Boundary-value problems for two-dimensional canonical systems
The authors consider a two-dimensional canonical system \(Jy'= \ell Hy\), where \(H\) is a real, nonnegative \(2\times 2\) matrix function on \(\mathbb{R}\) satisfying \(\text{tr }H(x)= 1\), and \(J\) is a signature matrix. If the system is definite on \(\mathbb{R}^+\), it gives rise to a symmetric relation in the Hilbert space \(L^2(H,\mathbb{R}^+)\), and this relation in turn induces a closed symmetric operator \(T^+_{\min,s}\) in some closed subspace \(L^2_s(H,\mathbb{R}^+)\) of \(L^2(H,\mathbb{R}^+)\). It is proved that \(T^+_{\min,s}\) has defect numbers \((1,1)\) and a characterization of all selfadjoint extensions is given. These results also apply to \(\mathbb{R}^-\) thus yielding a closed symmetric operator \(T^-_{\min,s}\) in \(L^2_s(H, \mathbb{R}^-)\). A one-to-one correspondence between the selfadjoint extensions of \(T^+_{\min, s}\oplus T^-_{\min,s}\) in \(L^2_s(H,\mathbb{R}^+)\oplus L^2_s(H, \mathbb{R}^-)\) is established which allows to study all possible realizations of the given system via an interface condition at the origin.
scientific article; zbMATH DE number 1484695
scientific article