Ec & Spec
We focus on problems in Quantum Mechanics (mostly related to the Schrödinger and Dirac equations) and magnetohydrodynamic that can be studied using fine techniques from harmonic analysis. Recently, some classic tools from Real and Harmonic Analysis turned out to be crucial in order to address answers to many relevant questions arising in Spectral Theory and PDEs. Spectral Theory has always been a fundamental ingredient to describe the stability properties of special waves. At the same time, the research about stability of nonlinear waves led to some challenging spectral problems and provided new real and harmonic analytical techniques to handle them. An increasing interest towards these two linked areas of Applied Mathematics and the consolidated experience of the PI1 in both of them motivate the first part of this project. The main objectives of the first two research lines are related with the spectral properties of relativistic Hamiltonians, and the stability of breathers, soliton-like waves evolving in an oscillatory fashion.
A new interest for non self-adjoint Hamiltonians, as mathematical models to describe Quantum Mechanics opened many questions on the nature of their spectrum. Although some relevant advances have been obtained for non relativistic models, a lot still needs to be understood in the relativistic case.
Moreover, starting by the pioneering results of Benjamin and Kato on the stability of the KdV-soliton, much has been done in the last years, to understand the underline dynamics leading to stability/instability, towards a kind of universal description of the large time behavior of dispersive waves, given by the soliton resolution. Breathers are a very peculiar type of waves, whose stability is known in a very few cases, and requires a deep use of sophisticated techniques from Spectral Theory.
The third research line concerns the pointwise behavior of solutions to the Schrödinger equation. This very natural problem, introduced by Carleson in the 80s, has motivated an extraordinary development of deep harmonic analysis techniques as multilinear Kakeya estimates and decoupling (it is indeed related to other fundamental questions, like the Stein restriction conjecture). Recently, a complete solution on R^n has been given in the breakthrough contributions of Du-Guth-Li and Du-Zhang. Relying on the consolidated experience of the PI2 on this problem, we aim to extend these results to the setting of compact manifolds, which (as it happens for the Strichartz estimates) is more challenging because of the presence of many resonances. A particular attention will be given to periodic solution, for which proving sharp results would represent a milestone.
We also consider probabilistic improvements, the idea being to exclude some null probability (pathological) sets of initial data where convergence fails. These probabilistic methods have their roots in the foundation of a statistical mechanics theory for the nonlinear Schrödinger equation and we find particularly interesting using them in the context of the Carleson problem. The fourth research line of investigation concerns the magnetohydrodynamic (MHD), which describes the motion and the behavior of electrically conducting fluids. Our goal will be to provide rigorous analytical examples of magnetic reconnection, namely of solutions of the MHD equation such that the magnetic lines change their topology during the flow.
