Zeroes of the spectral density of discrete Schrödinger operator with Wigner-von Neumann potential (Q1928564)

The author considers a Jacobi (tri-diagonal) matrix \[ J=\left(\begin{matrix} b_1 & 1 & 0 & \dots\\ 1 & b_2 & 1 & \dots\\ 0 & 1 & b_3 & \dots\\ \dots & \dots & \dots & \dots \end{matrix}\right) \] whose potential is the sum of a Wigner-von Neumann term and a summable term. Namely, the diagonal entries of \(J\) are given by \[ b_n=\frac{c\, \sin(2\omega n+\delta)}{n}+q_n\in\mathbb{R},\quad n\in\mathbb{N}, \] where \(c\neq 0\), \(\omega\notin \frac{\pi \mathbb{Z}}{2}\), and \(\{q_n\}_{n=1}^\infty\in l^1\). The essential spectrum of \(J\) coincides with the interval \([-2,2]\), \(\sigma_{\text{ess}}(J)=[-2,2]\). Moreover, \(\sigma_{\text{ac}}(J)=[-2,2]\) since \(\{b_n\}_{n=1}^\infty \in l^2\). If \(c=0\), i.e., \(\{b_n\}_{n=1}^\infty \in l^1\), then the spectrum on \((-2,2)\) is purely absolutely continuous. It is known that the presence (\(c\neq 0\)) of the Wigner-von Neumann potential with frequency \(\omega\) produces two critical (resonance) points \(2\cos\omega\) and \(-2\cos\omega\). The spectrum of \(J\) on the rest of the interval \((-2,2)\) is purely absolutely continuous. The main focus in the paper under review is on the behavior of the spectral density \(\rho'(\lambda)\) of \(J\) near the critical points \(\nu_{cr}\in \{-2\cos\omega, 2\cos\omega\}\). Namely, the author proves the existence of one-sided limits \[ \lim_{\lambda\to \nu_{cr}\pm0}\frac{\rho'(\lambda)}{|\lambda-\nu_{cr} |^\alpha},\qquad \alpha:=\frac{|c|}{2|\sin\omega|}, \] whenever \(\nu_{cr}\) is neither an eigenvalue of \(J\) nor a half-bound state. The proof relies on two main ingredients: the Weyl-Titchmarsh type formula from [\textit{J. Janas} and \textit{S. Simonov}, Stud. Math. 201, No. 2, 167--189 (2010; Zbl 1205.47029)] and the analysis of solutions of a certain model discrete linear system taken from [\textit{S. Naboko} and \textit{S. Simonov}, Math. Proc. Camb. Philos. Soc. 153, No. 1, 33--58 (2012; Zbl 1257.34072)].