The bitangential inverse spectral problem for canonical systems

Abstract

The theory of the direct and bitangential inverse input impedance problem is used to solve the direct and bitangential inverse spectral problem. The analysis of the direct spectral problem uses and extends a number of results that appear in the literature. Special attention is paid to the class of canonical integral systems with matrizants that are strongly regular J-inner matrix valued functions in the sense introduced in [7]. The bitangential inverse spectral problem is solved in this class. In our considerations, the data for this inverse problem is a given nondecreasing p× p matrix valued function σ( μ) on R and a normalized monotonic continuous chain of pairs {b 3 t(λ),b 4 t(λ)}, 0⩽t<d , of entire inner p× p matrix valued functions. Each such chain defines a class of canonical integral systems in which we find a solution of the inverse problem for the given spectral function σ( μ). A detailed comparison of our investigations of inverse problems with those of Sakhnovich is presented.