Abstract

This study presents an in-depth analysis of the nonconforming virtual element method (VEM) as a novel approach for approximating the eigenvalues of the Schrödinger equation. Central to the strategy is deploying the L2 projection operator to discretize potential terms within the model problem. Through compact operator theory, we rigorously establish the methodology's capability to achieve double-order convergence rates for the eigenvalue spectrum. Addressing the challenge posed by the nonconformity of the discrete space, we redefine the solution operator on a weaker space, which aligns with the Babuška-Osborn compactness framework. A comprehensive set of numerical experiments confirms the theoretical findings, showing the approximation qualities and computational efficiency of the method. A series of potential functions are used to illustrate the various challenges behind the choice of a potential for the simulation of the Schrödinger eigenvalue problem. These results confirm the potential of the nonconforming VEM as a robust and accurate tool for quantum mechanical eigenvalue problems. © 2025