Dynamics, spectral analysis, and renormalization in classical and quantum systems

本项目主要研究数学物理中的分析问题，包括三个主要部分。在第一部分中，PI建议致力于发展用于弱无序安德森模型谱分析的重整化群（RG）方法。 作为一个具体的应用，他计划研究态密度的性质。 技术能够改善目前的知识，这个问题几乎肯定是重要的各种其他基本问题有关的安德森模型。 在第二部分，PI建议继续他以前对非相对论量子电动力学（QED）数学基础的研究。 在各种合作中，他计划研究亚粒子散射理论的各个方面，并进一步发展等谱重整化群方法，用于分析量子场论中的谱问题。 在第三部分中，PI计划研究与高维Hamilton系统动力学相关的稳定性问题。该计划中提出的项目解决了数学物理中的三种类型的问题。安德森模型被广泛用于研究电子在随机介质（如半导体）中的量子动力学。 理解它对2维或更大维度的小无序输运性质的预测是一个主要的开放问题，拟议的项目旨在关注它的某些关键多尺度方面。非相对论QED描述非相对论量子力学物质（电子，原子，分子）与相对论量子化电磁辐射场（光子）相互作用。它以极高的精度模拟分子物理和化学中的低能过程。由于光子是无质量的，电子总是束缚无限多的低能（软）光子，从而形成一个被称为亚粒子的束缚态。 将后者纳入量子场论的标准框架是极其困难的，因为所谓的红外灾难（微扰计算通常会因软光子云而发散）。在PI及其合作者先前工作的基础上，拟议的项目旨在进一步发展亚粒子的散射理论。 此外，PI建议研究具有大量自由度的哈密顿系统的动力学和稳定性方面，例如描述经典晶体的晶格振动。

This project focuses on analytical problems in mathematical physics, and consists of three main parts. In the first part, the PI proposes to work on the development of renormalization group (RG) methods for the spectral analysis of the weakly disordered Anderson model. As a concrete application, he plans to study properties of the density of states. Techniques capable of improving the current knowledge about this problem will almost certainly be important for a variety of other essential questions concerning the Anderson model. In the second part, the PI proposes to continue his previous research on the mathematical foundations of non-relativistic Quantum Electrodynamics (QED). In various collaborations, he plans to study aspects of infraparticle scattering theory, and to further develop an isospectral renormalization group method for the analysis of spectral problems in quantum field theory. In the third part, the PI plans to investigate stability questions related to the dynamics of high-dimensional Hamiltonian systems.The projects presented in this proposal address three types of problems in mathematical physics. The Anderson model is widely used for the study of the quantum dynamics of electrons in random media, such as semiconductors. Understanding its predictions on transport properties at small disorders in dimensions 2 or larger poses a major open problem, and the proposed project intends to focus on certain key multiscale aspects of it. Non-relativistic QED describes non-relativistic quantum mechanical matter (electrons, atoms, molecules) interacting with the relativistic quantized electromagnetic radiation field (photons). It models low-energy processes in molecular physics and chemistry with excellent accuracy. Because photons are massless, electrons always bind an infinite number of low energetic (soft) photons, thus forming a bound state referred to as an infraparticle. Accommodating the latter into the standard framework of quantum field theory is extremely difficult, because of the so-called infrared catastrophe (perturbative computations typically diverge due to the soft photon cloud). Building on previous work of the PI and his collaborators, the proposed projects intend to further develop the scattering theory of infraparticles. Moreover, the PI proposes to investigate aspects of the dynamics and stability of Hamiltonian systems with a large number of degrees of freedom, which describe, for instance, the lattice vibrations of a classical crystal.