## Research themes

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

## Collaborators

Thank you so much to all of you.

## PhD Students

#### Current

• Antide Duraffour (with Yannick Guedes Bonthonneau and Nicolas Raymond)

## Research institutes

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.

## Recent manuscripts

Here are some works from the year 2021 on. You can access the complete list of publications here.

• #### The inverse spectral problem for quantum semitoric systems

(2021)

We prove that semitoric systems are completely spectrally determined : from the joint spectrum of a quantum semitoric system, in the semiclassical limit, one can recover all symplectic invariants of the underlying classical semitoric system, and hence the isomorphism class of the system. Moreover, by using specific algorithms and formulas, we obtain a constructive way of computing the invariants from the spectrum.

• #### Magnetic WKB constructions on surfaces

Reviews in Mathematical Physics (2021)

This article is devoted to the description of the eigenvalues and eigenfunctions of the magnetic Laplacian in the semiclassical limit via the complex WKB method. Under the assumption that the magnetic field has a unique and non-degenerate minimum, we construct the local complex WKB approximations for eigenfunctions on a general surface. Furthermore, in the case of the Euclidean plane, with a radially symmetric magnetic field, the eigenfunctions are approximated in an exponentially weighted space.

• #### Uniform spectral asymptotics for semiclassical wells on phase space loops

Indag. Math. 32 (1) pp. 3-32 (2021)

We consider semiclassical self-adjoint operators whose symbol, defined on a two-dimensional symplectic manifold, reaches a non-degenerate minimum $b_0$ on a closed curve. We derive a classical and quantum normal form which gives uniform eigenvalue asymptotics in a window $(−∞, b_0 + \varepsilon)$ for $\varepsilon>0$ independent on the semiclassical parameter. These asymptotics are obtained in two complementary settings: either an approximate invariance of the system under translation along the curve, which produces oscillating eigenvalues, or a Morse hypothesis reminiscent of Helffer-Sjöstrand’s “miniwell” situation.

• #### Quantum footprints of Liouville integrable systems

Reviews in Mathematical Physics 31 (1) pp. 2060014 (2021)

We discuss the problem of recovering geometric objects from the spectrum of a quantum integrable system. In the case of one degree of freedom, precise results exist. In the general case, we report on the recent notion of good labellings of asymptotic lattices.

• #### Exponential localization in 2D pure magnetic wells

Arkiv för Matematik (2021)

We establish a magnetic Agmon estimate in the case of a purely magnetic single non-degenerate well, by means of the Fourier-Bros-Iagolnitzer transform and microlocal exponential estimates à la Martinez-Sjöstrand.

• #### Long-time dynamics of coherent states in strong magnetic fields

Amer. J. Math. (2021)

We consider a charged particle on a plane, subject to a strong, purely magnetic external field. It is well known that the quantum evolution closely follows the classical dynamics for short periods of time, while for times larger than $\ln \frac{1}{\hbar}$, where $\hbar$ is Planck’s constant, purely quantum phenomena are expected to happen. In this paper we investigate the Schrödinger evolution of generalized coherent states for times of order $1/\hbar$. We prove that, when the initial energy is low, the initial states splits into multiple wavepackets, each one following the average dynamics of the guiding center motion but at its own speed.

## Editorial work

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 serve 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.