Isospectral sets for AKNS systems on the unit interval with generalized periodic boundary conditions
The authors consider the differential operator \(H(p,q)\) generated by the differential expression \[ \left(\left( \begin{matrix} 0 & -1 \\ 1 & 0\end{matrix} \right) {d\over dx} +\left( \begin{matrix} -q & p \\ p & q\end{matrix} \right) \right) F(x) \] and the generalized periodic boundary conditions \[ BF(1) =F(0) \] with square integrable potentials. The spectrum of \(H(p,q)\) for the boundary condition generated by \(B\) is the unbounded collection \(\{\lambda (p,q,B)\}_{k\in \mathbb{Z}}\). The objective of the paper is an inverse spectral theory for \(H(p,q)\). The set \[ \biggl\{ (p,q) \in L^2_{\mathbb{R}} \bigl([0,1] \bigr)^2,\;B\in SL(2,\mathbb{R}):\;\lambda_k(p,q,B)= \lambda_k(p_0,q_0,B_0),\;k\in \mathbb{Z} \biggr\} \] is called the isospectral set associated with \((p_0,q_0,B_0)\). The main results of the paper concern a description of the isospectral sets. The authors prove that if \(B_0\) is a rotation matrix then the generalized periodic spectrum is the usual periodic one and the isospectral set is equal to its section for \(B_0\). When \(B\) is not a rotation, the isospectral sets are cylindrical real analytic submanifolds of \(L^2_{\mathbb{R}}([0,1])^2SL(2,\mathbb{R})\) and their sections for fixed boundary conditions are real analytic submanifolds of \(L^2_{\mathbb{R}}([0,1])^2\).
generalized periodic boundary conditions
scientific article; zbMATH DE number 881243
scientific article