Localization Landscapes based on the Green's Matrix of Non-Hermitian Optical Scattering Systems
Abstract
The Filoche--Mayboroda (FM) localization landscape predicts where the spatially localized modes of a self-adjoint elliptic operator reside without diagonalizing it, but the construction fails for the complex, non-normal Green's matrix G of an open scattering array of electric dipoles. We propose three real-valued, positive landscapes built from the Feinberg--Zee hermitization of G: a regularised Gramian inverse (Gram), the diagonal Schur block that carries the Metz--Neri--Rogers spectral density (SY), and the full hermitized inverse contracted against a polarization-isotropic source pair (Herm). The three form a progression from spectrum-blind to fully hermitisation-faithful, agree in the limit z→ 0, and diverge where it matters: at nonzero spectral shift SY resolves modes that Gram collapses, and Herm resolves modes that SY collapses. We demonstrate the progression on random and deterministic aperiodic scattering arrays with a wide range of structural correlations, extend to the vectorial case of transverse electric polarization (TE), and provide rigorous justifications along with mathematical proofs and a complete numerical recipe in the appendices.
Related
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- Creation and Control of Scattering Singularities in non-Hermitian Systems
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BibTeX
@misc{negro2026localization,
title = {{Localization Landscapes based on the Green's Matrix of Non-Hermitian Optical Scattering Systems}},
author = {Luca Dal Negro and Hari Sundar},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.13413},
eprint = {2610.13413},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.13413},
}
RIS
TY - UNPB
TI - Localization Landscapes based on the Green's Matrix of Non-Hermitian Optical Scattering Systems
AU - Luca Dal Negro
AU - Hari Sundar
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2610.13413
UR - https://arxiv.org/abs/2610.13413
AB - The Filoche--Mayboroda (FM) localization landscape predicts where the spatially localized modes of a self-adjoint elliptic operator reside without diagonalizing it, but the construction fails for the complex, non-normal Green's matrix G of an open scattering array of electric dipoles. We propose three real-valued, positive landscapes built from the Feinberg--Zee hermitization of G: a regularised Gramian inverse (Gram), the diagonal Schur block that carries the Metz--Neri--Rogers spectral density (SY), and the full hermitized inverse contracted against a polarization-isotropic source pair (Herm). The three form a progression from spectrum-blind to fully hermitisation-faithful, agree in the limit z→ 0, and diverge where it matters: at nonzero spectral shift SY resolves modes that Gram collapses, and Herm resolves modes that SY collapses. We demonstrate the progression on random and deterministic aperiodic scattering arrays with a wide range of structural correlations, extend to the vectorial case of transverse electric polarization (TE), and provide rigorous justifications along with mathematical proofs and a complete numerical recipe in the appendices.
ER -