## 20/02/2023: Jan Derezinski @Polimi

February 20, 2023
Aula Consiglio VII Piano, Politecnico di Milano
Edificio 14 (Nave), Campus Leonardo

SPEAKER: Jan Derezinski (University of Warsaw)

Schrödinger operators with singular boundary conditions

In order to define a differential operator on a domain with boundary one often needs to specify boundary conditions. In the presence of singular terms this can be quite tricky. It may lead to interesting and surprising phase diagrams. I will illustrate this with the so-called perturbed Bessel operators, that is one-dimensional Schrödinger operators on the half-line of the form $– \partial_x^2 + (m^2 – 1/4) 1/x^2 + Q(x)$.

The talk will be based on my joint work with Jeremy Faupin.

## 5/12/2022: Davide Fermi @ PoliMi

December 5, 2022
Aula Seminario VI Piano, Politecnico di Milano
Edificio 14 (Nave), Campus Leonardo

SPEAKER: Davide Fermi (Politecnico di Milano)

Vacuum fluctuations with zero-range potentials

The Casimir effect is a manifestation of van der Waals polarization forces
acting between atoms and molecules, in a regime where relativistic retardation
effects cannot be neglected. An effective description can be given in terms of
quantum fields influenced by classical objects, such as boundaries or external
potentials. This leads to the emergence of ultraviolet divergences which can be
treated by means of the so-called zeta-function regularization. In this seminar
I first present a rigorous implementation of the zeta-function technique, using
the language of canonical quantization for a scalar field. To this purpose,
complex powers of the elliptic operator describing the dynamics are used to
define smeared versions of the Wightman field and renormalization of
expectation values is ultimately attained by analytic continuation. Next, I
address the issue of boundary anomalies, discussing some explicitly solvable
models with zero-range potentials modeling semi-transparent walls.

Based on joint works with Livio Pizzocchero (University of Milano)

## 26/10/2022: Jérémy Faupin @ PoliMi

October 26, 2022, 11:15 (new time this semester)
Aula Seminario - III piano
Politecnico di Milano
Edificio 14 (Nave), Campus Leonardo
P.zza da Vinci 32, Milano, Italy

SPEAKER: Jérémy Faupin (U de Lorraine @Metz)

Quasi-classical ground states in non-relativistic QED and related models

We will consider in this talk a non-relativistic particle bound by an external
potential and coupled to a quantized radiation field. This physical system is
mathematically described by a Pauli-Fierz Hamiltonian. We will study the energy
functional of product states of the form u⊗Ψ_f, where u is a normalized state
for the non-relativistic particle and Ψ_f is a coherent state in Fock space for
the field. This gives the energy of a Klein-Gordon-Schrödinger system in the
case of a spinless particle linearly coupled to a scalar field, or the energy
of a Maxwell-Schrödinger system in the case of an electron coupled to the
photon field. In both cases, we will discuss results concerning the existence
and uniqueness of a ground state, under general conditions on the external
potential and the coupling form factor. In particular, neither an ultraviolet
cutoff nor an infrared cutoff needs to be imposed. We will also discuss the
convergence in the ultraviolet limit and the second-order asymptotic expansion
in the coupling constant of the ground state energy.
This is joint work with J. Payet and S. Breteaux.

## 13/12/2021: Riccardo Adami @ PoliMi

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.

## 29/11/2021: Marcello Porta @ PoliMi

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.