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. Tickets (2,20 € for 90 minutes, or 13 € for 3 days) can be acquired by ATM app, in tobacco shops and newspapers stands, at machines in metro stations, or tap your credit card at the onboard readers. Another convenient option is to make a bike sharing subscription with the BikeMi app for 9 € per week, unlimited free rides included (up to 30 minutes per ride).

• Tuesday: Palazzo Brera, Via Brera 28
• 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: Wednesday, 19:30 at Trattoria Bertamè, Via Francesco Lomonaco, 13b, 20131 Milano. From Hotel Palazzo delle Stelline you can take the green metro (M2) or red metro (M1), both take 25-30 minutes. 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.
Quantum Field Theory is the framework in which our modern understanding of elementary particle physics is expressed, in particular through the so called standard model of particle physics. However, a precise mathematical modelling of many aspects of the theory is still incomplete. For example, the computation of the scattering cross section implies to deal with divergences. A procedure, called renormalisation, is then required to remove them and obtain physical reasonable quantities. In this talk, we will focus on the ultraviolet divergences and try to settle a mathematically well defined procedure on a toy model quantum field theory. This talk is based on a joint work with Jacob Schach Møller.

Bahhi, Meriem (University of Bourgogne): A mathematical study of a quasi-linear Schrödinger type-equation
We explore a Quasi-linear Schrödinger-type equation that is related to the describtion of the behavior of particles within atomic nuclei. Under some assumption on the parameters of the model we establish the existence, the uniqueness, and the non-degeneracy of positive radial solutions. Moreover we analyze the behavior of these solutions according to the parameters.

Barbieri, Santiago (Universitat de Barcelona): Semi-algebraic geometry and Generic Hamiltonian stability
As it is well-known, the steepness property is a local geometric transversality condition on the gradient of $C^2$ functions which proves fundamental in order to ensure the stability over long timespans of integrable Hamiltonian systems that undergo a small perturbation. Though steepness is generic - both in measure and in topological sense - among functions of high enough regularity, the original definition of this property is not constructive and, up to very recent times, the few existing criteria to check steepness were non-generically verified and applied only to functions depending on a low number of variables. By combining Yomdin's Lemma on the analytic reparametrization of semi-algebraic sets together with non-trivial estimates on the codimension of suitable real-algebraic varieties, in this talk I will state explicit algebraic criteria for steepness which are generically verified and apply to functions depending on any number of variables. This constitutes a very important result for applications, e.g. in celestial mechanics. The criteria can be constructed recursively and are based on algebraic equalities involving the derivatives of the studied function up to any given order and external real parameters, some of which belong to compact sets and some others to non-compact sets. Moreover, it can be shown that, generically, the non-compact external parameters can be eliminated from the equalities with the help of a linear quantifier elimination algorithm: this represents a crucial improvement for numerical implementations of the criteria.

References:
1) S. Barbieri, "Semi-algebraic Geometry and generic Hamiltonian stability", preprint. https://hal.science/hal-04213250/
2) S. Barbieri, "On the algebraic properties of exponentially stable integrable hamiltonian systems", Ann. Fac. Sci. Toulouse, 31(6): 1365-1390, 2022
3) N. N. Nekhoroshev, "Stable lower estimates for smooth mappings and for gradients of smooth functions", Math USSR Sb., 19(3):425–467, 1973
4) G. Schirinzi, M. Guzzo, "On the formulation of new explicit conditions for steepness from a former result of N.N. Nekhoroshev", J. Math. Phys, 54, 2013

Belloni, Andrea (Università degli studi di Milano): Non-relativistic limit of the KAM tori for the Klein-Gordon equation
It is heuristically well known that the non-relativistic limit of the nonlinear Klein-Gordon equation is the nonlinear Schrödinger equation. Several authors have proved rigorous results ensuring that solutions of the nonlinear Klein-Gordon equation, after a Gauge transformation, converge to solutions of the nonlinear Schrödinger equation uniformly on compact intervals of time. I will present a result proving existence of quasiperiodic solutions of the nonlinear Klein-Gordon equation on the one dimensional torus uniformly as $c\to \infty$. I will also prove that, after a Gauge transformation, such solutions converge uniformly with respect to $t\in \mathbb{R}$ to solutions of the nonlinear Schrödinger equation.

