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Borg's Periodicity Theorems for First-Order Self-Adjoint Systems with Complex Potentials

Part of: Ordinary differential operators General theory in ordinary differential equations Boundary value problems

Published online by Cambridge University Press:  19 December 2016

Sonja Currie ,
Thomas T. Roth  and
Bruce A. Watson
Sonja Currie
Affiliation:
School of Mathematics, University of the Witwatersrand, Private Bag 3, PO Wits 2050, South Africa (b.alastair.watson@gmail.com)
Thomas T. Roth
Affiliation:
School of Mathematics, University of the Witwatersrand, Private Bag 3, PO Wits 2050, South Africa (b.alastair.watson@gmail.com)
Bruce A. Watson
Affiliation:
School of Mathematics, University of the Witwatersrand, Private Bag 3, PO Wits 2050, South Africa (b.alastair.watson@gmail.com)
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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.

Keywords

Dirac systeminverse problemsspectral theory

MSC classification

Secondary: 34A55: Inverse problems 34L40: Particular operators (Dirac, one-dimensional Schrödinger, etc.) 34B05: Linear boundary value problems

Information

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 60 , Issue 3 , August 2017 , pp. 615 - 633
Copyright
Copyright © Edinburgh Mathematical Society 2016 

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