Strings, dual strings, and related canonical systems

✍ Scribed by Michael Kaltenbäck; Henrik Winkler; Harald Woracek

• Publisher: John Wiley and Sons
• Year: 2007
• Tongue: English
• Weight: 221 KB
• Volume: 280
• Category: Article
• ISSN: 0025-584X
• DOI: 10.1002/mana.200410562

✦ Synopsis

Abstract

A symmetric Nevanlinna function Q is of the form Q (z) = zQ~s~ (z^2^) where Qs and Q 0(z) = zQ~s~ (z) are also Nevanlinna functions. In such a situation Q~s~ and –Q^–1^~0~ are Stieltjes functions. An inverse result of L. de Branges implies that each Nevanlinna function is the Titchmarsh–Weyl coefficient of a uniquely determined canonical system with some nonnegative Hamiltonian matrix function H, and, according to M. G. Krein, each Stieltjes function is the Titchmarsh–Weyl coefficient of a uniquely determined string. The Hamiltonians corresponding to Q~s~ , Q~0~ and Q are constructed in terms of the string corresponding to Q~s~ and the dual string corresponding to –Q^–1^~0~. The relations between the associated Fourier transformations are described by commuting isometric isomorphisms between the considered spaces. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)