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

2024/09/16 by Vassilis G. Papanicolaou, Papanicolaou, Vassilis G.
Computer Science · Mathematics · #34B30 #34L40 #47A10 #47E05 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Holomorphic and Operator Theory #Matrix Theory and Algorithms #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2409.10266

openalex publication_date 2024/09/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let H = -d²/dx² + q(x), x ∈ ℝ, where q(x) is a periodic potential, and suppose that the spectrum σ(H) of H is the positive semi-axis [0, ∞). 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 σ(H) = [0, ∞). In this article we conjecture a characterization of all complex-valued entire potentials whose spectrum is [0, ∞). We also present an analog of Borg's result for complex potentials.

Related

• On half-line spectra for a class of non-self-adjoint Hill operators
• A necessary and sufficient condition for the Darboux-Treibich-Verdier potential with its spectrum contained in ℝ
• On the Spectrality and Spectral Expansion of the Non-self-adjoint\n Mathieu-Hill Operator in All Real Line
• Characterization of potential smoothness and Riesz basis property of Hill-Scrödinger operators with singular periodic potentials in terms of periodic, antiperiodic and Neumann spectra
• Hill's operators with the potentials analytically dependent on energy