Generalised canonical systems related to matrix string equations: corresponding structured operators and high-energy asymptotics of the Weyl functions

Abstract

We obtain high energy asymptotics of Titchmarsh-Weyl functions of the generalised canonical systems generalising in this way a seminal Gesztesy-Simon result. The matrix valued analog of the amplitude function satisfies in this case an interesting new identity. The corresponding structured operators are studied as well.

On the class of canonical systems corresponding to matrix string equations: general-type and explicit fundamental solutions and Weyl-Titchmarsh theory

“…x 0 β(t) * •dt to the operator (2.2) of squared integration as well as the form of the corresponding similarity transformation operator V (see Theorem C.1 and its proof in Appendix C). The representation of the fundamental solutions in Section 2 is important in itself, and it presents also a crucial step in the study of the high energy asymptotics of Weyl-Titchmarsh functions [48] and in solving the inverse problem to recover canonical system from the spectral or Weyl-Titchmarsh function [49]. Some basic results and notions on the Weyl-Titchmarsh theory of the generaltype canonical systems (1.4) are described in Appendix A.…”