Decay properties of spectral projectors with applications to electronic structure

Abstract

Motivated by applications in quantum chemistry and solid state physics, we\napply general results from approximation theory and matrix analysis to the\nstudy of the decay properties of spectral projectors associated with large and\nsparse Hermitian matrices. Our theory leads to a rigorous proof of the\nexponential off-diagonal decay ("nearsightedness") for the density matrix of\ngapped systems at zero electronic temperature in both orthogonal and\nnon-orthogonal representations, thus providing a firm theoretical basis for the\npossibility of linear scaling methods in electronic structure calculations for\nnon-metallic systems. We further discuss the case of density matrices for\nmetallic systems at positive electronic temperature. A few other possible\napplications are also discussed.\n