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

TL;DR

This paper derives improved decay bounds for spectral projectors of sparse Hermitian matrices, revealing their localization properties and the influence of eigenvalues, with implications for electronic structure computations.

Contribution

It introduces refined decay bounds for spectral projectors, utilizing the sign function and integral representation, and analyzes the effect of isolated eigenvalues on decay behavior.

Findings

Decay bounds are optimal asymptotically.

Spectral projectors exhibit exponential decay away from the diagonal.

Isolated eigenvalues can induce superexponential decay patterns.

Abstract

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.