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Spectral projectors of Hermitian matrices play a key role in manyapplications, and especially in electronic structure computations. Linearscaling methods for gapped systems are based on the fact that these specialmatrix functions are localized, which means that the entries decayexponentially away from the main diagonal or with respect to more generalsparsity patterns. The relation with the sign function together with anintegral representation is used to obtain new decay bounds, which turn out tobe optimal in an asymptotic sense. The influence of isolated eigenvalues in thespectrum on the decay properties is also investigated and a superexponentialbehaviour is predicted.

Refined decay bounds on the entries of spectral projectors associated with sparse Hermitian matrices