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 $\hat{\left\| f\right\| ^{2}d\rho}(t)$ of the spectral measure $\rho $ of $J$, whose sparse perturbations are at least separated by a distance $\exp \left(cj(\ln j)^{2}\right) /\delta ^{j}$, for some $c>1/2,$ $0<\delta <1$ and for a dense subset of $C_{0}^{\infty}(-2,2)$-functions $f$, decays as $t^{-1/2}\Omega (t)$, uniformly in the spectrum $[-2,2]$, $\Omega (t)$ 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 Schr\"{o}dinger 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.