Spectral Estimations for Can nical Systems

Two–dimensional canonical systems are boundary value problems of the form\documentclass{article}\usepackage{amssymb}\pagestyle{empty}\begin{document}$$Jy\prime (x) = -zH(x)y(x), \, x \in [0, \, L), \, L \le \infty, \, z \in {\mathbb{C}},$$\end{document}with y 1 (0) = 0 and Weyl's limit point case at L . The 2 × 2 matrix valued function H is real, symmetric and nonnegative, $J = \left ( \matrix {0 & -1 \cr 1 & 0 \cr } \right )$ . The correspondence between canonical systems and their Titchmarsh–Weyl coefficients Q is a bijection between the class of all matrix functions H with tr H ( x ) = 1 a.e. on [0, L ) and the class of the Nevanlinna functions ℕ augmented by the function Q ≡ 8. Each Titchmarsh–Weyl coefficient Q ∈ ℕ can be represented by means of a measure σ, the so–called spectral measure of the canonical system. In this note matrix functions H are specified whose corresponding spectral measures σ satisfy conditions of the form $\int ^{+ \infty} _{- \infty} {{d \sigma (\lambda)} \over {1 + \vert \lambda \vert \gamma}} < + \infty $ or $\int ^1 _{-1} {{d \sigma (\lambda)} \over {\vert \lambda \vert \gamma}} < + \infty, \, \gamma \in [0,2]$ . Herewith we generalize corresponding results of M.G. Krein and I. S. Kac for so–called vibrating strings.