Inverse problems for first‐order differential systems with periodic 2 × 2 matrix potentials and quasi‐periodic boundary conditions. (6th June 2018)

• Record Type: Journal Article
• Authors: Currie, Sonja; Roth, Thomas T.; Watson, Bruce A.
• Other Names: Bakouch Hassan guestEditor.
• Is Part Of: Mathematical methods in the applied sciences. Volume 41:Number 15(2018)
• Page Start: 5985
• Page End: 5988
• Publication Date: 2018-06-06
• Subjects: dirac system -- inverse problem -- quasi‐periodic eigenvalue problems; Mathematics -- Periodicals; Technology -- Mathematics -- Periodicals; 519
• DOI: 10.1002/mma.5113
• Languages: English
• ISSNs: 0170-4214

Abstract

Abstract : A generalisation is given of the inverse problem considered in S. Currie, B.A. Watson, and T.T. Roth. First‐order systems in C 2 on R with periodic matrix potentials and vanishing instability intervals, Math. Meth. Appl. Sci. 38 (2015), 4435‐4447. In particular, the self‐adjoint first‐order system, J Y ′ + Q Y = λ Y, with integrable, real, symmetric, and π ‐periodic, 2 × 2 matrix potential Q is considered, where J = 0 1 − 1 0 . It is shown that all eigenvalues to the above equation with boundary conditions Y ( π ) = ± R ( θ ) Y (0), where R ( θ ) is the rotation matrix cos θ sin θ − sin θ cos θ, θ ∈ [ 0, π ], are double eigenvalues if and only if Q = r I for some real scalar valued integrable function r .