Class Of Canonical Systems Research Articles

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

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.

On a class of canonical systems on half-axis

On a class of canonical systems on half-axis

Continual Analogues of Random Polynomials That are Orthogonal on a Circle

The paper obtains conditions providing absolute continuity almost surely for the spectral measure of the corresponding random differential operators for a class of canonical systems of ordinary differential equations with random coefficients. Estimates for the densities of the spectral measures are given. Corollaries corresponding to the deterministic case are formulated. Systems of stochastic differential equations with similar properties are considered.