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Inverse problems for first‐order differential systems with periodic 2 × 2 matrix potentials and quasi‐periodic boundary conditions
Author(s) -
Currie Sonja,
Roth Thomas T.,
Watson Bruce A.
Publication year - 2018
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.5113
Subject(s) - mathematics , eigenvalues and eigenvectors , mathematical analysis , inverse , periodic boundary conditions , boundary value problem , integrable system , scalar (mathematics) , matrix (chemical analysis) , matrix function , mathematical physics , pure mathematics , symmetric matrix , geometry , physics , materials science , quantum mechanics , composite material
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 matrixcos 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 .