Non-Self-Adjoint Hill Operators whose Spectrum is a Real Interval

Abstract

Let $H=-d^2/d{x}^2+q(x)$, $x\in \mathbb{R}$, where $q(x)$ is a periodic potential, and suppose that the spectrum $\ddot{I}ƒ(H)$ of $H$ is the positive semi-axis $[0,\infty )$. In the case where $q(x)$ is real-valued (and locally square-integrable) a well-known result of G. Borg states that $q(x)$ must vanish almost everywhere. However, as it was first observed by M.G. Gasymov, there is an abundance of complex-valued potentials for which $\ddot{I}ƒ(H)=[0,\infty )$. In this article we conjecture a characterization of all complex-valued entire potentials whose spectrum is $[0,\infty )$. We also present an analog of Borg's result for complex potentials.

22 pages