AF嵌入与无理旋转代数中谱的数值计算
AF embeddings and the numerical computation of spectra in irrational rotation algebras
AI总结：
本文利用Pimsner-Voiculescu的AF嵌入构造，为无理旋转代数中一大类算子（含准周期薛定谔算子）提供有限维谱逼近方法，并借助连分数理论给出尖锐收敛速率。
摘要：
离散化的一维薛定谔算子的谱分析是一个非常困难的问题，已有众多数学家对此进行研究。在数值分析与算子理论的交叉领域，一个自然的问题是寻找有限维矩阵，使其特征值逼近无穷维算子的谱。在本文中，我们注意到Pimsner-Voiculescu关于无理旋转代数的AF嵌入的开创性工作，为一大类算子（包括上述薛定谔算子的准周期情形）的有限维谱逼近问题提供了很好的解答。事实上，连分数理论不仅为谱计算提供了良好的矩阵模型（即Pimsner-Voiculescu构造），而且还能给出无理旋转代数中算子谱逼近的{\em 尖锐}收敛速率。
英文摘要：
The spectral analysis of discretized one-dimensional Schrödinger operators is a very difficult problem which has been studied by numerous mathematicians. A natural problem at the interface of numerical analysis and operator theory is that of finding finite dimensional matrices whose eigenvalues approximate the spectrum of an infinite dimensional operator. In this note we observe that the seminal work of Pimsner-Voiculescu on AF embeddings of irrational rotation algebras provides a nice answer to the finite dimensional spectral approximation problem for a broad class of operators including the quasiperiodic case of the Schrödinger operators mentioned above. Indeed, the theory of continued fractions not only provides good matrix models for spectral computations (i.e. the Pimsner-Voiculescu construction) but also yields {\em sharp} rates of convergence for spectral approximations of operators in irrational rotation algebras.
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