  Spectral projectors of Hermitian matrices play a key role in many
applications, and especially in electronic structure computations. Linear
scaling methods for gapped systems are based on the fact that these special
matrix functions are localized, which means that the entries decay
exponentially away from the main diagonal or with respect to more general
sparsity patterns. The relation with the sign function together with an
integral representation is used to obtain new decay bounds, which turn out to
be optimal in an asymptotic sense. The influence of isolated eigenvalues in the
spectrum on the decay properties is also investigated and a superexponential
behaviour is predicted.
