论文标题
固定随机schrödinger操作员的新光谱分析
A new spectral analysis of stationary random Schrödinger operators
论文作者
论文摘要
目前的工作是由量子波的长期运输特性的促进，使随机的schrödinger操作员陷入了新的光谱视角。基于Floquet类型振动的固定随机版本，我们将量子动力学的描述减少到基础概率空间上的抽象光谱扰动问题的纤维家族。我们对这些纤维操作员说明了自然的共振猜想：与周期性和准二体设置相反，这将需要布洛克波不在扩展状态下存在，而是作为共鸣模式，这将证明预期的指数衰减是合理的。尽管这种共鸣的猜想仍然保持开放，但我们开发了用于概率空间的光谱分析的新工具，特别是我们展示了Malliavin conculus的想法如何导致严格的Mourre型结果：我们获得了近似的动态共振结果，以及在动力学时间表上的衰减时间相关的第一个光谱证明。这种光谱方法提出了一种绕过扰动扩展和重新归一化技术的全新方法。
Motivated by the long-time transport properties of quantum waves in weakly disordered media, the present work puts random Schrödinger operators into a new spectral perspective. Based on a stationary random version of a Floquet type fibration, we reduce the description of the quantum dynamics to a fibered family of abstract spectral perturbation problems on the underlying probability space. We state a natural resonance conjecture for these fibered operators: in contrast with periodic and quasiperiodic settings, this would entail that Bloch waves do not exist as extended states, but rather as resonant modes, and this would justify the expected exponential decay of time correlations. Although this resonance conjecture remains open, we develop new tools for spectral analysis on the probability space, and in particular we show how ideas from Malliavin calculus lead to rigorous Mourre type results: we obtain an approximate dynamical resonance result and the first spectral proof of the decay of time correlations on the kinetic timescale. This spectral approach suggests a whole new way of circumventing perturbative expansions and renormalization techniques.
