Titchmarsh–Weyl theory for vector-valued discrete Schrödinger operators

Abstract

We develop the Titchmarsh–Weyl theory for vector-valued discrete Schrodinger operators. We show that the Weyl m functions associated with these operators are matrix valued Herglotz functions that map complex upper half plane to the Siegel upper half space. We discuss about the Weyl disk and Weyl circle corresponding to these operators by defining these functions on a bounded interval. We also discuss the geometric properties of Weyl disk and find the center and radius of the Weyl disk explicitly in terms of matrices.