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

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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.