论文标题
矩阵，复杂对数和多维菲涅尔积分的索引
Index of a matrix, complex logarithms, and multidimensional Fresnel integrals
论文作者
论文摘要
我们严格地讨论了找到$λ$ -INDEX $ \ MATHCAL {n}（λ）\ in [0,1，\ ldots，n] $的真实对称矩阵$ \ mathbf {m mathbf {m mathbf {m mathbf {m} $，定义为使用$λ$的eigenvalues的数量，仅使用$λ$，仅使用$ \ mathbf。我们表明，使用广泛使用的公式$$ \ MATHCAL {n}（λ）= \ lim_ {ε\ to 0^+} \ frac {1} {2π\ mathrm {i}} \ left [\ log \ det（\ MathBf {m}-λ+\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ m {i}ε） - \ log \ det（\ mathbf {m} -mathbf {m} - λ-\ mathrm）对数应谨慎处理，因为如果为两个对数选择了相同的分支，则通常无法产生正确的结果。 我们使用多维菲涅尔积分来改进公式，这表明即使是新版本，最多也提供了$ \ Mathcal {n}（λ）$的自搭配方程，其解决方案不能保证是唯一的。我们的结果通过显式示例和数值评估来证实。
We critically discuss the problem of finding the $λ$-index $\mathcal{N}(λ)\in [0,1,\ldots,N]$ of a real symmetric matrix $\mathbf{M}$, defined as the number of eigenvalues smaller than $λ$, using the entries of $\mathbf{M}$ as only input. We show that a widely used formula $$ \mathcal{N}(λ)=\lim_{ε\to 0^+}\frac{1}{2π\mathrm{i}}\left[\log\det(\mathbf{M}-λ+\mathrm{i}ε)-\log\det(\mathbf{M}-λ-\mathrm{i}ε)\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 $\mathcal{N}(λ)$, whose solution is not guaranteed to be unique. Our results are corroborated by explicit examples and numerical evaluations.