Cava, Giulia (Università degli studi Roma Tre): The scaling limit of boundary spin correlations in non-integrable Ising models
We consider non-integrable planar Ising models obtained by perturbing the nearest-neighbor model via a weak, even, finite range potential and we study its critical theory in the half-plane. We prove that the scaling limit of the multipoint boundary spin correlations is the same as for the nearest-neighbor model, up to an analytic multiplicative renormalization constant. The proof is based on an exact representation of the generating function of correlations in terms of a Grassmann integral and on a multiscale analysis thereof; it generalizes the methods developed by G. Antinucci, A. Giuliani and R.L. Greenblatt for the construction of the scaling limit of the bulk energy correlations in cylindrical geometry. (Joint work with A. Giuliani and R.L. Greenblatt.)

Corsi, Livia (Università degli studi Roma Tre): Infinite dimensional invariant tori for the 1d NLS Equation
In the study of close to integrable Hamiltonian PDEs, a fundamental question is to understand the behaviour of "typical" solutions. With this in mind it is natural to study the persistence of almost-periodic solutions and infinite dimensional invariant tori, which are indeed typical in the integrable case. Up to now, almost all results in the literature deal with very regular solutions for model PDEs with external parameters giving a large modulation. In this talk I shall discuss a new result, constructing Gevrey solutions for models with a weak parameter modulation. This is a joint work with G.Gentile and M.Procesi.

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
We consider a homogeneous Bose gas in the Gross-Pitaevskii limit at temperatures that are comparable to the critical temperature for Bose-Einstein condensation in the ideal gas. Our main result is an upper bound for the grand canonical free energy in terms of two new contributions: (a) the free energy of the interacting condensate is given in terms of an effective theory describing its particle number fluctuations, (b) the free energy of the thermally excited particles equals that of a temperature-dependent Bogoliubov Hamiltonian.

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.

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.

Giuliani, Filippo (Politecnico di Milano): Growth of Sobolev norms for the cubic NLS on irrational tori
An interesting problem concerning the analysis of Hamiltonian PDEs on compact manifolds is the study of solutions exhibiting a growth in their high order Sobolev norms as time evolves. Such problem is connected to the energy cascade phenomenon of weak wave turbulence theory, i.e. the transfer of energy among "high" and "low" Fourier modes of weakly nonlinear waves. In 2010 Colliander-Keel-Staffilani-Takaoka-Tao ( CKSTT) provided a breaktrough result on the existence of solutions of the defocusing cubic NLS, on the 2-d torus, undergoing arbitrarily large growth of Sobolev norms. Later, Staffilani and Wilson performed a quantitative analysis of the spread of energy from low to high modes of solutions of the cubic NLS on 2-d irrational tori. They observed that in such case the CKSTT strategy does not apply directly because of the lack of resonances. For this reason, it is also believed that the escape of energy should be weaker in the irrational setting. In this talk we show how to construct solutions of the cubic NLS on 2-d irrational tori undergoing a Sobolev norm explosion. We also show that the upper bound on the time at which the growth is achieved is the same in the rational and irrational setting.

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, Raphaël (Université de Cergy-Pontoise): Divergence and convergence of Birkhoff Normal Forms
Each real analytic symplectic diffeomorphism admitting a non resonant fixed point can be formally conjugated to a formal integrable symplectic diffeomorphism, its Birkhoff Normal Form (BNF). I shall discuss two questions. The first one is due to H. Eliasson: are there examples of divergent BNF? The second is the following: is it possible to perturb in the real analytic topology a given real analytic symplectic diffeomorphism so that its BNF converges?

