Séminaire: Problèmes Spectraux en Physique Mathématique

(ex-"séminaire tournant")

Prochain séminaire

Séminaires de l'année 2021-2022

Lundi 11 octobre 2021

14h - 15h Alexei Iantchenko (Malmö University)
Inverse problems in Seismology with spectral and resonance data

Semiclassical analysis can be employed to describe surface waves in an elastic half space which is quasi-stratified near its boundary. The propagation of such waves is governed by effective Hamiltonians on the boundary with a space-adiabatic behavior. Effective Hamiltonians of surface waves correspond to eigenvalues of ordinary differential operators, which, to leading order, define their phase velocities. In case of isotropic medium, the surface wave decouples up to principal parts, into Love and Rayleigh waves.

We present the conditional recovery of the Lamé parameters from spectral data, in two inverse problems approaches:
- semiclassical techniques using the semiclassical spectra as the data;
- exact methods for Sturm-Liouville operators, using the discrete and continuous spectra, or the Weyl function, as the data based on the solution of the Gel’fand-Levitan-Marchenko equation.

We conclude with discussion using resonances (leaking modes) as data.

 15h15 - 16h15 Bernard Helffer (Univ. de Nantes)
Spectral Theory for the Bloch-Torrey operator

We consider the semi-classical Bloch-Torrey operator in L2(Ω,Rk), where Ω is a domain in Rd and k=1,2,3.
If the simplest model is some closed realization of -h2 Δ + i x1, the original model of Bloch-Torrey corresponds to k=d=3 and reads -Δ⊗ I3 + b(x)×, where b(x) is the magnetic field.

After some general discussion of the qualitative spectral analysis of these models, our aim is to analyze, for various domains and in the limit h → 0, the left margin of the spectrum and to establish resolvent estimates.

The presented results have been obtained in collaboration with Y.Almog, D.S.Grebenkov and N.Moutal, in continuation of earlier works by Y.Almog and R.Henry devoted to the semi-classical analysis of
-h2Δ + i V(x), where V is a C potential.

Lundi 15 novembre 2021

14h - 15h Steven Flynn (Univ. of Bath)
Unraveling the Heisenberg X-ray transform

The classical X-ray Transform maps a function on Euclidean space to a function on the space of lines on this Euclidean space ,by integrating the function over the given line. Inverting the X-ray transform has wide-ranging applications, including to medical imaging and seismology. Much work has been done to understand this inverse problem in Euclidean space, Euclidean domains, and more generally, for symmetric spaces and
Riemannian manifolds with boundary, where the lines become geodesics. We formulate a sub-Riemannian version of the X-ray transform on the simplest sub-Riemannnian manifold, the Heisenberg group. Here serious geometric obstructions to classical inverse problems, such as existence of conjugate points, appear generically. With tools adapted to the geometry, such as an operator-valued Fourier Slice Theorem, we prove that an integrable function on the Heisenberg group is indeed determined by its line integrals over
sub-Riemannian (as well as over its compatible Riemannian and Lorentzian) geodesics.
We also pose an abundance of accessible follow-up questions, standard in the inverse problems community, concerning the sub-Riemannian case, and report progress answering some of them.
 15h15 - 16h15 Ayman Kachmar (Univ. libanaise)
Magnetic steps and semiclassical eigenvalue asymptotics

This talk adresses the Dirichlet magnetic Laplacian with a non-uniform magnetic field having a jump discontinuity along a smooth curve, known as Iwatsuka fields or magnetic steps. Under a geometric assumption on the curvature of the discontinuity curve, accurate eigenvalue asymptotics, displaying  the splitting between the consecutive eigenvalues, can be derived in the semiclassical limit (or equivalently,  the strong magnetic field limit). The proof is purely variational and does not involve pseudo-differential operators.
This is a joint work with W. Assaad and B. Helffer.

Lundi 13 décembre 2021

14h - 15h Caterina Vâlcu (Paris-Nord)
Solving initial data for Kaluza-Klein spacetimes

We study the constraint equations for Einstein equations on manifolds of the form Rn+1×Tm, where Tm is a flat m-dimensional torus. Spacetimes with compact directions were introduced almost a century ago by Theodor Kaluza and Oskar Klein as an early attempt of unifying electromagnetism and general relativity in a simple, elegant way. The aim of this article is to construct initial data for the Einstein equations on manifolds of the form Rn+1× Tm, which are asymptotically flat at infinity, without assuming any symmetry condition in the compact direction. We use the conformal method to reduce the constraint equations to a system of elliptic equation and work in the near CMC (constant mean curvature) regime. The main new feature of the proof is the introduction of new weighted Sobolev spaces, adapted to the inversion of the Laplacean on product manifolds. Classical linear elliptic results need to rigorously proved in this new setting. This is joint work with Cécile Huneau.
 15h15 - 16h15 Phan Thành Nam (LMU Munich)
Semiclassical approximation and Lieb-Thirring inequalities

In 1975, Lieb and Thirring proved that the sum of negative eigenvalues of a Schrödinger operator can be controlled by a phase-space expression, thus establishing a uniform Weyl's law up to a constant factor. In its dual form, their result extends Sobolev’s inequality to orthogonal functions and gives an important tool to understand large quantum systems. I will discuss some recent generalizations of the Lieb-Thirring inequality and also mention several open problems.

