A New Tool for Approaching Eigenvalues of the Quadratic Pencil of Schrödinger Operators

Abstract: The aim of this paper is to establish a robust theoretical andnumerical framework for implementing the generalized spectrumapproximation method in the context of quadratic spectralproblems. In order to facilitate the application of this method toquadratic problems, we introduce an innovative tool thatestablishes the convergence of the generalized quadratic spectrum,encompassing properties U and L, within the norms and collectivelycompact convergence modes. Furthermore, we conduct extensivenumerical experiments on the quadratic pencil of Schrödingeroperators. The effectiveness and precision of our approach aredemonstrated through a comprehensive comparison between the exactand approximate eigenvalues, highlighting the performance of ourmethod.