Spectral Analysis

具有慢衰减或遍历势的薛定谔算符的谱分析将被研究。利用Last-Simon、kiselevo -Last-Simon和Jitomirskaya-Last的最新成果，该项目研究了量子动力学的光谱测量的分形特性和长时间的相关问题。一个主要的推动力将是尝试将为一维系统开发的思想、技术和结果扩展到一般维度。其中一个有待研究的问题是在显式模型中是否存在混合的奇异连续和绝对连续谱。微分算子的谱分析一直是20世纪分析和理论物理学的一个重要主题。最近的研究表明，Weyl, Titchmarsh， Hartman和Wintner的经典作品所允许的潜能不仅存在，而且不是这些经典作品所暗示的病态。它与混沌科学的思想，特别是与豪斯多夫维度的概念联系在一起。我们建议进一步研究这些想法，将它们从早期作品的模型一维空间维度扩展到物理相关的三维空间，并探索量子系统长时间动力学的后果。

9707661 Simon The spectral analysis of Schrodinger operators with slowly decaying or ergodic potentials will be studied. Making use of recent results of Last-Simon, Kiselev-Last-Simon, and Jitomirskaya-Last, the project examines fractal properties of spectral measures and related issues of long time for quantum dynamics. A major thrust will be to attempt to extend to general dimension the ideas, techniques, and results developed for one-dimensional systems. Among open questions which will be examined is the existence of mixed singular continuous and absolutely continuous spectra in explicit models. Spectral analysis of differential operators has been a major theme of twentieth century analysis and theoretical physics. Recent studies have shown that potentialities that were allowed by classical works of Weyl, Titchmarsh, Hartman, and Wintner not only occur, but are not pathological as was suggested by these classical works. Links were made with ideas of chaos science, particularly with notions of Hausdorff dimension. It is proposed to study these ideas further, to extend them from the model one space dimension of earlier works to physically relevant three dimensions, and to explore the consequences for the long-time dynamics of quantum systems.