The inverse spectral problem for canonical systems. (English) Zbl 0843.34031

Winkler, H.

Integral Equations Oper. Theory 22, No. 3, 360-374 (1995).

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 It is mentioned that the basic ideas for the proof are contained in an unpublished article of M. G. Krejn and H. Langer.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.

Reviewer: A.Strauss (Ul’yanovsk)

MSC:

34B20	Weyl theory and its generalizations for ordinary differential equations
34A55	Inverse problems involving ordinary differential equations
47A75	Eigenvalue problems for linear operators
47B25	Linear symmetric and selfadjoint operators (unbounded)

Keywords:

two dimensional canonical system; Weyl’s limit point case; Weyl coefficient; spectral measure

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