Two-dimensional trace-normed canonical systems of differential equations and selfadjoint interface conditions
The paper deals with the system \(Jf'=-zH(x)f\) on the line, where \(z\) is a complex parameter, \(H(x)\) is a real, nonnegative \(2\times 2\)-matrix with \(\operatorname{tr} H(x)=1\), and \[ J=\left( \begin{matrix} 0 & -1 \\ 1 & 0 \end{matrix}\right). \] The system is considered with selfadjoint interface conditions at zero. It is shown that the generalized Fourier transforms on \(\mathbb R_{\pm}\) are partially isometric. Compression to the halfline \((0,\infty)\) results in boundary conditions which depend on the eigenvalue parameter involving a fractional linear transform of the Titchmarsh-Weyl coefficient of the halfline \((0,\infty)\). The corresponding generalized Fourier transforms are isometric except possibly on a one-dimensional subspace where they are contractive.
scientific article; zbMATH DE number 2160359