Spectra of non-selfadjoint Hill's operators and a class of Riemann surfaces (Q1912824)

The paper concerns a spectral analysis of the nonselfadjoint Hill's operator \(L= -{d^2\over dx^2}+ q\) in \(L^2(\mathbb{R})\) with scalar complex-valued \(\Pi\)-periodic function \(q\in L^2[0, \Pi]\). The spectrum \(\sigma(L)\) is formed by a countable system of analytic arcs. The aim of the paper is to describe all systems of analytic arcs which may be spectra of Hill's operators. The author shows that all possible spectra of Hill's operators are one-to-one parametrized by Riemann surfaces of a certain class.