For more information, go to https://sites.google.com/view/iqm22/workshop-i
February 21, 2022, 16:00 (UNUSUAL TIME!) Zoom only (online talk)
SPEAKER: Ian Jauslin (Rutgers University)
An effective equation to study Bose gases at all densities
I will discuss an effective equation, which is used to study the ground state of the interacting Bose gas. The interactions induce many-body correlations in the system, which makes it very difficult to study, be it analytically or numerically. A very successful approach to solving this problem is Bogolubov theory, in which a series of approximations are made, after which the analysis reduces to a one-particle problem, which incorporates the many-body
correlations. The effective equation I will discuss is arrived at by making a very different set of approximations, and, like Bogolubov theory, ultimately reduces to a one-particle problem. But, whereas Bogolubov theory is accurate only for very small densities, the effective equation coincides with the many-body Bose gas at both low and at high densities. I will show some theorems which make this statement more precise, and present numerical evidence that this effective equation is remarkably accurate for all densities, small, intermediate, and large. That is, the analytical and numerical evidence suggest that this effective equation can capture many-body correlations in a one-particle picture beyond what Bogolubov can accomplish. Thus, this effective equation gives an alternative approach to study the low density behavior of the Bose gas (about which there still are many important open questions). In addition, it opens an avenue to understand the physics of the Bose gas at intermediate densities, which, until now, were only accessible to Monte Carlo simulations.
February 7, 2022, 14:00 Sala di Rappresentanza Mathematics Department, University of Milan Via Cesare Saldini 50, Milano, Italy
SPEAKER: Ngoc Nhi Nguyen (U Paris Saclay)
Fermionic semiclassical L^p estimates
Spectral properties of Schrödinger operators are studied a lot in mathematical physics. They can give the description of trapped fermionic particles. Researches on the spatial concentration of semiclassical Schrödinger operators’ eigenfunctions are still carried out, whether in physics or in mathematics. There are very precise results in special cases like the harmonic oscillator. However, it is not always possible to obtain explicitly point wise information for more general potentials. We can measure the concentration by estimating these functions with L^p bounds.
December 20-22, 2021 Sala di Rappresentanza Università degli Studi di Milano Via Cesare Saldini 50, Milano, Italy
Website: Quantum Before Christmas
This workshop will take place from Mon. 20 Dec. @14h, to Wed. 22 Dec. @13h.
Twelve speakers will present their current research, covering topics from many-body quantum mechanics to PDEs.
For more information, see the conference webpage above.
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.
November 8, 2021, 14:00 "Aula di Rappresentanza" Mathematics Department, University of Milan Via Cesare Saldini 50, Milano, Italy
SPEAKER: Per Moosavi (ETH Zürich)
Non-local Luttinger model out of equilibrium: Exact results and emergence of generalized hydrodynamics
The non-local Luttinger model is an exactly solvable 1+1D quantum field theory with finite-range interactions that lies somewhere between conformal and Bethe-ansatz integrable models. Using bosonization, I will show how exact analytical results can be computed for the time evolution of this model following an inhomogeneous quantum quench from initial states defined by smooth inverse-temperature and chemical-potential profiles. These results demonstrate that the finite-range interactions give rise to dispersive effects, not present in the conformal case of point-like interactions. Combining the same methods with the recent proposal of generalized hydrodynamics, one finds that this model allows for fully explicit yet non-trivial solutions of the resulting Euler-scale hydrodynamic equations. These results are shown to emerge from the exact analytical ones at the relevant time and length scales. As such, the non-local Luttinger model provides a tractable example to analytically study the emergence of hydrodynamics in a quantum many-body system.