[12] Tunneling between magnetic wells in two dimensions.
With S. Fournais, Y. Guedes Bonthonneau, N. Raymond. 2025. arXiv:2502.17290.
The two-dimensional magnetic Laplacian is considered. We calculate the leading term of the splitting between the first two eigenvalues of the operator in the semiclassical limit under the assumption that the magnetic field does not vanish and has two symmetric magnetic wells with respect to the coordinate axes. This is the first result of quantum tunneling between purely magnetic wells under generic assumptions. The proof, which strongly relies on microlocal analysis, reveals a purely magnetic Agmon distance between the wells. Surprisingly, it is discovered that the exponential decay of the eigenfunctions away from the magnetic wells is not crucial to derive the tunneling formula. The key is microlocal exponential decay inside the characteristic manifold, with respect to the variable quantizing the classical center guide motion.
[11] Quantum unique ergodicity for the magnetic Laplacian on T^2.
With G. Rivière. 2024. arXiv:2411.18449.
Given a smooth integral two-form and a smooth potential on the flat torus of dimension 2, we study the high energy properties of the corresponding magnetic Schrödinger operator. Under a geometric condition on the magnetic field, we show that every sequence of high energy eigenfunctions satisfies the quantum unique ergodicity property even if the Liouville measure is not ergodic for the underlying classical flow (the Euclidean geodesic flow on the 2-torus).
[10] Magnetic tunneling between disc-shaped obstacles.
With S. Fournais. Communications in Mathematical Physics. 2025.
In this paper we derive formulae for the semiclassical tunneling in the presence of a constant magnetic field in two dimensions. The `wells' in the problem are identical discs with Neumann boundary conditions, so we study the magnetic Neumann Laplacian in the complement of a set of discs. We provide a reduction method to an interaction matrix, which works for a general configuration of obstacles. When there are two discs, we deduce an asymptotic formula for the spectral gap. When the discs are placed along a regular lattice, we derive an effective operator which gives rise to the famous Harper's equation. Main challenges in this problem compared to recent results on magnetic tunneling are the fact that one-well ground states have non-trivial angular momentum which depends on the semiclassical parameter, and the existence of eigenvalue crossings.
[9] The free energy of dilute Bose gases at low temperatures interacting via strong potentials.
With S. Fournais, T. Girardot, L. Junge, M. Olivieri and A. Triay. 2024. arXiv:2408.14222.
We consider a dilute Bose gas in the thermodynamic limit and prove a lower bound on the free energy for low temperatures which is in agreement with the conjecture of Lee-Huang-Yang on the excitation spectrum of the system. Combining techniques of Fournais-Solovej and Haberberger-Hainzl-Nam-Seiringer-Triay, we give a simpler and shorter proof resolving the case of strong interactions, including the hard-core potential.
[8] Purely magnetic tunneling between radial magnetic wells.
With S. Fournais and N. Raymond. Revista Matematica Iberoamericana. 2024.
This article is devoted to the semiclassical spectral analysis of the magnetic Laplacian in two dimensions. Assuming that the magnetic field is positive and has two symmetric radial wells, we establish an accurate tunneling formula, that is a one-term estimate of the spectral gap between the lowest two eigenvalues. This gap is exponentially small when the semiclassical parameter goes to zero, but positive.
[7] Tunneling effect between radial electric wells in a homogeneous magnetic field.
Letters in Mathematical Physics. 2024.
We establish a tunneling formula for a Schrödinger operator with symmetric double-well potential and homogeneous magnetic field in two dimensions. Each well is assumed to be radially symmetric and compactly supported. We obtain an asymptotic formula for the difference between the two first eigenvalues of this operator, that is exponentially small in the semiclassical limit.
[6] Quantitative magnetic isoperimetric inequality.
With R. Ghanta and L. Junge. Journal of Spectral Theory. 2024.
In 1996 Erdoes showed that among planar domains of fixed area, the smallest principal eigenvalue of the Dirichlet Laplacian with constant magnetic field is uniquely achieved on the disk. We establish a quantitative version of this inequality, with an explicit remainder term depending on the field strength that measures how much the domain deviates from the disk.
