Whereas much work in the mathematical literature on quantum chaos has focused on phenomena such as quantum ergodicity and scarring, relatively little is known at the rigorous level about the existence of eigenfunctions whose morphology is more complex. Quantum systems whose dynamics is intermediate between certain regimes - for example, at the transition between Anderson localized and delocalized eigenfunctions, or in systems whose classical dynamics is intermediate between integrability and chaos - have been conjectured in the physics literature to have eigenfunctions exhibiting multifractal, self-similar structure. To-date, no rigorous mathematical results have been obtained about systems of this kind in the context of quantum chaos. We give here the first rigorous proof of the existence of multifractal eigenfunctions for a widely studied class of intermediate quantum systems. Specifically, we derive an analytical formula for the Renyi entropy associated with the eigenfunctions of arithmetic Seba billiards, in the semiclassical limit, as the associated eigenvalues tend to infinity. We also prove multifractality of the ground state for more general, non-arithmetic billiards and show that the fractal exponent in this regime satisfies a symmetry relation, similar to the one predicted in the physics literature, by establishing a connection with the functional equation for Epstein's zeta function.
我们研究高能量的Dirichlet拉普拉斯算子域台球流表现出混合动力学行为的特征函数。(More一般地，我们考虑半经典薛定谔算子与混合假设的哈密顿流。具体地说，我们假设台球流有一个不变的遍历组件，U，并研究缺陷措施，亩，正密度连续性的本征函数（更一般地说，几乎正交准模）。我们表明，任何缺陷措施相关联的这样一个子序列，当限制到U，满足μ = c μ_L，其中μ_L是刘维尔措施。这证明了珀西瓦尔猜想的一部分。
Quantum Ergodicity and the Analysis of Semiclassical Pseudodifferential Operators
这篇本科生论文关注的是发展微分几何和半经典分析的工具，以理解Schnirelman（1974），Zelditch（1987）和Colin de Verdière（1985）的量子遍历性定理以及Rudnick和Sarnak（1994）的量子唯一遍历性猜想。前者指出，在任何具有负曲率或遍历测地线流的黎曼流形上，拉普拉斯-贝尔特拉米算子的本征函数在密度为1的相空间中均匀分布。在相同的假设下，后者指出，本征函数在相空间中的哈密顿量的纤维上诱导出一系列Wigner概率测度，并且这些测度在弱-* 拓扑中收敛到一致的Liouville测度。如果是真的，猜想意味着这样的本征函数均匀分布在高本征值极限没有特殊的“疤痕”模式。这在物理上意味着混沌哈密顿系统的最精细的细节永远不能反映它们的量子力学行为，即使在半经典极限下也是如此。本论文的主要贡献是将量子遍历性和量子唯一遍历性问题置于初等解析和几何的框架中。除了提出和总结许多重要的证明，如科林·德·维迪埃的量子遍历定理的证明，我们进行图形模拟某些台球流和conventiorily讨论几个主题的量子混沌研究。
