Meromorphic Inner Functions and their Applications

Abstract

In this dissertation we study an important class of functions in complex analysis, known as Meromorphic Inner Functions (MIF) and we exploit their properties to solve problems from mathematical physics.

In the first part, we answer an old problem, first studied by Louis de Branges about a property of the derivative of MIFs on the real line. We use the theory of Clark measures to solve this problem with the aid of complex and harmonic analysis theory.

In the second part, we study the spectral properties of the Schrödinger operator. Certain inverse spectral problems in this area can be translated into the language of complex analysis and we use the injectivity of Toeplitz operators to solve these problems.