On half-line spectra for a class of non-self-adjoint Hill operators

TL;DR

This paper offers a new, straightforward proof that certain non-self-adjoint Hill operators with complex periodic potentials have spectra covering the entire non-negative real line.

Contribution

It provides an alternative, elementary proof of Gasymov's 1980 result on the spectra of specific non-self-adjoint Hill operators.

Findings

Confirmed the spectrum is [0, ∞) for the class of operators.

Simplified the proof method for Gasymov's theorem.

Enhanced understanding of spectral properties of non-self-adjoint operators.

Abstract

In 1980, Gasymov showed that non-self-adjoint Hill operators with complex-valued periodic potentials of the type $V(x)=\sum_{k=1}^{\infty} a_k e^{ikx}$, with $\sum_{k=1}^{\infty}|a_k|<\infty$, have spectra $[0, \infty)$. In this note, we provide an alternative and elementary proof of this result.