Abstracts of the talks

In this talk I will consider the 2d linear Schrödinger equation with a time-dependent point interaction. Such model appears in the effective description of a quantum particle interacting with a quantized bosonic scalar field, in the classical limit for the degrees of freedom of the latter. We will describe how to prove a general global well-posedness result for the Cauchy problem and study the long time behavior, proving complete ionization for suitable assumptions on the initial datum and on the point-interaction. Based on a joint work with Raffaele Carlone (Università di Napoli) and Lorenzo Tentarelli (Politecnico di Torino).


Originally arisen to understand characterising properties connected with dispersive phenomena, in the last decades the method of multipliers has been recognized as a useful tool in Spectral Theory, in particular in connection with proof of absence of point spectrum for self-adjoint and non self-adjoint operators. In this seminar we will see the developments of the method reviewing some recent results concerning self-adjoint and non self-adjoint Schrödinger operators in different settings and relativistic Pauli and Dirac operators.  Moreover we will show how this technique can be fruitful developed to obtain similar results for higher order operators. In particula special emphasis will be given to the case of perturbed bilaplacian. The talk is based on joint works with L. Fanelli and D. Krejcirik.

This talk is divided in two parts. The first one consists in defining the Dirac operator and its physical properties. We explain the notion of spinors through irreducible group representations and recall the historical derivation of the Dirac equation and its physical interpretation. The second part will review some of the different types of analytical questions one may ask when considering the Dirac operator, among which : self-adjointness issues, derivation of efficient operators, and dispersion.


Ever since the existence of Dark Matter was first conjectured, physicists have been using of phenomenological models to extract information from experimental data, with the aim of describing how Dark Matter may evolve and distribute. Defining phenomenological models with fully understood mathematical properties represents a robust way to get reliable information, addressing a physical phenomenon that might otherwise remain mainly  unknown. In this talk I will illustrate a novel Klein-Gordon-Wave model for Dark Matter, outlining what scenarios it may suggest for Dark Matter distributions. First, I will describe a perturbative regime, where the application of  a normal form approximation leads to the Schrödinger-Wave model, the most used phenomenological model in this field. Then, I will focus on excited stationary states, setting a comparison with experimental data and illustrating what new properties they may predict for Dark Matter.

In the study of electronic structure of heavy atoms, relativistic effects cannot be neglected anymore and the Dirac operator naturally appears in place of the Schrödinger operator, raising up a number of additional difficulties which will be deeply discussed at the beginning of this talk. The complexity of these systems has been addressed by various approximations. In this seminar, we will consider a model consisting of differential equations coupling the time evolution of a finite number of relativistic electrons with the Newtonian dynamics of finitely many nuclei, where the former is described by the Bogoliubov-Dirac-Fock equation of quantum electrodynamics. A global well-posedness result will be achieved by addressing the Cauchy problem for this model. We think that this system can be seen as a first step in the study of molecular dynamics phenomena in relativistic quantum chemistry.