[5] The ground state energy of a two-dimensional Bose gas.
With S. Fournais, T. Girardot, L. Junge and M. Olivieri. Communications in Mathematical Physics. 2024.
We prove a three-terms asymptotic formula for the ground state energy density of a dilute Bose gas in 2 dimensions, in the thermodynamic limit. This result in 2 dimensions corresponds to the famous Lee-Huang-Yang formula in 3 dimensions. The proof is valid for essentially all positive potentials with finite scattering length, in particular it covers the crucial case of the hard core potential.
[4] Eigenvalue asymptotics for confining magnetic Schrödinger operators with complex potentials.
With N. Raymond and S. Vu Ngoc. International Mathematics Research Notices. 2023.
This article is devoted to the spectral analysis of the electro-magnetic 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 selfadjoint case are proved (or recovered) in the proposed unifying framework, but new results are established when the electric potential is complex-valued. In such situations, when the non-selfadjointness comes with its specific issues (lack of a "spectral theorem", resolvent estimates), the analogue of the "low-lying eigenvalues" of the selfadjoint case are still accurately described and the spectral gaps estimated.
[3] 2D random magnetic Laplacian with white noise magnetic field. With A. Mouzard. Stochastic Processes and their applications. 2022.
We define the random magnetic Laplacian with spatial white noise as magnetic field on the two-dimensional torus using paracontrolled calculus. It yields a random self-adjoint operator with pure point spectrum and domain a random subspace of non-smooth functions in L2. We give sharp bounds on the eigenvalues which imply an almost sure Weyl-type law.
[2] A semiclassical Birkhoff normal form for constant-rank magnetic fields. Analysis & PDE. 2024.
In this paper, we consider the semiclassical magnetic Laplacian on a Riemannian manifold, with constant-rank and non-vanishing magnetic field. We construct several Birkhoff normal forms to describe the spectrum. In particular, we prove eigenvalue asymptotics in powers of h^{1/2}. The non-zero kernel of B (generalization of the magnetic field lines to higher dimensions) induces a specific motion in the semiclassical limit, which influences the eigenvalue asymptotics.
[1] A semiclassical Birkhoff normal form for symplectic magnetic wells. Journal of Spectral Theory. 2022.
In this paper, we construct a Birkhoff normal form for the semiclassical magnetic Laplacian (or magnetic Schrödinger operator) on a d-dimensional Riemannian manifold. We assume that the magnetic field is symplectic, and admits a simple well. In this construction, the operator becomes a function of suitable harmonic oscillators, up to an error of order depanding on resonance phenomenons. It gives eigenvalues and Weyl asymptotics in the semiclassical limit.
[2] Lower bounds on the energy of the Bose gas.
With S. Fournais, T. Girardot, L. Junge and M. Olivieri. Reviews in Mathematical Physics. 2023.
We present an overview of the approach to establish a lower bound to the ground state energy for the dilute, interacting Bose gas in a periodic box. In this paper, the size of the box is larger than the Gross-Pitaevskii lengthscale. The presentation includes both the two and three-dimensional cases, and catches the second order correction, i.e. the Lee-Huang-Yang term. The calculation on a box of this lengthscale is the main step to calculate the energy in the thermodynamic limit. However, the periodic boundary condition simplifies many steps of the argument considerably compared to the localized problem coming from the thermodynamic case.
[1] Review on spectral asymptotics for the semiclassical Bochner Laplacian of a line bundle.
Confluentes Mathematici. 2022.
We first give a short introduction to the Bochner Laplacian on a Riemannian manifold, and explain why it acts locally as a magnetic Laplacian. Then we review recent results on the semiclassical properties of semi-excited spectrum with inhomogeneous magnetic field, including Weyl estimates and eigenvalue asymptotics. These results show under specific assumptions that the spectrum is well described by a familly of operators whose symbols are space-dependent Landau levels. Finally we discuss the strength and limitations of these theorems, in terms of possible crossings between Landau levels.