## 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 2022 on. You can access the complete list of publications here.

• #### Eigenvalue Asymptotics for Confining Magnetic Schrödinger Operators with Complex Potentials

International Mathematics Research Notices (2022)

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.

• #### Asymptotic lattices, good labellings, and the rotation number for quantum integrable systems

Discrete and Continuous Dynamical Systems 42 (12) pp. 5683-5735 (2022)

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.

## 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.