Quantum and Dynamical 2023 Christmas

Workshop in Milan, Italy

December 19-22, 2023 (Tuesday to Friday)

The workshop is composed by two parts, one mainly devoted to dynamical systems that will occupy the days 19 and 20 December and one on quantum systems and many-body problems that will occupy the days 21 and 22 December. The main focus of the part on dynamical systems is on the stability/instability properties of finite and infinite dimensional Hamiltonian systems, but we are more generally interested in discussing recent advances in the theory of dynamical systems and in applications to some important problems coming from applications. The part on quantum systems will primarily focus on interacting Fermi and Bose gases, effective theories, and emergent phenomena and phase transitions. We hope to welcome colleagues from both fields over the whole workshop and look forward to generating synergies and new collaborations across the fields.

Program

Venue

The conference is spread over three different venues! The first two are in the center of Milan, the last one in the Città Studi quarter. From the center, Città Studi is best reached by the green metro to the stop "Piola" or by Tram 19.

• Tuesday: Palazzo Brera, Via Brera 26
• Wednesday and Thursday: Sala Napoleonica, Università degli Studi di Milano, Via Sant'Antonio 12
• Friday: Sala Rappresentanza, Dipartimento di Matematica, Via Cesare Saldini 50

Most international and national rail tickets to Milano can be bought on Trainline or check Seat61.com.

Social dinner: Location TBA. The dinner has to be payed individually by each participant, but for speakers the expense can be added to the reimbursement after the conference if a fiscal receipt is presented.

Hotel: For all speakers we booked rooms at Hotel Palazzo delle Stelline according to the dates indicated on the registration form.

Abstracts

Adami, Riccardo (Politecnico di Torino): Ground states for the Nonlinear Schrödinger Equation on hybrids
Motivated by the recent tecnological development on ultracold gases and atomtronic, the study of the Nonlinear Schrödinger Equation on exotic domains is nowadays a well-established topic in Mathematical Physics. Here we focus on the problem of finding the Ground States in the so-called hybrid plane, made of a plane attached to the origin of a halfline. By Ground States we mean minimizers of the NLS energy among the states with the same mass. We show that Ground States exist for small and for large mass, while in the intermediate region there are intervals of nonexistence. This is a joint project with Filippo Boni, Raffaele Carlone, and Lorenzo Tentarelli.

Alvarez, Benjamin (Centre de physique théorique Toulon): Ultraviolet renormalisation for a quantum field toy model.
TBA

Bahhi, Meriem (University of Bourgogne): Title TBA
TBA

Barbieri, Santiago (TBA): Title TBA
TBA

Cava, Giulia (Università degli studi Roma Tre): The scaling limit of boundary spin correlations in non-integrable Ising models
TBA

Corsi, Livia (Università degli studi Roma Tre): Title TBA
TBA

De Palma, Giacomo (University of Bologna): The quantum Wasserstein distance of order 1
We propose a generalization of the Wasserstein distance of order 1 to the quantum states of n qudits. The proposal recovers the Hamming distance for the vectors of the canonical basis and more generally the classical Wasserstein distance for quantum states diagonal in the canonical basis. We prove a continuity bound for the von Neumann entropy with respect to the proposed distance, which significantly strengthens the best continuity bound with respect to the trace distance. We also propose a generalization of the Lipschitz constant to quantum observables. The notion of quantum Lipschitz constant allows us to compute the proposed distance with a semidefinite program. We prove a Gaussian concentration inequality for the spectrum of quantum Lipschitz observables and a quadratic concentration inequality for quantum Lipschitz observables measured on product states. We generalize the proposed quantum Wasserstein distance of order 1 to quantum spin systems on the lattice $\mathbb{Z}^d$. Finally, we prove that local quantum commuting interactions above a critical temperature satisfy a transportation-cost inequality, which implies the uniqueness of their Gibbs states.

De Suzzoni, Anne Sophie (Ecole Polytechnique): Stability of thermodynamic equilibria for the Hartree-Fock equation with exchange term
In the Hartree-Fock equation, which models the evolution of a particle system under symmetry assumptions, the energy exchange term between particles is often neglected in favor of the so-called mean field term. Indeed, under certain structural assumptions about the interaction between particles, these two terms corroborate each other, as is the case for point interaction potentials, i.e., when the Hartree-Fock equation reduces to the Schrödinger equation. On the other hand, the distinction between these two terms has no impact on the locally well-posedness of the equation. Nevertheless, for the global problem, and especially for the problem of asymptotic stability of non-localized equilibria of the equation, the two terms play a very different role and modify the analysis of the linearized equation around the equilibrium under consideration. This talk will present the Hartree-Fock equation with exchange term, its heuristic derivation and its associated equilibria. Finally, a result on the asymptotic stability of thermodynamic equilibria will be presented. This is a collaborative result with Charles Collot (CYU), Elena Danesi (Padova) and Cyril Malézé (Ecole Polytechnique).

Deleporte, Alix (University Paris-Saclay): Semiclassical analysis of free fermions
To each orthogonal projector of finite rank N on $L^2(R^d)$ is associated a point process on $R^d$ with N points, which gives the joint probability density of fermions that fill the image of the projector. The study of the statistical properties of these fermions, in the large N limit, is linked to semiclassical spectral theory problems, some of them well studied (the Weyl law gives a law of large numbers), some of them new. In particular, the behaviour of the variance is linked with the properties of commutators involving spectral projectors, which are not so well understood. In this talk, I will present my work in collaboration with Gaultier Lambert (KTH) on this topic.

