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Real eigenvalues in the non-Hermitian Anderson model
The eigenvalues of the Hatano--Nelson non-Hermitian Anderson matrices, in the
spectral regions in which the Lyapunov exponent exceeds the non-Hermiticity
parameter, are shown to be real and exponentially close to the Hermitian
eigenvalues. This complements previous results, according to which the
eigenvalues in the spectral regions in which the non-Hermiticity parameter
exceeds the Lyapunov exponent are aligned on curves in the complex plane.Comment: 21 pp., 2 fig; to appear in Ann. Appl. Proba
Decay properties of spectral projectors with applications to electronic structure
Motivated by applications in quantum chemistry and solid state physics, we
apply general results from approximation theory and matrix analysis to the
study of the decay properties of spectral projectors associated with large and
sparse Hermitian matrices. Our theory leads to a rigorous proof of the
exponential off-diagonal decay ("nearsightedness") for the density matrix of
gapped systems at zero electronic temperature in both orthogonal and
non-orthogonal representations, thus providing a firm theoretical basis for the
possibility of linear scaling methods in electronic structure calculations for
non-metallic systems. We further discuss the case of density matrices for
metallic systems at positive electronic temperature. A few other possible
applications are also discussed.Comment: 63 pages, 13 figure
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