Specral Theory of Canonical Systems

Abstract

The main purpose of this dissertation is to give an alternate proof of de Branges' theorem on canonical systems and to prove Remling's theorem on canonical systems. In order to prove de Branges theorem, rst we show that, in the limit-circle case, the defect index of a symmetric relation induced by a canonical system is constant on complex plane. Then this follows de Branges' theorem that a canonical system with trH = 1 implies the limit-point case. As such, we develop spectral theory of a linear relation in a Hilbert space as a tool and use the theory to discuss spectral theory of a relation induced by a canonical system. Next, we prove Remling's theorem on canonical systems. We follow the similartechniques of Remling from [14]. More precisely, we rst prove Breimesser-Pearson theorem on canonical systems, following the similar techniques from [3]. Then, we present the proof of Remling's theorem on canonical systems. We also show the connection between Jacobi and Schr odinger equations and canonical systems.