Deuchert, Andreas (University of Zürich): Upper bound for the grand canonical free energy of the Bose gas in the Gross–Pitaevskii limit
TBA

Diaz, Jaime Paradela (University of Maryland): Arnold diffusion the Restricted 3 Body Problem
A major challenge in dynamical systems is to understand the mechanisms driving global instability in the 3 Body Problem (3BP), which models the motion of three bodies under Newtonian gravitational interaction. The 3BP is called restricted if one of the bodies has zero mass and the other two, the primaries, have strictly positive masses $m_0$, $m_1$. In the region of the phase space where the massless body is far from the primaries, the problem can be studied as a (fast) periodic perturbation of the 2 Body Problem (2BP), which is integrable. We prove that the restricted 3BP exhibits topological instability: for any values of the masses $m_0$, $m_1$ (except $m_0 = m_1$), we build orbits along which the angular momentum of the massless body (conserved along the flow of the 2BP) experiences an arbitrarily large variation. In order to prove this result we show that a degenerate Arnold diffusion mechanism takes place in the restricted 3BP. Our work extends previous results by Delshams, Kaloshin, De la Rosa and Seara for the a priori unstable case $m_1 \ll m_0$, to the case of arbitrary masses $m_0, m_1 > 0$, where the model displays features of the so-called a priori stable setting. This is joint work with Marcel Guardia and Tere Seara.

Fayad, Bassam (TBA): Title TBA
TBA

Florio, Anna (Ceremade - Université Paris Dauphine): Birkhoff attractor of dissipative billiards
In a joint work with Olga Bernardi and Martin Leguil, we study the dynamics of dissipative convex billiards. In these billiards, the usual elastic reflection law is replaced with a new law where the angle bends towards the normal after each collision. For such billiard dynamics there exists a global attractor; we are interested in the topological and dynamical complexity of an invariant subset of this attractor, the so-called Birkhoff attractor, whose study goes back to Birkhoff, Charpentier, and, more recently, Le Calvez. We show that for a generic convex table, on one hand, the Birkhoff attractor is simple, i.e., a normally contracted submanifold, when the dissipation is strong; while, on the other hand, the Birkhoff attractor is topologically complicated and presents a rich dynamics when the dissipation is mild.

Grébert, Bénoit (Université de Nantes): Discrete pseudo-differential operators and applications to numerical schemes
We consider a class of discrete operators introduced by O. Chodosh, acting on infinite sequences and mimicking the standard properties of pseudo-differential operators. By using a new approach, we extend this class to finite or periodic sequences, allowing a general representation of discrete pseudo-differential operators obtained by finite differences approximations and easily transfered to time discretizations. In particular we can define the notion of order and regularity, and we recover the fundamental property, well known in pseudo-differential calculus, that the commutator of two discrete operators gains one order of regularity. As examples of practical applications, we revisit standard error estimates for the convergence of splitting methods, obtaining in some Hamiltonian cases no loss of derivative in the error estimates, in particular for discretizations of general waves and/or water-waves equations. Moreover, we give an example of preconditioner constructions inspired by normal form analysis to deal with the similar question for more general cases. (In collaboration with E. Faou)

Krikorian, Raphael (TBA): Title TBA
TBA

Lucia, Angelo (Universidad Complutense de Madrid): Spectral gap of decorated AKLT models
TBA

Massetti, Jessica (TBA): Title TBA
TBA

Moscolari, Massimo (Politecnico di Milano): From decay of correlations to locality and stability of Gibbs states
TBA

Noja, Diego (Università degli studi di Milano-Biccoca): Title TBA
TBA

Paradela, Jaime (TBA): Title TBA
TBA

Roos, Barbara (University of Tübingen): Boundary Superconductivity in BCS Theory
I will discuss the influence of boundaries on the critical temperature of superconductors in Bardeen-Cooper-Schrieffer (BCS) theory. First, I will explain the linear criterion commonly used to study the critical temperature. Second, I will present recent results for superconductors in dimensions $d=1,2,3$ showing that the critical temperature on half-spaces is strictly higher than on $\mathbb{R}^d$ and even higher on a quadrant, at least at weak coupling.

Seara, Tere (TBA): Title TBA
TBA

Zang, Ke (TBA): Title TBA
TBA

Registration

Speakers have been invited directly. If you would like to participate in the workshop, please write an email to the organizers so that we can estimate the number of participants (see contact information below).

Speakers should keep all their receipts (train tickets, boarding passes, fiscal receipts of meals) for reimbursement after the conference. The hotel for speakers has been already booked and payed by us.

Contact

For questions, please write an email to your favorite organizer.

Organizing committee:
Dario Bambusi
Niels Benedikter
Chiara Boccato
Beatrice Langella
Sascha Lill
Ngoc Nhi Nguyen
Laurent Niederman
Simone Paleari
Shulamit Terracina

We gratefully acknowledge support by Università degli Studi di Milano, Istituto Lombardo Accademia di Scienze e Lettere, the MIUR PRIN project 2020XB3EFL, and the European Union through the ERC StG FermiMath nr. 101040991.