LABORATOIRE J.A. DIEUDONNE

UMR CNRS-UNS N°7351
Maxime Ingremeau


Research areas

All my research is somewhat related to Partial differential equations (PDEs), which are equations satisfied by a function of several variables, relating some of its derivatives. A lot of phenomena in physics and other natural sciences can be described using PDEs. The questions I am interested in mainly concern the wave equation and the Schrödinger equation (which is an analogue of the wave equation describing the evolution of quantum objects). I am mostly interested in solutions which are periodic with respect to time, and do thus solve an equation of the form

Δu + k² u = f.     (1)
More precisely, I work on the following topics:

ℏ Semiclassical analysis: Semiclassical analysis is a set of techniques used to study PDEs (mostly linear) depending on a small (or large) parameter. For instance, semiclassical analysis can be used to describe solutions of the wave equation at high frequencies (or solutions of Schrödinger equation for very massive or large objects) in a given geometry : for instance, studying equation (1) when k → ∞ is a typical semiclassical problem. The properties of these solutions are often related with the properties of the classical trajectories (for instance trajectories of billiard balls bouncing on obstacles, trajectories of particles following Newton’s equations, or trajectories of someone walking in straight line on a surface).

♥ Quantum chaos: Quantum Chaos is the study of the wave equation or the Schrödinger equation in situations where the underlying classical dynamics is chaotic. A typical problem of quantum chaos is to describe the properties of eigenfunctions of the Laplacian (that is, solutions of (1) when f=0), in the large eigenvalue limit (i.e. in the semiclassical regime), on manifolds of negative sectional curvature, or in domains whose billiard dynamics is chaotic.
Other situations which can be related Quantum Chaos (though the classical dynamics is less natural) is the study of eigenfunctions of the Laplacian on large graphs, or on quantum graphs.

⏧ Quantum graphs: The simplest examples of Quantum Graphs are metric graphs, which are just graphs where each edge has a length. On these one dimensional objects, one can define a Laplacian (or more generally, Schrödinger operators), and study the properties of its eigenvalues or eigenfunctions. Although they are very simple objects, quantum graphs have properties characteristic of quantum chaos.
I am mostly interested in the large graph limit, i.e., taking sequences of (larger and larger) quantum graphs and looking at their asymptotic properties.

∢ Scattering Theory: Scattering Theory is the study of propagation of waves in unbounded media. In scattering theory, space is divided into two parts, an "interaction region", where the waves can bounce on obstacles, be deviated by force fields or by geometry, and a "free" region, where the waves just escape to infinity.
When studying scattering theory in the high-frequency regime (in other words, when studying semiclassical scattering theory), a key role is played by the set of trajectories which remain in the interaction region forever, and do not escape towards infinity. When these trapped trajectories have a chaotic behavior, this is a problem of open quantum chaos.

∿ Random waves: Random waves is the study of random solutions of equation (1) in ℝd with f=0. A natural way of building such random solutions is to take a sum of a large number of plane waves having all the same wavelength, with random phases, amplitudes, and directions of propagation. When rescaling this sum and taking the limit, we obtain a random function (the monochromatic stationary isotropic Gaussian field). The properties of the zero set of this function has been the subject of much interest lately.
One motivation to study such random functions in Berry’s conjecture central in quantum chaos, saying that eigenfunctions of the Laplacian in chaotic geometries, in the semiclassical limit, behave like random waves.

• Numerical analysis: Given a Partial Differential Equation having a unique solution, it is not always possible to express this solution using an explicit expression; however, it is generally possible to write an algorithm which will approach the solution with an arbitrary precision. Designing such algorithms and studying their properties is the aim of Numerical Analysis.
I am mostly interested in Numerical Analysis of equations similar to (1) when the wave number k becomes large. Since the solutions oscillate rapidly, usual discretization techniques are not very efficient, and one must une tools from semiclassical analysis.

