重分形与离散薛定谔算子中的几个问题

项目名称： 重分形与离散薛定谔算子中的几个问题

项目编号： No.11201256

项目类型： 青年科学基金项目

立项/批准年度： 2013

项目学科： 数理科学和化学

项目作者： 瞿燕辉

作者单位： 清华大学

项目金额： 22万元

中文摘要： 本项目拟研究重分形与离散薛定谔算子中的几个基本问题。在重分形方面，我们拟研究最一般框架下的重分形分析，得到推广的重分形公式。在离散薛定谔算子方面，我们拟研究具Toeplitz势的高维薛定谔算子的谱结构和谱测度的谱型。作为结合两个领域的尝试，我们拟研究具Fibonacci势的薛定谔算子的谱上自然支撑的Gibbs测度与谱测度的关系，以及它们的重分形分析。本项目的研究有助于加深对重分形和离散薛定谔算子的理论认识，同时对他们应用于各个领域也具有指导意义。

中文关键词： Sturm 哈密尔顿；Thue-Morse哈密尔顿；Hausdorff维数；传播指数；谱

英文摘要： This project is devoted to the study of several problems in multifractal and discrete Schrodinger operators. For multifractal, we plan to study it in the most general setting, and obtain the generalized multifractal formalism. For discrete Schrodinger operator, we plan to study the higher dimensional operators with Toeplitz potentials, we are mainly concerned with the spectral structure and the spectral type of the related spectral measures. As an attempt to combine these two fields, we will study the relationship between the spectral measure and certain Gibbs measure naturally supported on the spectrum of the one dimensional operator with Fibonacci potential. We will also study their multifractal property. The study of this project will be useful, not only for a better understanding of their theories, but for a better instruction for their applications to various fields.

英文关键词： Sturm Hamiltonian；Thue-Morse Hamiltonian；Hausdorff dimension；transport exponent；spectrum