Finite section method for aperiodic Schrödinger operators

Abstract

We consider 1D discrete Schrödinger 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 Schrödinger 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.

References in corpus (7)

• Uniform spectral properties of one-dimensional quasicrystals, I. Absence of eigenvalues
• The Fibonacci Hamiltonian
• An Affirmative Answer to a Core Issue on Limit Operators
• Limit Operators, Collective Compactness, and the Spectral Theory of Infinite Matrices
• Spectral analysis of tridiagonal Fibonacci Hamiltonians
• Transport Exponents of Sturmian Hamiltonians
• Finite Sections of Periodic Schrödinger Operators