Publications

  1. ℏ♥∢ Distorted plane waves in chaotic scattering, Analysis and PDE, Vol. 10 (2017), No. 4, 765-816.
  2. ℏ♥∢ Distorted plane waves on manifolds of nonpositive curvature, Communications in Mathematical Physics, Mars 2017, Volume 350, Issue 2, pp 845-891
  3. ℏ♥∢ Sharp resolvent bounds and resonance-free regions, Communications in Partial Differential Equations, 2018, vol. 43, no 2, p. 286-291.
  4. ℏ∢ Equidistribution of phase shifts in trapped scattering, Journal of Spectral Theory, Volume 8, Issue 4, 2018, pp. 1199–1220.
  5. ℏ∢ The semi-classical scattering matrix from the point of view of Gaussian states, Methods and Applications of Analysis, Volume 25, 2018, Number 2, pp. 117-132.
  6. ℏ∢ Equidistribution of phase shifts in obstacle scattering (joint work with Jesse Gell-Redman), Communications in Partial Differential Equations, vol. 44, no 1, p. 1-19.
  7. ℏ♥∢∿ Lower bounds for the number of nodal domains for sums of two distorted plane waves in non-positive curvature, Asian Journal of Mathematics, 2020, vol. 24, no 3, p. 417-436.
  8. ∿ A lower bound for the Bogomolny-Schmit constant for random monochromatic plane waves (joint work with Alejandro Rivera), Mathematical Research Letters, Volume 26 (2019), Number 4, pp 1179 – 1186
  9. ♥⏧ Quantum ergodicity for large equilateral quantum graphs (joint work with Mostafa Sabri et Brian Winn), Journal of the London Mathematical Society, 2020, vol. 101, no 1, p. 82-109.
  10. ℏ♥∿ Local Weak Limits of Laplace Eigenfunctions, Tunisian Journal of Mathematics, 2021, vol. 3, no 3, p. 481-515.
  11. ⏧∿ Absolutely Continuous Spectrum for Quantum Trees , Communications in Mathematical Physics, 2021, vol. 383, no 1, p. 537-594. (joint work with Nalini Anantharaman, Mostafa Sabri et Brian Winn).
  12. ⏧Empirical spectral measures of quantum graphs in the Benjamini-Schramm limit , Journal of Functional Analysis, 2021, vol. 280, no 12, p. 108988. (joint work with Nalini Anantharaman, Mostafa Sabri et Brian Winn).
  13. ♥⏧ Quantum ergodicity for expanding quantum graphs in the regime of spectral delocalization , Journal de Mathématiques Pures et Appliquées, 2021, vol. 151, p. 28-98. (joint work with Nalini Anantharaman, Mostafa Sabri et Brian Winn).
  14. ℏ♥∢ Semiclassical limits of distorted plane waves in chaotic scattering without a pressure condition, International Mathematical Research Notices, Vol 16 (2022) pp 12030–12071.
  15. ℏ♥∿ How Lagrangian states evolve into random waves, Journal de l’École polytechnique — Mathématiques, Volume 9 (2022), pp. 177-212. Joint work with Alejandro Rivera.
  16. ⏧∢ Scattering resonances of large weakly open quantum graphs , Pure and Applied Analysis, Vol. 4 (2022), No. 1, pp 49–83.
  17. ⏧∢ A trace formula for scattering resonances of unbalanced quantum graphs , North-Western European Journal of Mathematics, Vol 9, pp 77-99.

Preprints

  1. ≈ ℏ Efficient approximation of high-frequency Helmholtz solutions by Gaussian coherent states (joint work with Théophile Chaumont-Frelet and Victorita Dolean) .
    2. (The previous paper uses several classical technical facts about Gabor frames. Some elementary proofs of these results are given in the self-contained note Decay of coefficients and approximation rates in Gabor Gaussian frames , written Théophile Chaumont-Frelet.
  2. ℏ♥∿ Emergence of Gaussian fields in noisy quantum chaotic dynamics (joint work with Martin Vogel) .
  3. ℏ♥∿ Improved L∞ bounds for eigenfunctions under random perturbations in negative curvature (joint work with Martin Vogel) .

Collaborators (past and present)

Other scientific texts

  • A talk (in French) at the Bourbaki seminar, about Alexander Logunov's work on Yau's conjecture. The notes (also in French) can be found here.
  • My PhD, in French.
  • My Master's thesis, in French as well.