Schroedinger difference equation with deterministic ergodic potentials

TL;DR

This paper reviews recent rigorous mathematical results on the one-dimensional Schrödinger equation with deterministic ergodic potentials, focusing on substitutional sequences like Fibonacci and Thue-Morse, highlighting advances in understanding spectral properties.

Contribution

It provides a comprehensive overview of recent developments and rigorous results in the spectral theory of Schrödinger operators with deterministic ergodic potentials generated by substitution sequences.

Findings

Spectral properties of Schrödinger operators with Fibonacci potentials elucidated

Rigorous results on spectral measures for Thue-Morse potentials presented

Mathematical background for ergodic potentials discussed

Abstract

We review the recent developments in the theory of the one-dimensional tight-binding Schr\"odinger equation for a class of deterministic ergodic potentials. In the typical examples the potentials are generated by substitutional sequences, like the Fibonacci or the Thue-Morse sequence. We concentrate on rigorous results which will be explained rather than proved. The necessary mathematical background is provided in the text.