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- W3108724586 abstract "We critically discuss the problem of finding the $lambda$-index $mathcal{N}(lambda)in [0,1,ldots,N]$ of a real symmetric matrix $mathbf{M}$, defined as the number of eigenvalues smaller than $lambda$, using the entries of $mathbf{M}$ as only input. We show that a widely used formula $$ mathcal{N}(lambda)=lim_{epsilonto 0^+}frac{1}{2pi mathrm{i}}left[logdet(mathbf{M}-lambda+mathrm{i}epsilon)-logdet(mathbf{M}-lambda-mathrm{i}epsilon)right] $$ based on the branch-cut structure of the complex logarithm should be handled with care, as it generically fails to produce the correct result if the same branch is chosen for the two logarithms. We improve the formula using multidimensional Fresnel integrals, showing that even the new version provides at most a self-consistency equation for $mathcal{N}(lambda)$, whose solution is not guaranteed to be unique. Our results are corroborated by explicit examples and numerical evaluations." @default.
- W3108724586 created "2020-12-07" @default.
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- W3108724586 date "2020-12-22" @default.
- W3108724586 modified "2023-09-27" @default.
- W3108724586 title "Index of a matrix, complex logarithms, and multidimensional Fresnel integrals" @default.
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- W3108724586 doi "https://doi.org/10.1088/1751-8121/abccf9" @default.
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