2026/07/29 by Ahmadreza Azimifard
Mathematics · #math.FA #msc:42B10 #msc:47A10 #msc:47B10 #msc:47B35
36 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
arxiv created 2026/07/29 · arxiv updated 2026/07/30
Let S=PcA0QB0PcA0 be the spatio-spectral concentration operator of bounded sets cA0,B0⊂ℝd, and let Λε=#\n:ε<λn(S)<1-ε\ be its plunge count. For A0 and B0 finite disjoint unions of bounded axis-parallel open boxes, we prove an explicit uniform upper bound on Λε, valid for every d≥1, c>0, and 0<ε<1/2, with all constants written in terms of the side lengths. On the range α≥4, c≥2, and α-c<ε<1/2, it gives Λε≤ Ccd-1log(1/ε)log (αc/log(1/ε)). 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(cA0)cQB0PcA0 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 ε<4-d, Λε≥ Mad=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 Tr((S-S2)m)=βmπ-2log c+Om(1) for each fixed m, where βm=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.