AI 中文总结
本文针对乘积盒集中算子，通过张量分解方法得到其 plunge 计数的显式谱界，补充了 Kulikov 等人的结果，并给出模型立方体对的相关下界与迹估计。
AI 中文摘要
设 $S=P_{cA_0}Q_{B_0}P_{cA_0}$ 为有界集 $cA_0,B_0\subset\mathbb{R}^d$ 的空间-谱集中算子，令 $\u039B_\u03B5=\\#\{n:\u03B5<\lambda_n(S)<1-\u03B5\}$ 为其 plunge 计数。当 $A_0$ 和 $B_0$ 是有界轴对齐开盒的有限不交并时，我们对所有 $d\geq1$、$c>0$ 和 $0<\u03B5<1/2$，证明了 $\u039B_\u03B5$ 的显式一致上界，所有常数由边长表示。在 $\u03B1\geq4$、$c\geq2$ 且 $\u03B1^{-c}<\u03B5<1/2$ 的范围内，该上界给出 $\u039B_\u03B5\leq Cc^{d-1}\log(1/\u03B5)\log\bigl(\u03B1 c/\log(1/\u03B5)\bigr)$。Kulikov 和 Dam Larsen 此前已针对更广泛的类证明了该范围的阶，本文的贡献是独立证明并得到乘积盒的显式全参数估计。证明采用 $P_{(cA_0)^c}Q_{B_0}P_{cA_0}$ 的 telescoping 张量分解为 $d$ 个初等张量算子，含一个一维非对角因子和 $d-1$ 个局域化因子。Schatten 拟范数沿张量因子相乘，单个对数仅来自法向。对模型立方体对，还证明当 $\u03B5<4^{-d}$ 时，$\u039B_\u03B5\geq M_a^d=\Omega((\log c)^d)$。利用精确迹恒等式、显式三次下界以及 Basor 和 Widom 的正弦核行列式渐近，进一步对每个固定 $m$ 得到 $\operatorname{Tr}((S-S^2)^m)=\u03B2_m\pi^{-2}\log c+O_m(1)$，其中 $\u03B2_m=B(m,m)$，同时得到阶为 $\log c$ 的双侧固定深度窗口估计。下界不匹配，且固定阶陈述对 $m$ 不一致。
英文摘要
Let $S=P_{cA_0}Q_{B_0}P_{cA_0}$ be the spatio-spectral concentration operator of bounded sets $cA_0,B_0\subset\mathbb{R}^d$, and let $Λ_\varepsilon=\#\{n:\varepsilon<λ_n(S)<1-\varepsilon\}$ be its plunge count. For $A_0$ and $B_0$ finite disjoint unions of bounded axis-parallel open boxes, we prove an explicit uniform upper bound on $Λ_\varepsilon$, valid for every $d\geq1$, $c>0$, and $0<\varepsilon<1/2$, with all constants written in terms of the side lengths. On the range $α\geq4$, $c\geq2$, and $α^{-c}<\varepsilon<1/2$, it gives $Λ_\varepsilon\leq Cc^{d-1}\log(1/\varepsilon)\log\!\bigl(αc/\log(1/\varepsilon)\bigr)$. Kulikov and Dam Larsen previously proved this order on that range for a broader class; the present contribution is an independent proof and an explicit all-parameter estimate for product boxes. The proof uses a telescoping tensorization of $P_{(cA_0)^c}Q_{B_0}P_{cA_0}$ into $d$ elementary tensor operators, with one one-dimensional off-diagonal factor and $d-1$ localization factors. Schatten quasi-norms then multiply across tensor factors, and the single logarithm arises only from the normal direction. For the model cube pair, we also prove that, when $\varepsilon<4^{-d}$, $Λ_\varepsilon\geq M_a^d=Ω((\log c)^d)$. Using an exact trace identity, an explicit cubic minorant, and the sine-kernel determinant asymptotics of Basor and Widom, we further obtain $\operatorname{Tr}((S-S^2)^m)=β_mπ^{-2}\log c+O_m(1)$ for each fixed $m$, where $β_m=B(m,m)$, together with a two-sided fixed-depth window estimate of order $\log c$. The lower bound is not matching, and the fixed-order statements are not uniform in $m$.
Comments36 pages. Explicit spectral bounds for finite unions of axis-parallel product boxes; includes tensor-factorization arguments, cube-model lower bounds, and fixed-order trace asymptotics
