Null-Space-Free 6D Spectral Embedding with Local Rayleigh Quotient Recovery for 3D Quasiperiodic Maxwell's Eigenproblems
Abstract
We develop a numerical framework for three-dimensional quasiperiodic Maxwell eigenvalue problems obtained through a six-dimensional periodic embedding. A projected Bloch--Fourier discretization yields a structured generalized eigenvalue problem with a large gradient-type kernel. Explicit orthonormal bases for the longitudinal and transverse subspaces remove this kernel exactly and reduce the original generalized eigenvalue problem to a null-space-free standard eigenvalue problem containing only the positive spectrum. An explicit inverse representation of the reduced operator avoids nested inner--outer linear solves and leads to an inverse Lanczos method whose main inner computation is a Hermitian positive definite conjugate-gradient solve with condition number bounded by that of the mass matrix; a residual estimate quantifies the effect of inner solves on the inverse Ritz pairs. To recover the computed modes in physical space, the six-dimensional Fourier eigenvectors are reconstructed on a three-dimensional Yee grid by a separated multi-center Taylor expansion, which avoids the dense Fourier-to-grid phase matrix and remains practical when direct dense reconstruction becomes prohibitively expensive. Local weighted Rayleigh quotients provide an independent physical-space validation, and their mass-weighted expectation is proved to equal the cropped Yee Rayleigh quotient under a partition-of-unity condition. Numerical experiments confirm the accuracy and computational effectiveness of the proposed framework and its physical-space recovery of three-dimensional quasiperiodic Maxwell modes.
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BibTeX
@misc{sun2026nullspacefree,
title = {{Null-Space-Free 6D Spectral Embedding with Local Rayleigh Quotient Recovery for 3D Quasiperiodic Maxwell's Eigenproblems}},
author = {Teng-Chao Sun and Tiexiang Li and Wen-Wei Lin and Xing-Long Lyu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.06965},
eprint = {2609.06965},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.06965},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Null-Space-Free 6D Spectral Embedding with Local Rayleigh Quotient Recovery for 3D Quasiperiodic Maxwell's Eigenproblems
AU - Teng-Chao Sun
AU - Tiexiang Li
AU - Wen-Wei Lin
AU - Xing-Long Lyu
PY - 2026
DA - 2026/09/22
PB - arXiv
DO - 10.48550/arxiv.2609.06965
UR - https://arxiv.org/abs/2609.06965
AB - We develop a numerical framework for three-dimensional quasiperiodic Maxwell eigenvalue problems obtained through a six-dimensional periodic embedding. A projected Bloch--Fourier discretization yields a structured generalized eigenvalue problem with a large gradient-type kernel. Explicit orthonormal bases for the longitudinal and transverse subspaces remove this kernel exactly and reduce the original generalized eigenvalue problem to a null-space-free standard eigenvalue problem containing only the positive spectrum. An explicit inverse representation of the reduced operator avoids nested inner--outer linear solves and leads to an inverse Lanczos method whose main inner computation is a Hermitian positive definite conjugate-gradient solve with condition number bounded by that of the mass matrix; a residual estimate quantifies the effect of inner solves on the inverse Ritz pairs. To recover the computed modes in physical space, the six-dimensional Fourier eigenvectors are reconstructed on a three-dimensional Yee grid by a separated multi-center Taylor expansion, which avoids the dense Fourier-to-grid phase matrix and remains practical when direct dense reconstruction becomes prohibitively expensive. Local weighted Rayleigh quotients provide an independent physical-space validation, and their mass-weighted expectation is proved to equal the cropped Yee Rayleigh quotient under a partition-of-unity condition. Numerical experiments confirm the accuracy and computational effectiveness of the proposed framework and its physical-space recovery of three-dimensional quasiperiodic Maxwell modes.
ER -