Abstract

We study the operator acting in \(L_2(\mathbb{R})\) by the formula \(( \mathcal{A} \psi)(x)=\psi(x+\omega)+\psi(x-\omega)+ \lambda e^{-2\pi i x} \psi(x)\), where \(x\in\mathbb R\) is a variable, and \(\lambda>0\) and \(\omega\in(0,1)\) are parameters. It is related to the simplest quasi-periodic operator introduced by P. Sarnak in 1982. We investigate \( \mathcal{A} \) using the monodromization method, the Buslaev–Fedotov renormalization approach, which arose when trying to extend the Bloch–Floquet theory to difference equations on \( \mathbb{R} \). Within this approach, the analysis of \( \mathcal{A} \) turns out to be very natural and transparent. We describe the geometry of the spectrum and calculate the Lyapunov exponent.

中文翻译：

---

单数化和 $$ \mathcal{P} \mathcal{T} $$ -对称非自伴随准周期算子

摘要

我们通过公式\(( \mathcal{A} \psi)(x)=\psi(x+\omega)+\psi(x-\) 来研究作用于\(L_2(\mathbb{R})\) 的算子omega)+ \lambda e^{-2\pi ix} \psi(x)\)，其中\(x\in\mathbb R\)是一个变量，而\(\lambda>0\)和\(\ omega\in(0,1)\)是参数。它与 P. Sarnak 在 1982 年引入的最简单的准周期算子有关。我们使用单函数化方法（Buslaev-Fedotov 重整化方法）研究\( \mathcal{A} \) ，该方法是在尝试扩展 Bloch– Floquet 理论到\( \mathbb{R} \)上的差分方程。在这种方法中，分析\( \mathcal{A} \)事实证明非常自然和透明。我们描述了光谱的几何形状并计算了李亚普诺夫指数。