For more information, go to https://sites.google.com/view/iqm22/workshop-ii
For more information, go to https://sites.google.com/view/iqm22/workshop-i
December 13, 2021, 14:00 Sala Consiglio - VII Piano Politecnico di Milano Building 14 (Nave), Campus Leonardo P.zza da Vinci 32, Milano, Italy
SPEAKER: Riccardo Adami (Politecnico di Torino)
Ground states for the two-dimensional NLS in the presence of point interactions
We prove the existence of ground states, i.e. minimizers of the energy at fixed mass, for the focusing, subcritical Nonlinear Schroedinger equation in two dimensions, with a linear point interaction, or defect. Ground states turn out to be positive up to a phase, and to show a logaritmico singularity at the defect. The analogous problem has been widely treated in the one dimensional setting, including the case of graphs. The two dimensional version is more complicated because of the structure of the energy space, that is larger than the standard one. This result opens the way to the study of nonlinear hybrids. This is a joint work with Filippo Boni, Raffaele Carlone, and Lorenzo Tentarelli.
Important Notice: To access the seminar room, please wait at the entrance of the Mathematics Department, Building 14. One of the organizers will let you in using the dedicated elevator for staff. As per internal regulations of Politecnico, the COVID19 green certificate will be checked before entering the room.
November 29, 2021, 14:00 Aula Seminario III Piano Politecnico di Milano Building 14 (Nave), Campus Leonardo P.zza da Vinci 32, Milano, Italy
SPEAKER: Marcello Porta (SISSA Trieste)
Correlation energy of mean-field Fermi gases
In this talk I will discuss the ground state properties of homogeneous, interacting Fermi gases, in the mean-field scaling. In this regime, Hartree-Fock theory provides a good approximation for the ground state energy of the system; this approximation is based on the replacement of the space of fermionic wave functions with the smaller set of Slater determinants, where the only correlations among the particles are those induced by the Pauli principle. I will discuss a rigorous approach that allows to go beyond the Hartree-Fock approximation, and that in particular allows to compute the leading order of the correlation energy, defined as the difference between the many-body and Hartree-Fock ground state energies. The expression we obtain reproduces the ground state energy of a non-interacting Bose gas, and agrees with the prediction of the random-phase approximation. The proof is based on a rigorous bosonization method, that allows to describe the particle-hole excitations around the Fermi surface in terms of a quasi-free Bose gas. Joint work with N. Benedikter, P. T. Nam, B. Schlein and R. Seiringer.