Final Report Summary - MAQD (Mathematical Aspects of Quantum Dynamics)
This project was devoted to the mathematical understanding of the time evolution of quantum mechanical systems.
The goal of the first part of the project was the derivation of effective evolution equations approximating the full many-body quantum dynamics in certain limiting regimes. Motivated by applications to physics, we focus mainly on three different regimes; the bosonic mean field limit, the Gross-Pitaevskii regime, and the fermionic mean field limit. The mean field regime is realized when particles experience a large number of weak collisions, so that the total force acting on each of them can be effectively approximated by an average mean field potential. For systems of bosons, the many body evolution can be approximated by the Hartree equation in the mean field limit. In a series of paper, we proved convergence towards the Hartree equation, and, more importantly, we studied fluctuations around the limiting Hartree dynamics. A more subtle regime concerning bosonic systems is the so called Gross-Pitaevskii limit. In this case, collisions among particles are rare but strong. The Gross-Pitaeavskii limit is important for the description of Bose-Einstein condensates. It turns out that also in this case one can approximate the solution of the Schroedinger equation governing the many-body quantum evolution by a simpler nonlinear effective equation, known as the Gross-Pitaevskii equation. During the duration of the ERC project, we were able to prove convergence towards the limiting Gross-Pitaevskii equation with precise bounds on the rate of the convergence. In the last months of the project, we also started moving towards the next natural question, namely the description of the fluctuations around the Gross-Pitaevskii dynamics. We expect to be able to report soon on progress in this direction. Compared with the bosonic mean field regime, the Gross-Pitaevskii limit is much more subtle; in particular, correlation among particles are important in this case. To take into account correlations, one needs to appropriately modify the tools used in the analysis of mean field systems. Finally, we considered the mean field regime for fermions (a different type of particles encountered in nature, characterized by antisymmetric wave functions). Because of the antisymmetry of the wave functions, the fermionic mean field regime is naturally linked with a semiclassical regime; this makes the analysis much more involved, from the technical point of view. In a series of works completed during the duration of the ERC project, we studied this limit, proving the convergence towards the Hartree-Fock dynamics and then towards the classical Vlasov equation. To reach this goal, we had to develop new mathematical tools.
In the second part of the project, the objective was the study of the spectral properties of random matrices, which are directly related with the properties of the time evolution generated by random matrices. The main goal was to obtain a proof of the universality of local correlations of Wigner matrices (hermitean random matrices whose entries are independent and identically distributed). We were able to reach this goal shortly before the start of the ERC project (but after I inserted this important question in the proposal). As a consequence, we worked on different questions related with the spectrum of Wigner matrices. The two main results that we obtained in this direction were i) the convergence of the average density of states towards the famous semicircle law, uniformly on every scale and ii) optimal bounds on the rate of convergence of the Stieltjes transform of Wigner matrices.
The goal of the first part of the project was the derivation of effective evolution equations approximating the full many-body quantum dynamics in certain limiting regimes. Motivated by applications to physics, we focus mainly on three different regimes; the bosonic mean field limit, the Gross-Pitaevskii regime, and the fermionic mean field limit. The mean field regime is realized when particles experience a large number of weak collisions, so that the total force acting on each of them can be effectively approximated by an average mean field potential. For systems of bosons, the many body evolution can be approximated by the Hartree equation in the mean field limit. In a series of paper, we proved convergence towards the Hartree equation, and, more importantly, we studied fluctuations around the limiting Hartree dynamics. A more subtle regime concerning bosonic systems is the so called Gross-Pitaevskii limit. In this case, collisions among particles are rare but strong. The Gross-Pitaeavskii limit is important for the description of Bose-Einstein condensates. It turns out that also in this case one can approximate the solution of the Schroedinger equation governing the many-body quantum evolution by a simpler nonlinear effective equation, known as the Gross-Pitaevskii equation. During the duration of the ERC project, we were able to prove convergence towards the limiting Gross-Pitaevskii equation with precise bounds on the rate of the convergence. In the last months of the project, we also started moving towards the next natural question, namely the description of the fluctuations around the Gross-Pitaevskii dynamics. We expect to be able to report soon on progress in this direction. Compared with the bosonic mean field regime, the Gross-Pitaevskii limit is much more subtle; in particular, correlation among particles are important in this case. To take into account correlations, one needs to appropriately modify the tools used in the analysis of mean field systems. Finally, we considered the mean field regime for fermions (a different type of particles encountered in nature, characterized by antisymmetric wave functions). Because of the antisymmetry of the wave functions, the fermionic mean field regime is naturally linked with a semiclassical regime; this makes the analysis much more involved, from the technical point of view. In a series of works completed during the duration of the ERC project, we studied this limit, proving the convergence towards the Hartree-Fock dynamics and then towards the classical Vlasov equation. To reach this goal, we had to develop new mathematical tools.
In the second part of the project, the objective was the study of the spectral properties of random matrices, which are directly related with the properties of the time evolution generated by random matrices. The main goal was to obtain a proof of the universality of local correlations of Wigner matrices (hermitean random matrices whose entries are independent and identically distributed). We were able to reach this goal shortly before the start of the ERC project (but after I inserted this important question in the proposal). As a consequence, we worked on different questions related with the spectrum of Wigner matrices. The two main results that we obtained in this direction were i) the convergence of the average density of states towards the famous semicircle law, uniformly on every scale and ii) optimal bounds on the rate of convergence of the Stieltjes transform of Wigner matrices.