Lundi 17 janvier 2022

14h - 15h Alix Deleporte (Orsay)
Processus ponctuels déterminantaux et projecteurs spectraux pour Schrödinger

Les processus ponctuels déterminantaux (DPP) sont une famille de modèles probabilistes qui représentent les propriétés statistiques des fermions libres. Leur étude est également motivée par d'autres modèles naturels, comme les matrices aléatoires ou les représentations aléatoires des groupes finis.

À chaque (suite de ) projecteurs localement de rang fini, on peut associer une (suite de) DPP. Ceci nous motive à étudier la limite semiclassique de projecteurs spectraux naturels.

Dans cet exposé, je parlerai de résultats obtenus récemment avec G. Lambert (UZH) sur l'asymptotique des projecteurs spectraux pour les opérateurs de Schrödinger en limite semiclassique et les applications à la convergence des DPP à différentes échelles.

 15h15 - 16h15 Andrea Mantile (Reims) On the origin of the Minnaert resonances

It is well known that the presence, in a homogeneous acoustic medium, of a small inhomogeneity (of size ε), enjoying a high contrast of both its mass density and bulk modulus, amplifies the generated total fields. This amplification is more pronounced when the incident frequency is close to the Minnaert frequency ωM.
Here we explain the origin of such a phenomenon: at first we show that the scattering of an incident wave of frequency ω is described by a self-adjoint ω-dependent Schrödinger operator with a singular δ-like potential supported at the inhomogeneity interface. Then we show that, in the low energy regime (corresponding in our setting to ε<<1) such an operator has a nontrivial limit (i.e., it asymptotically differs from the Laplacian) if and only if ω = ωM.
The limit operator describing the nontrivial scattering process is explicitly determined and belongs to the class of point perturbations of the Laplacian. When the frequency of the incident wave approaches ωM, the scattering process undergoes a transition between an asymptotically trivial behaviour and a nontrivial one. As a byproduct, we get the existence of a local maximum of the scattering amplitude occurring at the Minnaert frequency. It is this last property that is experimentally observed and used in the applications.
In collaboration with: A. Posilicano and M. Sini.

Lundi 7 mars 2022

14h - 15h Louis Garrigue (ENPC)
The inverse problem of Density Functional Theory

The map from electric potentials to the one-body density of ground states is important in the quantum many-body theory, because of its injectivity. Considering Density Functional Theory as an inverse problem, we will present the mathematical specificities of its dual formulation, and show how to construct a potential producing some target one-body density, in ground and excited states cases.
 15h15 - 16h15 Louis Gass (Rennes) L'approche Salem-Zygmund pour prouver la convergence presque-sûre du volume nodal aléatoire.

Soit (M,g) une variété riemannienne compacte sans bord, et λn, φn la suite ordonnée des valeurs propres et fonctions propres du Laplacien, qui satisfont l'égalité Δ φn = -λn2 φn. On considère le modèle des "Riemannian random waves" défini par fλ = ∑ λn≤λ an φn(x), où (an)n est une suite idd de variables aléatoires gaussiennes. Presque sûrement sous les coefficients gaussiens, on prouve que le processus fλ, évalué en un point aléatoire X qui suit une loi uniforme sur la variété, converge localement en loi vers un champ gaussien universel, lorsque λ tend vers l'infini. Ce résultat est réminiscent de la conjecture de Berry que nous détaillerons. En utilisant la continuité du volume nodal en topologie C1 nous en déduiront que presque sûrement sous les coefficients gaussiens, la mesure nodale converge vers la mesure Riemannienne.

Lundi 11 avril 2022

14h - 15h Mokdad Mokdad (Dijon)
Scattering of test fields in the interior of black holes

Scattering theories are of great importance for many problems in General Relativity and Quantum physics, and scattering in the interior of a black hole is particularly relevant in the context of the cosmic censorship conjecture and the related Cauchy horizon instability problem. This instability is thought to be directly linked to a notion of gravitational blue-shift at the horizon, which manifest itself as a blow-up in some observed quantity.
By constructing the scattering channels, one aims to recover information about the behavior of the field near the horizons, where the instabilities might be seen as the unboundedness of the scattering operators (energy blow-up) or from the lack of regularity at the horizon of the propagating field (C1-blow-up). In this talk I will present the recent development of the scattering of waves and Dirac fields in the interior of spherically symmetric black holes between the Cauchy horizon and the event horizon. From a mathematical point of view, these two fields exhibit contrasting scattering phenomena. The scattering theory for Dirac fields is robust thanks to the conserved Dirac current and the well posedness of the characteristic Cauchy problem, even though some care must be exercised when the ambient charge is taken into account. On the other hand, the scattering of linear waves is very delicate and interesting and surprising breakdowns of scattering happens in generic situations. The physical implications and causes for these breakdowns are not fully understood yet, and some of them may be related to possibly new phenomena.

