Abstract
The purpose of this paper is to demonstrate the efficacy of the generalized quadratic spectrum approximation in addressing the issue of spectral pollution that arises in the approximation of unbounded operator spectra. To achieve this, we investigate key analytical properties of the generalized quadratic resolvent function, such as holomorphicity and Fréchet differentiability, which are essential for developing the numerical framework. Furthermore, we extend the concept of \(\nu\) -convergence, originally introduced for the classical spectrum, to establish property \(U\) convergence, a novel approach that effectively mitigates spectral pollution. This extension provides a more robust framework for addressing spectral contamination. To illustrate the practical applicability of our method, we focus on the quadratic pencil of Schrödinger’s operator. By combining the finite differences method with the generalized quadratic spectral approximation, we estimate the eigenvalues of the selected operator. Extensive numerical experiments have been conducted to validate the efficiency and accuracy of our approach. The results confirm the effectiveness of our method in resolving spectral pollution and demonstrate its capability to deliver precise eigenvalue approximations. These findings highlight the potential of our technique for broader applications in spectral analysis.
