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

Abstract

Using linear similarity of a certain class of Volterra operators to the squared integration, we derive an important representation of the general-type fundamental solutions of the canonical systems corresponding to matrix string equations. Explicit fundamental solutions of such canonical systems are constructed (via the GBDT version of Darboux transformation) as well. Examples and applications to dynamical canonical systems are given. Explicit solutions of the dynamical canonical systems are constructed. Three appendices are dedicated to the Weyl-Titchmarsh theory for canonical systems, to the transformation of a subclass of canonical systems into matrix string equations (and of a smaller subclass of canonical systems into matrix Schrödinger equations), and to a linear similarity problem for Volterra operators.