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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 -
A mathematical theory of resolution limits for super-resolution of positive sources
(2023)SAM Research ReportThe superresolving capacity for number and location recoveries in the super-resolution of positive sources is analyzed in this work. Specifically, we introduce the computational resolution limit for respectively the number detection and location recovery in the one-dimensional super-resolution problem and quantitatively characterize their dependency on the cutoff frequency, signal-to-noise ratio, and the sparsity of the sources. As a ...Report -
An Operator Theory for Analyzing the Resolution of Multi-illumination Imaging Modalities
(2023)SAM Research ReportBy introducing a new operator theory, we provide a unified mathematical theory for general source resolution in the multi-illumination imaging problem. Our main idea is to transform multi-illumination imaging into single-snapshot imaging with a new imaging kernel that depends on both the illumination patterns and the point spread function of the imaging system. We therefore prove that the resolution of multi-illumination imaging is ...Report -
Topological phenomena in honeycomb Floquet metamaterials
(2022)SAM Research ReportBeing driven by the goal of finding edge modes and of explaining the occurrence of edge modes in the case of time-modulated metamaterials in the high-contrast and subwavelength regime, we analyse the topological properties of Floquet normal forms of periodically parameterized time-periodic linear ordinary differential equations. In fact, our main goal being the question whether an analogous principle as the bulk-boundary correspondence ...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 -
Non-reciprocal wave propagation in space-time modulated media
(2021)SAM Research ReportWe prove the possibility of achieving non-reciprocal wave propagation in space-time modulated media and give an asymptotic analysis of the non-reciprocity property in terms of the amplitude of the time-modulation. Such modulation causes a folding of the band structure of the material, which may induce degenerate points. By breaking time-reversal symmetry, we show that these degeneracies may open into non-symmetric, unidirectional band ...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 -
High order topological asymptotics: reconciling layer potentials and compound asymptoticexpansions
(2021)SAM Research ReportA systematic two-step procedure is proposed for the derivation of full asymptotic expansions of the solution of elliptic partial differential equations set on a domain perforated with a small hole on which a Dirichlet boundary condition is applied. First, an integral representation of the solution is sought, which enables to exploit the explicit dependence with respect to the small parameter to predict the correct form of a two-scale ...Report -
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