Lucia, Angelo (Universidad Complutense de Madrid): Spectral gap of decorated AKLT models
The AKLT model is a SU(2)-invariant quantum spin model which has played a major role in the development of connections between quantum information and quantum many body theory. A major open problem is to understand for which lattices the model has a spectral gap, a non-vanishing difference between the first excited energy level of the Hamiltonian and the ground state energy. In this talk, I will consider the family of AKLT models defined on decorated versions of simple, connected graphs. The decoration consists of replacing each edge with a chain of n sites, for a fixed n called the decoration parameter. I will show that, when the decoration parameter is larger than a linear function of the maximal degree of the graph, the model has a nonvanishing spectral gap above the ground state energy. (Joint work with A. Young)

Mastropietro, Vieri (Università degli Studi di Milano): Lattice QFT, Anomalies and RG
The Standard Model is expected to exist only as an effective theory with a finite uv cut-off. It is a mathematical challenging problem to construct it up to cut-off much higher than the experiments scales. Such construction is rigorously achieved for cut-offs of the order of the inverse coupling in the Weak sector, where the theory reduces to the Fermi theory. To get more realistic exponentially high cut-offs one needs to face many problems, among which is the issue of the non-renormalization and cancellation of anomalies with a lattice. We present some theorems proving such properties by RG methods in lattice models in $d=4$ up to cut-off of the order of the inverse coupling and in $d=2$ in the continuum limit. Open problems and directions will be discussed.

Moscolari, Massimo (Politecnico di Milano): From decay of correlations to locality and stability of Gibbs states
I will show that whenever a Gibbs state satisfies decay of correlations, then it is stable, in the sense that local perturbations influence the Gibbs state only locally, and it is local, namely it satisfies local indistinguishability. These implications hold true in any dimensions, only require locality of the Hamiltonian and rely on Lieb-Robinson bounds. A central role in the proofs is played by the quantum belief propagation for Gibbs states, which I will briefly review. Furthermore, I will discuss how our results can be applied to quantum spin systems in any dimension with short-range interactions at high enough temperature, and to one-dimensional quantum spin chains with translation-invariant and exponentially decaying interactions above a threshold temperature that goes to zero in the limit of finite range interactions. The talk is based on a joint work with Ángela Capel, Stefan Teufel and Tom Wessel.

Noja, Diego (Università degli studi di Milano-Biccoca): On some asymptotic properties of NLS equation with a point interaction
I will discuss some asymptotic properties of the solution of the time dependent NLS equation with a point interaction in dimension two and three. In particular, it will be shown that blow-up of the solution develops in dimension two and for supercritical nonlinearity, and that there is absence of scattering in both dimension two and three for low ("long-range") nonlinearities.

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 (UPC, Barcelona): Computation of the asymptotic wavenumber of spiral waves of the Ginzburg Landau equation
It is well known that Archimedian spiral wave solutions of the complex Ginzburg-Landau equation exist when the asymptotic wavenumber K is determined by the twist parameter q. Since the eighties, different heuristic perturbation techniques, like formal asymptotic expansions, have conjectured an asymptotic expression of K(q). In this work, using a functional analysis approach, we prove the validity of the asymptotic formula for $K(q)$, providing a rigorous bound for its relative error.

Zhang, Ke (TBA): A bumpy metric theorem for co-rank-1 sub-Riemannian classical systems
A sub-Riemannian Hamiltonian system is a convex Hamiltonian that is flat on a rank-m subbundle of the cotangent bundle. They are the Legendre dual of the sub-Riemannian Lagrangians, which are only defined on a co-rank-m distribution of the tangent bundle. A special feature for sub-Riemannian structures is the existence of singular curves, which are (in a sense) singular points in the space of admissible curves.Consider the analog of classical mechanical systems, which is the sum of a potential energy and a sub-Riemannian kinetic energy of degree 2. We show that in the co-rank-1 case, generically all periodic are non-degenerate provided that they don't project to singular curves. This is a sub-Riemannian analog of the bumpy metric theorem. We also obtain the analogous result for sub-Riemannian and sub-Finsler metrics as corollaries. This is a joint work with Shahriar Aslani.

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.