Abstract.
The class of two-dimensional trace-normed canonical systems of differential equations on \(\mathbb{R}\) is considered with selfadjoint interface conditions at 0. If one or both of the intervals around 0 are H-indivisible the interface conditions which give rise to selfadjoint relations (multi-valued operators) are determined. It is shown that the corresponding generalized Fourier transforms are partially isometric. Compression to the halfline (0, ∞) results in boundary conditions which depend on the eigenvalue parameter involving a fractional linear transform of the Titchmarsh-Weyl coefficient of the halfline (−∞, 0). The corresponding generalized Fourier transforms are isometric except possibly on a one-dimensional subspace where they are contractive.
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de Snoo, H., Winkler, H. Two-Dimensional Trace-Normed Canonical Systems of Differential Equations and Selfadjoint Interface Conditions. Integr. equ. oper. theory 51, 73–108 (2005). https://doi.org/10.1007/s00020-003-1243-y
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DOI: https://doi.org/10.1007/s00020-003-1243-y