A Novel Computational and Analytical Framework for 2D Quasiperiodic Helmholtz Eigenvalue Problems via the Projection Method
For researchers solving quasiperiodic eigenvalue problems in continuous media, this provides a more reliable eigenpair solution than existing methods.
The paper introduces a spectral framework that embeds quasiperiodic Helmholtz eigenvalue problems into higher-dimensional periodic spaces using the projection method, and validates eigenvalues via weighted expectation of Rayleigh quotients. Numerical experiments show the method is accurate, robust, and scalable.
In this paper, we propose a spectral framework that embeds 1D and 2D quasiperiodic Helmholtz eigenvalue problems into higher-dimensional (2D and 4D) periodic spaces via the projection method \cite{jiang2014numerical, jiang2024numerical}. To effectively map the elevated high-dimensional states back to the original physical space, we establish a novel validation framework based on the weighted expectation of pointwise Rayleigh quotients. Supported by comprehensive error and spectral analysis, we demonstrate that the eigenvalues derived from this expectation align more authentically with the original quasiperiodic model, ultimately yielding a more appropriate and reliable eigenpair solution. Numerical experiments on continuous media demonstrate that our approach offers an accurate, robust, and scalable tool for solving quasiperiodic Helmholtz eigenvalue problems.