一致正弦核行列式渐近、尾部分位数与长椭本征值界
Uniform sine-kernel determinant asymptotics, tail-side quantiles, and prolate eigenvalue bounds
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中文总结 AI 辅助
该研究推导了一维sinc核集中算子的尾部分位数公式、单侧张量积本征值界，通过IIKS最速下降法证明了正弦核行列式的负耦合渐近，还得到了移动深度下半桥界。
中文摘要 AI 辅助
设$S_c=P_{(0,c)}QP_{(0,c)}$为一维sinc核集中算子，令$N_a(c)=\\#\{n:\lambda_n(c)>a\}$，并设$\bar L=\log((1-\delta)/\delta)$。我们证明，对每个固定的$A>0$，在$6\le\bar L\le A\log c$范围内，尾部分位数公式$N_\delta(c)=c+\pi^{-2}\bar L\log(4\pi^2c/\bar L)+O_A(\log c+\bar L)$一致成立。该公式给出了下半 plunge 和全 plunge 的相应加性公式，主项分别为$\pi^{-2}\bar L\log(4\pi^2c/\bar L)$和该量的两倍。对固定的$A>0$、$d\ge1$及$q\in(1/2,1)$，精确的单侧坐标选取给出单侧张量积界$\Lambda_\delta(c;d)\ge\pi^{-2}d c^{d-1}\bar L\log(4\pi^2c/\bar L)-O_{A,d,q}(c^{d-1}(\log c+\bar L))$，其适用范围为$L_{d,q}\le\bar L\le A\log c$，其中$L_{d,q}=\log(q^{-(d-1)}(e^6+1)-1)$；当$d\ge2$时，张量内容是非平凡的。分析输入为带符号的增长参数正弦核行列式渐近：对$0\le\omega\le A\log s$一致成立，$\log\det(I+(e^{2\omega}-1)K_s)=4\omega s/\pi+2\pi^{-2}\omega^2\log(4s)+2\log|G(1+i\omega/\pi)|^2+O_A((1+\omega)^4\log^2s/s)$，其中$G$为Barnes G函数。我们通过直接IIKS最速下降法证明了该Bothner–Deift–Its–Krasovsky定理的负耦合对应结果。我们还保留了一致头部侧结果，并利用双向行列式约化得到了移动深度下的下半桥界，其常数为$1/(32\pi^2)$；将其扩展到更深的范围需用到Kulikov–Dam Larsen的结果，且可能需要更小的常数。这些计数公式是加性的，当$\bar L$一致趋于无穷时，它们的误差成为一致相对误差；固定阈值的情况由Landau–Widom分别处理。Lambert-$W_{-1}$公式仅记录用于连续主项，而非单个特征值。
英文摘要
Let $S_c=P_{(0,c)}QP_{(0,c)}$ be the one-dimensional sinc-kernel concentration operator, let $N_a(c)=\#{n:λ_n(c)>a}$, and set $\bar L=\log((1-δ)/δ)$. We prove, uniformly for each fixed $A>0$, the tail-side quantile formula $N_δ(c)=c+π^{-2}\bar L\log(4π^2c/\bar L)+O_A(\log c+\bar L)$ for $6\le\bar L\le A\log c$. It yields corresponding additive formulas for the lower half and full plunge, with main terms respectively $π^{-2}\bar L\log(4π^2c/\bar L)$ and twice this quantity. An exact one-tail-coordinate selection gives, for fixed $A>0$, $d\ge1$, and $q\in(1/2,1)$, the one-sided tensor-product bound $Λ_δ(c;d)\geπ^{-2}d c^{d-1}\bar L\log(4π^2c/\bar L)-O_{A,d,q}(c^{d-1}(\log c+\bar L))$ for $L_{d,q}\le\bar L\le A\log c$, where $L_{d,q}=\log(q^{-(d-1)}(e^6+1)-1)$; the tensor content is nontrivial for $d\ge2$. The analytic input is a signed growing-parameter sine-kernel determinant asymptotic: uniformly for $0\leω\le A\log s$, $\log\det(I+(e^{2ω}-1)K_s)=4ωs/π+2π^{-2}ω^2\log(4s)+2\log|G(1+iω/π)|^2+O_A((1+ω)^4\log^2s/s)$, where $G$ is the Barnes $G$-function. We prove this negative-coupling counterpart of the Bothner--Deift--Its--Krasovsky theorem by direct IIKS steepest descent. We also retain the uniform head-side results and use a two-way determinant reduction to obtain the moving-depth lower-half bridge bound with constant $1/(32π^2)$; extending it to the deeper range uses Kulikov--Dam Larsen and may require a smaller constant. These counting formulas are additive. Their errors become uniformly relative when $\bar L$ tends uniformly to infinity; fixed thresholds are covered separately by Landau--Widom. A Lambert-$W_{-1}$ formula is recorded only for the continuous main term, not for individual eigenvalues.
