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Boundary-value problems for two-dimensional canonical systems. (English) Zbl 0966.47012

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.

MSC:

47B25 Linear symmetric and selfadjoint operators (unbounded)
47A20 Dilations, extensions, compressions of linear operators
47E05 General theory of ordinary differential operators
34B20 Weyl theory and its generalizations for ordinary differential equations
34A55 Inverse problems involving ordinary differential equations
34L05 General spectral theory of ordinary differential operators
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