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Motivated by applications in quantum chemistry and solid state physics, weapply general results from approximation theory and matrix analysis to thestudy of the decay properties of spectral projectors associated with large andsparse Hermitian matrices. Our theory leads to a rigorous proof of theexponential off-diagonal decay ("nearsightedness") for the density matrix ofgapped systems at zero electronic temperature in both orthogonal andnon-orthogonal representations, thus providing a firm theoretical basis for thepossibility of linear scaling methods in electronic structure calculations fornon-metallic systems. We further discuss the case of density matrices formetallic systems at positive electronic temperature. A few other possibleapplications are also discussed.

Decay properties of spectral projectors with applications to electronic structure