Minimization Principle for Linear Response Eigenvalue Problem with Applications

We present a minimization principle for the sum of the first few smallest positive eigenvalues and Cauchy-like interlacing inequalities for the linear response eigenvalue problem arising from the calculation of excitation states of many-particle systems, a hot topic among computational material scientists today for materials design to advance energy science. Subsequently, we develop the best approximations of these smallest positive eigenvalues by a structure-preserving subspace projection. Based on these newly established theoretical results, we outline conjugate gradient-like algorithms for simultaneously computing the first few smallest positive eigenvalues and associated eigenvectors. Finally, we present numerical examples to illustrate essential convergence behaviors of the proposed conjugate gradient-like methods with and without preconditioning. 2000 Mathematics Subject Classification. Primary 65L15. Secondary 15A18, 81Q15