Institut de Recherche Mathématiques de Rennes
My research lies at the crossroads of analysis and geometry, mainly because I like to consider the phase space geometry and its relationships with Quantum mechanics. While I’m interested in all aspects of semiclassical analysis, I’ve been specializing in integrable systems, classical or quantum, and their singularities. This is a nice door to differential geometry and normal forms. It gives rise to interesting inverse problems: how can you recover symplectic or Hamiltonian invariants from quantum spectra?
A quite important class of integrable systems are obtained by two degree of freedom systems with an $S^1$ symmetry. Under additional non-degeneracy conditions, I called them semitoric systems, because they share some rigidity aspects of toric systems and some nice flexibility from general Hamiltonian systems. In the last ten years, they have been studied by various teams and continue to reveal intriguing properties, both at classical and quantum levels.
Both symplectic and semiclassical ideas turn out to be relevant for intense magnetic fields; the general ideas from semiclassical analysis and (near) integrable systems apply remarkably and give accurate descriptions of quantum dynamics and eigenvalues of magnetic laplacians. While Magnetic hamiltonians do have an (approximate) $S^1$ symmetry, the relationship with semitoric systems remains to be elucidated!
I’m also exploring other applications of symplectic and semiclassical techniques: quantum propagation, tunnel effet, holography, …
Here are a few keywords that tend to show up again and again in my works.
microlocal analysis, spectral theory, integrable systems, semitoric systems, inverse spectral theory for integrable systems, symplectic geometry, mathematical physics, Hamiltonian dynamics, Birkhoff normal forms, Morse theory, pseudodifferential operators, Berezin-Toeplitz operators, classical and quantum magnetic fields, analytic symbols, Fourier integral operators, holography
Thank you so much to all of you.
While a mathematician seemingly does not need much to get working, we all recognize the importance of the support from our research institutes, and most importantly the invitation programs that allow us to travel around the world and meet our collaborators in person. Well, this was before covid-era. But I still find it the most efficient way to advance my research. For this reason I warmly thank the following institutions for their generous invitations.
Please contact me if I was invited to work at your institute and it's not listed below!
Here are some works from the year 2022 on. You can access the complete list of publications here.
This article tackles the spectral analysis of the Robin Laplacian on a smooth bounded two-dimensional domain in the presence of a constant magnetic field. In the semiclassical limit, a uniform description of the spectrum located between the Landau levels is obtained. The corresponding eigenfunctions, called edge states, are exponentially localized near the boundary. By means of a microlocal dimensional reduction, our unifying approach allows on the one hand to derive a very precise Weyl law and a proof of quantum magnetic oscillations for excited states, and on the other hand to refine simultaneously old results about the low-lying eigenvalues in the Robin case and recent ones about edge states in the Dirichlet case.
Consider the pair $(H, \omega)$ consisting of a Hamiltonian and a symplectic structure on $\mathbb R^2$ such that $H$ has an $A_{k-1}$ singularity at the origin with $k\geq 2$. We give a symplectic classification of such pairs up to symplectic transformations (in the $C^\infty$-smooth and real-analytic categories) in two cases: a) the transformations preserve the fibration and b) the transformations preserve the Hamiltonian of the system. The classification is obtained by bringing the pair $(H, \omega)$ to a symplectic normal form $(H = \xi^2 \pm x^k, \omega = \mathrm{d} (f \mathrm{d} \xi)), \ f = \sum_{i=1}^{k-1} x^i f_i(x^k),$ modulo some relations which are explicitly given. For example, in the case of the quartic oscillator $H = \xi^2 + x^{4}$, the germs of $f_{1}$ and $f_3$ and the Taylor series of $\pm f_{2}$ at the origin classify germs of symplectic structures up to $H$-preserving $C^\infty$-diffeomorphisms. In the case when $H = \xi^2 - x^{4}$, the Taylor series of $f_{1}, f_3$ and $\pm f_{2}$ at the origin are the only symplectic invariants. We also show that the group of $H$-preserving symplectomorphisms of an $A_{k-1}$ singularity with $k$ odd consists of symplectomorphisms that can be included into a $C^\infty$-smooth (resp., real-analytic) $H$-preserving flow, whereas for $k$ even the same is true modulo the $\mathbb Z_2$-subgroup given by the maps $\{\mathrm{Id}, \mathrm{Inv}\}$, $\mathrm{Inv}(x,\xi) = (-x,-\xi)$. The paper also briefly discusses the conjecture that the symplectic invariants of $A_{k}$ singularities are spectrally determined.
This article is devoted to the spectral analysis of the electromagnetic Schrödinger operator on the Euclidean plane. In the semiclassical limit, we derive a pseudo-differential effective operator that allows us to describe the spectrum in various situations and appropriate regions of the complex plane. Not only results of the self-adjoint case are proved (or recovered) in the proposed unifying framework, but also new results are established when the electric potential is complex-valued. In such situations, when the non-self-adjointness comes with its specific issues (lack of a “spectral theorem”, resolvent estimates), the analogue of the “low-lying eigenvalues” of the self-adjoint case are still accurately described and the spectral gaps estimated.
This article introduces the notion of good labellings for asymptotic lattices in order to study joint spectra of quantum integrable systems from the point of view of inverse spectral theory. As an application, we consider a new spectral quantity for a quantum integrable system, the quantum rotation number. In the case of two degrees of freedom, we obtain a constructive algorithm for the detection of appropriate labellings for joint eigenvalues, which we use to prove that, in the semiclassical limit, the quantum rotation number can be calculated on a joint spectrum in a robust way, and converges to the well-known classical rotation number. The general results are applied to the semitoric case where formulas become particularly natural.
There is now plenty of evidence to suggest that we all should stop wasting the taxpayer’s money, and instead turn to non-profit editors for handling our beloved manuscripts. If you’re not convinced, please have a look at this post by Tim Gowers, or, if you can read French, at the very complete Frédéric Hélein webpage. For this reason, I have decided to give a lot of my time and energy to the launching of a new journal, free for authors and readers, the Annales Henri Lebesgue. I also served in the editorial board of Annales Scientifiques de l’École Normale Supérieure which, albeit not totally non-profit, operates now in a very reasonable way.