Tensor factorization and explicit spectral bounds for product-box concentration operators
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 $\Lambda_\varepsilon=\#\{n:\varepsilon<\lambda_n(S)<1-\varepsilon\}$ 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 $\Lambda_\varepsilon$, 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 $\alpha\geq4$, $c\geq2$, and $\alpha^{-c}<\varepsilon<1/2$, it gives $\Lambda_\varepsilon\leq Cc^{d-1}\log(1/\varepsilon)\log\!\bigl(\alpha c/\log(1/\varepsilon)\bigr)$. 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 $\varepsilon<4^{-d}$, $\Lambda_\varepsilon\geq M_a^d=\Omega((\log c)^d)$. Using an exact trace identity, an explicit cubic minorant, and the sine-kernel determinant asymptotics of Basor and Widom, we further obtain $\operatorname{Tr}((S-S^2)^m)=\beta_m\pi^{-2}\log c+O_m(1)$ for each fixed $m$, where $\beta_m=B(m,m)$, 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 $m$.