The inverse spectral problem for canonical systems
The two dimensional canonical system is connected with the Hamiltonian \(H\), a real symmetric nonnegative trace normed \(2 \times 2\) matrix function on \([0, \infty)\). At \(\infty\) the Weyl's limit point case prevails. The corresponding Weyl coefficient belongs to \(\widetilde N : = N \cup \{\infty\}\), where \(N\) is the set of Nevanlinna functions. It is shown that any \(Q \in \widetilde N\) is a Weyl coefficient of a unique canonical system and that this proposition follows from some results of L. de Branges concerning Hilbert spaces of entire functions. It is mentioned that the basic ideas for the proof are contained in an unpublished article of M. G. Krejn and H. Langer. The construction of \(H\) is described when the given spectral measure of \(\mathbb{Q}\) has a density \(p (\lambda)^{-1}\), where \(p\) is a polynomial, positive on \(\mathbb{R}\). This is analogous to a method proposed by M. G. Krejn of constructing the mass distribution function of a string, if the main spectral function has a density of the same form.
scientific article; zbMATH DE number 845916
scientific article