Two-Dimensional Trace-Normed Canonical Systems of Differential Equations and Selfadjoint Interface Conditions

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.