We prove explicit upper bounds for the number of eigenvalues of a one-dimensional time-frequency localization operator in an open transition window. An abstract residual-energy principle combines a rank-N approximant with the Hilbert-Schmidt energy of its remainder and retains the strict integer correction imposed by the open threshold. For the Fourier kernel, centered least-squares polynomials, optimized moments, and Chebyshev-Bessel truncations improve or match the diameter-Taylor certificate; on interval blocks the moment envelope gains the exact factor 1/(2N+1). Finite measurable partitions yield best-polynomial matrix and residual-energy bounds, together with two-level threshold allocations that exploit spatial or Fourier-side orthogonality and remain stable under large empty gaps. A complementary defect channel gives a componentwise variational Schatten envelope for finite interval unions. For arbitrary finite-measure hard windows, exact cross-boundary and symmetric-difference formulas give strict transition counting and a certified near-one cluster that can be subtracted from direct counts. We also treat integrable soft masks, a root-sum-square core-tail decomposition, and unbounded windows controlled by moments. An explicit unbounded finite-measure set has all polynomial moments and infinite fractional translation perimeter for every order in (0,1], yet admits an O(log(1/epsilon)/log log(1/epsilon)) certificate. Finally, kernels with finitely many bounded separated phase interactions and finite separated amplitude rank admit factorial, least-squares, Chebyshev-Bessel, span-compressed, and anisotropic-degree bounds. For bounded hard Fourier windows, the final hybrid is no larger than the corresponding earlier certificates proved here; no universal comparison with all regular-domain estimates is asserted.
Let $S_c=P_{(0,c)}QP_{(0,c)}$ be the one-dimensional sinc-kernel concentration operator, let $N_a(c)=\#{n:\lambda_n(c)>a}$, and set $\bar L=\log((1-\delta)/\delta)$. We prove, uniformly for each fixed $A>0$, the tail-side quantile formula $N_\delta(c)=c+\pi^{-2}\bar L\log(4\pi^2c/\bar L)+O_A(\log c+\bar L)$ 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 $\pi^{-2}\bar L\log(4\pi^2c/\bar L)$ 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 $\Lambda_\delta(c;d)\ge\pi^{-2}d c^{d-1}\bar L\log(4\pi^2c/\bar L)-O_{A,d,q}(c^{d-1}(\log c+\bar L))$ 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 $0\le\omega\le A\log s$, $\log\det(I+(e^{2\omega}-1)K_s)=4\omega s/\pi+2\pi^{-2}\omega^2\log(4s)+2\log|G(1+i\omega/\pi)|^2+O_A((1+\omega)^4\log^2s/s)$, 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 $1/(32\pi^2)$; 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.
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 $m$.
A. Azimifard· 0 citations
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