Inverse problems for first‐order differential systems with periodic 2 × 2 matrix potentials and quasi‐periodic boundary conditions

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 .