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

6 pages; no figure; to appear in Math. Nachr