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.

翻译：Hermitian 矩阵的光谱投影仪在许多应用中,特别是在电子结构计算中发挥着关键作用。 偏差系统的线性缩放方法基于以下事实:这些特殊的矩阵功能是本地化的,这意味着条目从主对角或更一般的宽度模式中迅速衰减。与标志函数和整体表示法的关系被用来获得新的衰减线,从非现成的意义上看,这些线性线性调整是最佳的。还调查了孤立的光谱中叶素值对衰变特性的影响,并预测了超显性行为。