Borg's Periodicity Theorems for First-Order Self-Adjoint Systems with Complex Potentials. Issue 3 (19th December 2016)

• Record Type: Journal Article
• Authors: Currie, Sonja; Roth, Thomas T.; Watson, Bruce A.
• Is Part Of: Proceedings of the Edinburgh Mathematical Society. Volume 60:Issue 3(2017)
• Page Start: 615
• Page End: 633
• Publication Date: 2016-12-19
• Subjects: Dirac system, -- inverse problems, -- spectral theory; Primary 34A55, -- Secondary 34L40, -- 34B05; Mathematics -- Periodicals; 510
• DOI: 10.1017/S0013091516000389
• Languages: English
• ISSNs: 0013-0915

Abstract

Abstract: A self-adjoint first-order system with Hermitian π -periodic potential Q ( z ), integrable on compact sets, is considered. It is shown that all zeros of are double zeros if and only if this self-adjoint system is unitarily equivalent to one in which Q ( z ) is π /2-periodic. Furthermore, the zeros of are all double zeros if and only if the associated self-adjoint system is unitarily equivalent to one in which Q ( z ) = σ 2 Q ( z ) σ 2 . Here, Δ denotes the discriminant of the system and σ 0, σ 2 are Pauli matrices. Finally, it is shown that all instability intervals vanish if and only if Q = rσ 0 + qσ 2, for some real-valued π -periodic functions r and q integrable on compact sets.