non-Hermitian skin effect
(2023)SAM Research ReportThis paper shows that the skin effect in systems of non-Hermitian subwavelength resonators is robust with respect to random imperfections in the system. The subwavelength resonators are highly contrasting material inclusions that resonate in a low-frequency regime. The non-Hermiticity is due to the introduction of an imaginary gauge potential, which leads to a skin effect that is manifested by the system's eigenmodes accumulating at one ...Report -
Mathematical foundations of the non-Hermitian skin effect
(2023)SAM Research ReportWe study the skin effect in a one-dimensional system of finitely many subwavelength resonators with a non-Hermitian imaginary gauge potential. Using Toeplitz matrix theory, we prove the condensation of bulk eigenmodes at one of the edges of the system. By introducing a generalised (complex) Brillouin zone, we can compute spectral bands of the associated infinitely periodic structure and prove that this is the limit of the spectra of the ...Report -
Spectral convergence in large finite resonator arrays: the essential spectrum and band structure
(2023)SAM Research ReportWe show that resonant frequencies of a system of coupled resonators in a truncated periodic lattice converge to the essential spectrum of corresponding infinite lattice. We use the capacitance matrix as a model for fully coupled resonators with long-range interactions in three spatial dimensions. For one-, two- or three-dimensional lattices embedded in three-dimensional space, we show that the (discrete) density of states for the finite ...Report -
Spectral convergence of defect modes in large finite resonator arrays
(2023)SAM Research ReportWe show that defect modes in infinite systems of resonators have corresponding modes in finite systems which converge as the size of the system increases. We study the generalized capacitance matrix as a model for three-dimensional coupled resonators with long-range interactions and consider defect modes that are induced by compact perturbations. If such a mode exists, then there are elements of the discrete spectrum of the corresponding ...Report -
Anderson localization in the subwavelength regime
(2022)SAM Research ReportRandom media are ubiquitous in both natural and artificial structures and there are many important problems related to understanding their wave-scattering properties. In particular, the phenomenon of wave localization in random media, known as Anderson localization in certain settings, has proved difficult to understand, particularly in physically derived models and systems with long-range interactions. In this article, we show that the ...Report -
Bernhard Riemann, the ear, and an atom of consciousness
(2021)SAM Research ReportWhy did Bernhard Riemann (1826–1866), arguably the most original mathematician of his generation, spend the last year of life investigating the mechanism of hearing? Fighting tuberculosis and the hostility of eminent scientists such as Hermann Helmholtz, he appeared to forsake mathematics to prosecute a case close to his heart. Only sketchy pages from his last paper remain, but here we assemble some significant clues and triangulate from ...Report -
Functional analytic methods for discrete approximations of subwavelength resonator systems
(2021)SAM Research ReportWe survey functional analytic methods for studying subwavelength resonator systems. In particular, rigorous discrete approximations of Helmholtz scattering problems are derived in an asymptotic subwavelength regime. This is achieved by re-framing the Helmholtz equation as a non-linear eigenvalue problem in terms of integral operators. In the subwavelength limit, resonant states are described by the eigenstates of the generalized capacitance ...Report