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 15 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 effect, 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

  • Nathan Reguer (with Alix Deleporte)
  • Antide Duraffour (with Yannick Guedes Bonthonneau and Nicolas Raymond)

Former

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.

Please contact me if I was invited to work at your institute and it's not listed below!

Recent manuscripts

Here are some works from the year 2024 on. You can access the complete list of publications here.

  • Accurate semiclassical analysis of light propagation on tilted hyperplanes

    • Patrick Gioia
    • San Vũ Ngọc
    (2025)

  • Semiclassical Birkhoff–Gustavson Normal Forms and Spectral Asymptotics for Nearly Resonant Schrödinger Operators

    • Abdelkader Bourebai
    • Kaoutar Ghomari
    • San Vũ Ngọc
    Asymptotic Analysis 141 (1) pp. 3-28 (2025)

    The concept of near resonances for harmonic approximations of semiclassical Schrödinger operators is introduced and explored. Combined with a natural extension of the Birkhoff–Gustavson normal form, we obtain formulas for approaching the discrete spectrum of such operators which are both accurate and easy to implement. We apply the theory to the physically important case of the near Fermi (i.e. 1:2) resonance, for which we propose explicit expressions and numerical computations.

  • Phase Space Formulation of Light Propagation on Tilted Planes

    • Patrick Gioia
    • Antonin Gilles
    • Anas El Rhammad
    • San Vũ Ngọc
    Photonics 11 pp. 1034 (2024)

  • Linear canonical transformations in phase space: the Gabor frames approach

    • Patrick Gioia
    • Antonin Gilles
    • Antoine Lagrange
    • Anas El Rhammad
    • San Vũ Ngọc
    Optics, Photonics and Digital Technologies for Imaging Applications VIII pp. 18 SPIE (2024)

  • Un entretien avec Yves Colin de Verdière

    • Frédéric Faure
    • San Vũ Ngọc
    Gaz. Math. (181) pp. 59-66 (2024)

  • Boundary states of the Robin magnetic Laplacian

    • R. Fahs
    • L. Le Treust
    • N. Raymond
    • S. Vũ Ngọc
    Doc. Math. 29 pp. 1157-1200 (2024)

    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.

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 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.