Bitangential direct and inverse problems for systems of differential equations

Summary

A number of results obtained by the authors on direct and inverse problems for canonical systems of differential equations, and their implications for certain classes of systems of Schro¨dinger equations and systems with po- tential are surveyed. Connections with the theory of J -inner matrix valued and reproducing kernel Hilbert spaces, which play a basic role in the original developments, are discussed.

1. Introduction

In this paper we shall present a brief survey of a number of results on direct and inverse problems for canonical integral and differential systems that have been obtained by the authors over the past several years. We shall not attempt to survey the literature, which is vast, or to compare the methods surveyed here with other approaches. The references in [Arov and Dym 2004; 2005b; 2005c] (the last of which is a survey article) may serve at least as a starting point for those who wish to explore the literature.

where H(t) is an m x m locally summable mvf (matrix valued function) that is positive semidefinite a.e. on the interval [0, d], j is an m x m signature matrix, i.e., J * J and J * J = I_m, and y(t, ƛ) is a k x m mvf.