非周期薛定谔算子的有限截面法
Finite section method for aperiodic Schrödinger operators
AI总结：
针对势由Sturmian字给出的一维离散非周期薛定谔算子，研究其谱的豪斯多夫收敛性与有限截面法，证明可通过周期逼近元分析非周期情形，并以斐波那契哈密顿量验证结论。
摘要：
我们研究势函数由Sturmian字（斯特姆字，是斐波那契哈密顿量的自然推广）给出的一维离散薛定谔算子。通过周期势的标准逼近方法，我们证明了轴上薛定谔算子及其压缩到半轴上的对应谱的豪斯多夫收敛性。基于半轴结果，我们研究了有限截面法——这是另一种算子逼近方法，通过将算子压缩到有限但不断增长的区间上实现，常用于近似求解算子方程。我们发现，同样针对该目的，非周期情形也可以通过其周期逼近元来研究。我们通过验证斐波那契哈密顿量这一特殊情形的有限截面相关结果，阐释了非周期算子有限截面法的相关结论。
英文摘要：
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.
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