On the Point Spectrum of a Non-Self-Adjoint Quasiperiodic Operator

Abstract

We consider a difference operator acting in \(l^2(\mathbb Z)\) by the formula \(( \mathcal{A} \psi)_n=\psi_{n+1}+\psi_{n-1}+\lambda e^{-2\pi \mathrm{i} (\theta+\omega n)} \psi_n\), \(n\in \mathbb{Z}\), where \(\omega\in(0,1)\), \(\lambda>0\), and \(\theta\in [0,1]\) are parameters. This operator was introduced by P. Sarnak in 1982. For \(\omega\not\in \mathbb Q\), the operator \( \mathcal{A} \) is quasiperiodic. Previously, within the framework of a renormalization approach (monodromization method), we described the location of the spectrum of this operator. In the present work, we first establish the existence of the point spectrum for different values of parameters, and then study the eigenfunctions. To do so, using ideas of the renormalization approach, we study the difference operator on the circle obtained from the original one by the Fourier transform. This allows us, first, to obtain a new type condition guaranteeing the existence of point spectrum and, second, to describe in detail a multi-scale self-similar structure of the Fourier transforms of the eigenfunctions.

DOI 10.1134/S106192082403004X