On the Fourier transformation of spectral measures of discrete Schrödinger operators with sparse potentials (Q2734842)

The author considers the operator \(H_\varphi\) defined by NEWLINE\[NEWLINE(H_\varphi u)(n)= \begin{cases} u(n+1)+u(n-1) +V(n)u(n),\;n\geq 2\\ u(2)+\bigl(V(1)-\text{cot} (\varphi)\bigr) u(1),\;n=1\end{cases}NEWLINE\]NEWLINE for \(u\in l_2(\mathbb{N})\) if \(H_\varphi u\in l_2(\mathbb{N})\). It is assumed that \(V\) is a sparse potential, i.e., NEWLINE\[NEWLINEV(l)=\begin{cases} v_n,\;l=x_n\\ 0,\text{ otherwise }\end{cases}NEWLINE\]NEWLINE where the minimal assumption on \(x_n\) is: \(\liminf_{n\to\infty}{x_{m+1}\over x_n}>1\). The author considers the cases when \(V\) is bounded and \(V\) is unbounded. Denote by \(\rho_\varphi\) the spectral measure of the selfadjoint operator \(H_\varphi\). There are formulated sufficient conditions on \(V\) which imply that for the Fourier transform of \(g\rho_\varphi\), \(\lim_{t\to\infty} \widehat{g\rho_\varphi} (t)=0\), where \(g\in C^\infty_0 (-2,2)\) is a fixed function. The asymptotics of \(\widehat {g\rho_\varphi}\) at infinity are studied, too.