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Pointwise Decay of Fourier-Stieltjes transform of the Spectral Measure for Jacobi Matrices with Faster-than-Exponential Sparse Perturbations

We consider off-diagonal Jacobi matrices $J$ with (faster-than-exponential) sparse perturbations. We prove (Theorem \ref{onehalf}) that the Fourier transform $\hat{\left\| f\right\| ^{2}dρ}(t)$ of the spectral measure $ρ$ of $J$, whose sparse perturbations are at least separated by a distance $\exp \left(cj(\ln j)^{2}\right) /δ^{j}$, for some $c>1/2,$ $0<δ<1$ and for a dense subset of $C_{0}^{\infty}(-2,2)$-functions $f$, decays as $t^{-1/2}Ω(t)$, uniformly in the spectrum $[-2,2]$, $Ω(t)$ increasing less rapidly than any positive power of $t$, improving earlier results obtained by Simon (Commun. Math. Phys. \textbf{179}, 713-722 (1996)) and by Krutikov-Remling (Commun. Math. Phys. \textbf{223}, 509-532 (2001)) for Schrödinger operators with sparse potential that increases as fast as exponential-of-exponential. Applications to the spectrum of the Kronecker sum of two (or more) copies of the model are given.