24/11/2022: Luca Fresta @ UniMi

November 24, 2022, 15:45
Sala di Rappresentanza
Mathematics Department, University of Milan
Via Cesare Saldini 50, Milano, Italy

SPEAKER: Luca Fresta (Universität Bonn)

Stochastic Analysis of Subcritical Euclidean Fermionic Field Theories

In my talk, I will introduce a forward-backward stochastic differential equation
which provides a stochastic quantisation of subcritical Grassmann measures. The method is inspired by the so-called continuous renormalisation group, but
avoids the technical difficulties encountered in the direct study of the Polchinski’s flow equation for the effective potentials. If time permits, I will also show how to prove the exponential decay of correlations by a coupling method.
Work in collaboration with De Vecchi and Gubinelli.

21/11/2022: Nikolai Leopold @ UniMi

November 21, 2022, 11:15
Sala di Rappresentanza
Mathematics Department, University of Milan
Via Cesare Saldini 50, Milano, Italy

SPEAKER: Nikolai Leopold (Universität Basel)

Norm approximations for the Fröhlich dynamics

In this talk I will discuss recent results about the time evolution of the Fröhlich Hamiltonian in a mean-field limit in which many particles weakly couple to the quantized phonon field. For large particle number and initial data in which the particles are in a Bose-Einstein condensate and the excitations of the phonon field are in a coherent state I will show that the time evolved many-body state can be approximated in norm by an effective dynamics. The approximation is given by a product state which evolves according to the Landau–Pekar equations and which is corrected by a Bogoliubov dynamics.
If time permits I will, in addition, present a joint work with D. Mitrouskas, S. Rademacher, B. Schlein and R. Seiringer about the Fröhlich model in the strong coupling limit and compare the Bogoliubov dynamics in the strong coupling and mean-field regime.

12/12/2022: Vojkan Jakšić @ UniMi

December 12, 2022, 11:15
- Aula Dottorato, 1st floor -
Mathematics Department, University of Milan
Via Cesare Saldini 50, Milano, Italy

SPEAKER: Vojkan Jakšić (McGill University Montreal)

Some remarks on adiabatic time evolution and quasi-static processes in translation-invariant quantum systems

This talks concerns slowly varying, non-autonomous quantum dynamics of a translation invariant spin or fermion system on the lattice Z^d . This system is assumed to be initially in thermal equilibrium, and we consider realizations of quasi-static processes in the adiabatic limit. By combining the Gibbs variational principle with the notion of quantum weak Gibbs states, we will discuss a number of general structural results regarding such realizations.

This talk is based on a joint work with C-A Pillet and C. Tauber.

30/5/2022: Laurent Lafleche @ UniMi

May 30, 2022, 16:30 (non-standard time)
Sala di Rappresentanza
Mathematics Department, University of Milan
Via Cesare Saldini 50, Milano, Italy

SPEAKER: Laurent Lafleche (U Texas at Austin)

Semiclassical regularity and mean-field limit
with singular potentials

In this talk I will present several techniques and concepts used in the context of the mean-field and the classical limit allowing to go from the N -body Schrödinger equation with singular potential to the Hartree–Fock and Vlasov equations, linked to works in collaboration with Chiara Saffirio and Jacky Chong. At the level of the Vlasov–Poisson equation, typical mean-field techniques from quantum mechanics for pure states can be translated to a weak-strong stability estimate in L^1 for the Vlasov equation. Another weak-strong stability can be obtained for the difference of the square roots of the solutions in L^2. They allow to better understand the mean-field and semiclassical estimates. These estimates are weak-strong in the sense that they require only the regularity of one of the solutions. This requires the propagation of a semi-classical notion of regularity uniformly in N and h. A typical obstacle is the lack of positivity of the Wigner transform and its few conserved quantities. A solution to this problem is to consider operators as the right generalization of the phase space distribution, and a quantum analogue of Sobolev spaces defined using Schatten norms. The advantage of these techniques is that they allow to obtain regularity estimates without higher order error terms.

16/5/2022: Robin Reuvers @ UniMi

May 16, 2022, 16:30 (non-standard time)
Sala di Rappresentanza
Mathematics Department, University of Milan
Via Cesare Saldini 50, Milano, Italy

SPEAKER: Robin Reuvers (U Roma Tre)

Ground state energy of dilute Bose gases in 1D

In 1963, Lieb and Liniger formulated an exactly solvable model for interacting bosons in 1D. Thanks to its exact, Bethe ansatz solution, the model and its generalizations soon became popular objects of study in mathematical physics. Later, when new techniques allowed for the creation of (quasi-)1D systems in the lab, the Lieb-Liniger model found experimental use and became even better known.

In the meantime, Lieb and collaborators had moved on, and were rigorously studying interacting bosons in 2 and 3D. Without the availability of exact solutions, rigorous results were much more difficult to acquire, and a popular goal was the rigorous derivation of the ground state energy of gases of bosons in various settings in 2 and 3D. Many of the results focused on the dilute limit, in which the density of the boson gas is very low.

Somehow, Bose gases in 1D were excluded from this development. Of course, the original Lieb-Liniger model provided a solvable example, but we can nevertheless use insights from the 2 and 3D approaches to prove new results about the ground state energy of dilute Bose gases in 1D.

In the talk, I will review the developments above, and explain the new results.

21/2/2022: Ian Jauslin @ Zoom

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.

7/2/2022: Ngoc Nhi Nguyen @ UniMi

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.

20-22/12/2021: Quantum Before Christmas @ UniMi

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.

8/11/2021: Per Moosavi @ UniMi

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.