GBDT, multiplicative integrals and linear similarity

TL;DR

This work develops a generalized Bäcklund–Darboux transformation (GBDT) framework for the non-isospectral canonical system with a hidden spectral parameter $\lambda=(z-x)^{-1}$, enabling systematic construction of transformed systems, multiplicative integrals, and characteristic data. It establishes how transformed Hamiltonians $\widetilde{H}(x)=w_0(x)^*H(x)w_0(x)$ and transformed fundamental solutions $\widetilde{W}(x,z)$ arise, and derives how multiplicative integrals transform through $\widetilde{W}(x,z)=v(x,z)W(x,z)v(\xi,z)^{-1}$ with explicit limit relations. The paper connects these constructions to spectral theory via Livšic–Brodskii characteristic functions, showing that the simple part of the associated operator is linearly similar to a self-adjoint operator with absolutely continuous spectrum, and that transformed characteristic functions satisfy $\widetilde{\mathcal{W}}(z)=v(b,z)\mathcal{W}(z)v(a,z)^{-1}$. Explicit rank-one examples illustrate the method and provide closed-form expressions for the transformed objects, highlighting the framework’s practical computability and its relevance to triangular models and RH-type problems.

Abstract

A GBDT version of the Bäcklund-Darboux transformation for a non-isospectral canonical system is considered. Applications to multiplicative integrals and their limit values, to characteristic matrix functions and to linear similarity problems are obtained. Some interesting examples are constructed as well.