All articles are available either by clicking on the title, or by sending me a kind email.
Searching for my papers on other websites is not always obvious, because of the various (mis)spellings of my name (including accented characters). If you have access to zbMATH or MathSciNet, this is your best bet, they are good at this. Here are some links with incomplete data you may try: ArXiv, HAL.
The list below is produced from my BibTeX file. Here is also a more traditional PDF version.
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.
Given a quantum semitoric system composed of pseudodifferential operators, Berezin-Toeplitz operators, or a combination of both, we obtain explicit formulas for recovering, from the semiclassical asymptotics of the joint spectrum, all symplectic invariants of the underlying classical semitoric system. Our formulas are based on the possibility to obtain good quantum numbers for joint eigenvalues from the bare data of the joint spectrum. In the spectral region corresponding to regular values of the momentum map, the algorithms developed by Dauge, Hall and the second author (27) produce such labellings. In our proof, it was crucial to extend these algorithms to the boundary of the spectrum, which led to the new notion of asymptotic half-lattices, and to globalize the resulting labellings. Using the construction given by Pelayo and the second author in (79), our results prove that semitoric systems are completely spectrally determined in an algorithmic way : from the joint spectrum of a quantum semitoric system one can construct a representative of the isomorphism class of the underlying classical semitoric system. In particular, this recovers the uniqueness result obtained by Pelayo and the authors in (62,61), and completes it with the explicit computation of all invariants, including the twisting index. In the cases of the spin-oscillator and the coupled angular momenta, we implement the algorithms and illustrate numerically the computation of the invariants from the joint spectrum.
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.
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 $(−{\infty}, b_0 + \varepsilon)$ 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.
We present a new continuous automata based computer simulation of virus propagation in human populations, and apply it to the Covid-19 outbreak, in various scales and situations. We also take the opportunity to propose various mathematical questions, and ask about their biological relevance.
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.
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.
We explain why Theorem B in the original article does not follow from the main result of this paper (Theorem A). While we conjecture that Theorem B should nevertheless be true, in this erratum we prove a slightly weaker version of it.
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.
Transposing the Berezin quantization into the setting of analytic microlocal analysis, we construct approximate semiclassical Bergman projections on weighted $L^2$ spaces with analytic weights, and show that their kernel functions admit an asymptotic expansion in the class of analytic symbols. As a corollary, we obtain new estimates for asymptotic expansions of the Bergman kernel on $\CM^n$ and for high powers of ample holomorphic line bundles over compact complex manifolds.
We prove that the Schwartz class is stable under the magnetic Schrödinger flow when the magnetic 2-form is non-degenerate and does not oscillate too much at infinity.
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.
We study the Hamiltonian dynamics of a charged particle submitted to a pure magnetic field in a two-dimensional domain. We provide conditions on the magnetic field in a neighbourhood of the boundary to ensure the confinement of the particle. We also prove a formula for the scattering angle in the case of radial magnetic fields.
These notes are an expanded version of a mini-course given at the Poisson 2016 conference in Geneva. Starting from classical integrable systems in the sense of Liouville, we explore the notion of non-degenerate singularities and expose recent research in connection with semi-toric systems. The quantum and semiclassical counterpart are also presented, in the viewpoint of the inverse question: from the quantum mechanical spectrum, can one recover the classical system?
This paper deals with semiclassical asymptotics of the three-dimensional magnetic Laplacian in presence of magnetic confinement. Using generic assumptions on the geometry of the confinement, we exhibit three semiclassical scales and their corresponding effective quantum Hamiltonians, by means of three microlocal normal forms *à la Birkhoff*. As a consequence, when the magnetic field admits a unique and non degenerate minimum, we are able to reduce the spectral analysis of the low-lying eigenvalues to a one-dimensional $\hbar$-pseudo-differential operator whose Weyl’s symbol admits an asymptotic expansion in powers of $\hbar^{\frac1 2}$.
Using an abstract notion of semiclassical quantization for self-adjoint operators, we prove that the joint spectrum of a collection of commuting semiclassical self-adjoint operators converges to the classical spectrum given by the joint image of the principal symbols, in the semiclassical limit. This includes Berezin-Toeplitz quantization and certain cases of ℏ-pseudodifferential quantization, for instance when the symbols are uniformly bounded, and extends a result by L. Polterovich and the authors. In the last part of the paper we review the recent solution to the inverse problem for quantum integrable systems with periodic Hamiltonians, and explain how it also follows from the main result in this paper.
Quantum semitoric systems form a large class of quantum Hamiltonian integrable systems with circular symmetry which has received great attention in the past decade. They include systems of high interest to physicists and mathematicians such as the Jaynes-Cummings model (1963), which describes a two-level atom interacting with a quantized mode of an optical cavity, and more generally the so-called systems of Jaynes-Cummings type. In this paper we consider the joint spectrum of a pair of commuting semiclassical operators forming a quantum integrable system of Jaynes-Cummings type. We prove, assuming the Bohr-Sommerfeld rules hold, that if the joint spectrum of two of these systems coincide up to O(ℏ²), then the systems are isomorphic.
This paper is devoted to the classical mechanics and spectral analysis of a pure magnetic Hamiltonian in $\mathbb{R}^2$. It is established that both the dynamics and the semiclassical spectral theory can be treated through a Birkhoff normal form and reduced to the study of a family of one dimensional Hamiltonians. As a corollary, recent results by Helffer-Kordyukov are extended to higher energies.
We prove, assuming that the Bohr-Sommerfeld rules hold, that the joint spectrum near a focus-focus singular value of a quantum integrable system determines the classical Lagrangian foliation around the full focus-focus leaf. The result applies, for instance, to $\hbar$-pseudodifferential operators on cotangent bundles and Berezin-Toeplitz operators on prequantizable compact symplectic manifolds.
A natural way of generalising Hamiltonian toric manifolds is to permit the presence of generic isolated singularities for the moment map. For a class of such “almost-toric 4-manifolds” which admits a Hamiltonian $S^1$-action we show that one can associate a group of convex polygons that generalise the celebrated moment polytopes of Atiyah, Guillemin–Sternberg. As an application, we derive a Duistermaat–Heckman formula demonstrating a strong effect of the possible monodromy of the underlying integrable system.