Monodromization and a \( \mathcal{P} \mathcal{T} \)-Symmetric Nonself-Adjoint Quasi-Periodic Operator

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.