Spectral theory of Schrödinger operators and the almost Mathieu operator

Mikkel Topsøe

Studenteropgave: Speciale

Abstrakt

My thesis gives an introduction to functional analysis with main focus on spectral analysis. The main focus is the spectral analysis of a very special quasi-periodic discrete Schr\"{o}dinger operator called the almost Mathieu operator, and the spectrum differences between itself, the continuous, and the discrete non-ergodic Schr\"{o}dinger operators. In the text it is shown that the almost Mathieu operator is an ergodic bounded self-adjoint operator. Furthermore it is shown that the operator satisfies the Herman estimate, Aubry duality, and Anderson localization. Spectral analysis of fundamental and essential Schr\"{o}dinger operators is given in order to outline the introduced spectral theory and in the scope to compare and discuss main differences between the almost Mathieu operator and the non-ergodic Schr\"{o}dinger operators.

UddannelserMatematik, (Bachelor/kandidatuddannelse) Kandidat
SprogEngelsk
Udgivelsesdato29 jun. 2015
VejledereCarsten Lunde

Emneord

  • almost Mathieu
  • spectral theory