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.

翻译：我们考虑一维离散薛定谔算子，其非周期势由Sturmian词给出，这是Fibonacci哈密顿量的自然推广。通过标准的周期势近似，我们建立了轴上薛定谔算子及其在半轴压缩对应谱的Hausdorff收敛。基于半轴结果，我们研究了有限截面方法——另一种算子近似方法，即通过压缩至有限但增长的区间，常用来近似求解算子方程。我们发现，对于这一目的，非周期情形也可通过其周期逼近来研究。我们关于非周期算子有限截面方法的结果，通过验证Fibonacci哈密顿量特例有限截面的一个结果得以说明。