 15h15 - 16h15 Nils Berglund (Orléans) An Eyring-Kramers law for slowly oscillating bistable diffusions

We consider two-dimensional stochastic differential equations, describing the motion of a slowly and periodically forced overdamped particle in a double-well potential, subjected to weak additive noise. We give sharp asymptotics of Eyring-Kramers type for the expected transition time from one potential well to the other one. It is expected that the inverse of this mean transition time is close to the spectral gap of the system's infinitesimal generator, given by a non-self-adjoint second-order differential operator.
The main difficulty of the analysis is that the forced system is non-reversible, so that standard methods from potential theory used to obtain Eyring-Kramers laws for reversible diffusions do not apply. Instead, we use results by Landim, Mariani and Seo that extend the potential-theoretic approach to non-reversible systems.

Nils Berglund, An Eyring-Kramers law for slowly oscillating bistable
diffusions, Probability and Mathematical Physics 2-4:685-743 (2021)
(hal-02901432, arXiv/2007.08443)

Lundi 9 mai 2022

14h - 15h Alessandro Olgiati (University of Zurich)
Bosons in a double well: two-mode approximation and fluctuations

I will discuss the ground state properties of a system of bosonic particles trapped by a double-well potential, in a joint limit of large inter-well separation and high potential barrier. The bosons mutually interact via a two-body potential in the mean-field regime. The leading-order physics of the model is governed by a Bose-Hubbard Hamiltonian coupling two low-energy modes, each supported in the bottom of one well. Fluctuations beyond these two modes are ruled by two independent Bogoliubov Hamiltonians, one for each well. Our main result is that, when the system is in the ground state, the variance of the number of particles occupying the low-energy modes is suppressed. This is a violation of the central limit theorem that holds in the occurrence of Bose-Einstein condensation, and therefore a signature of the emergence of strong correlations in the ground state.
We achieve the result by proving a precise ground state energy expansion in terms of Bose-Hubbard and Bogoliubov energies.
Joint work with Nicolas Rougerie and Dominique Spehner.

 15h15 - 16h15 Cécilia Lancien (Grenoble) Spectrum of random quantum channels 

The main question that we will investigate in this talk is the following: what does the spectrum of a quantum channel typically look like? We will see that various natural models of random quantum channels generically exhibit a large spectral gap, between their first and second largest eigenvalues. This is in tight analogy with what is observed for the spectral gap of transition matrices associated to random graphs. In both the classical and the quantum settings, establishing results of this kind is interesting as it has important consequences regarding the speed of convergence to equilibrium of the corresponding dynamics. We will also present implications of the quantum result in terms of typical decay of correlations in so-called matrix product states (which are used to describe the states of 1D many-body quantum systems with local interactions, appearing for instance in quantum condensed matter physics).

Lundi 13 juin 2022

14h - 15h Clotilde Fermanian Kammerer (Paris-Est Créteil)
Quelques résultats sur la dynamique d'un électron dans un cristal

Dans cet exposé je discuterai un résultat obtenu en collaboration avec Fabricio Macia (Politecnico Madrid) et Victor Chabu (Sao Paulo). Nous avons étudié une équation de Schrödinger modélisant la dynamique d'un électron dans un régime asymptotique où la longueur d'onde est comparable à la taille des cellules du cristal. Nous avons obtenu une description du comportement asymptotique de la densité d'énergie moyennée en temps sous des hypothèses assez souples pour permettre des croisements entre les modes de Bloch impliqués par une réduction du problème de type Bloch-Floquet.

 15h15 - 16h15 Amirali Hannani (Paris-Dauphine) Hydrodynamic limit for a disordered quantum harmonic chain

I will talk about a one-dimensional unpinned disordered chain of quantum harmonic oscillators: a hydrodynamic limit in the hyperbolic scaling of time and space is proven; distribution of the elongation, momentum and energy converges to the solution of the Euler equation in this scaling. Two physical phenomena are behind this proof: Anderson localization decouples the mechanical and thermal energy, providing the closure of the macroscopic equation out of thermal equilibrium, and indicating that the temperature profile does not evolve in time. Macroscopic evolution of the mechanical energy is a result of divergence of the localization length at the bottom of the spectrum. 
Decay of correlation-type phenomena facilitates dealing with the quantum nature of the system. To the best of our knowledge, this is among the first examples where one can prove the hydrodynamic limit for a quantum system rigorously. We also strengthen the above convergence in the sense of ”higher moments” in recent joint work with Francois Huveneers.

Dernière mise à jour: 27 mai 2022
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