Finite section method for aperiodic Schr\"odinger operators

TL;DR

This paper investigates the spectral properties of 1D aperiodic Schr"odinger operators with Sturmian potentials, establishing convergence of spectra via periodic approximations and analyzing the finite section method for solving related operator equations.

Contribution

It introduces a method to analyze aperiodic Schr"odinger operators using periodic approximants, extending spectral convergence results and finite section method analysis.

Findings

Hausdorff convergence of spectra for aperiodic operators

Finite section method applies to aperiodic operators via periodic approximants

Confirmed finite section results for Fibonacci Hamiltonian

Abstract

We consider 1D discrete Schr\"odinger operators with aperiodic potentials given by a Sturmian word, which is a natural generalisation of the Fibonacci Hamiltonian. Via a standard approximation by periodic potentials, we establish Hausdorff convergence of the corresponding spectra for the Schr\"odinger operators on the axis as well as for their compressions to the half-axis. Based on the half-axis results, we study the finite section method, which is another operator approximation, now by compressions to finite but growing intervals, that is often used to solve operator equations approximately. We find that, also for this purpose, the aperiodic case can be studied via its periodic approximants. Our results on the finite section method of the aperiodic operator are illustrated by confirming a result on the finite sections of the special case of the Fibonacci Hamiltonian.