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Frédéric Hérau

Publications and source records attributed to Frédéric Hérau.

At least 19 recordsLinked to original sources
arXiv2026

Semiclassical tunneling for some 1D Schrödinger operators with complex-valued potentials

We consider the non-selfadjoint, semiclassical Schrödinger operator $\mathscr{L}(h) := -h^2\partial_x^2+e^{iα}V$, where $α\in (-π,π)$ and $V: \mathbb{R}\to \mathbb{R}_+$ is even and vanishes at exactly two (symmetric) non-degenerate minima. We establish a semiclassical tunneling result: the spectrum of $\mathscr{L}(h)$ near the origin is given by a sequence of algebraically simple eigenvalues which come in exponentially close pairs (within a $\mathscr{O}(e^{-S/h})$ distance where $S > 0$ is explicit), each pair being separated from the others by a distance $\mathscr{O}(h)$. A one-term estimate of the gap between the two smallest eigenvalues in magnitude is derived; it reveals that, when $α\neq 0$, they quickly rotate around each other as $h$ goes to $0$.

math-ph↗
arXiv2025

Localization of the eigenfunctions of a Bloch-Torrey operator on the half-plane

We consider a non-self adjoint operator of the form $-h^2 Δ+ i(V(x) + α(x)y)$ on the upper half plane $y > 0$ with Dirichlet boundary conditions on $\{y = 0\}$ with $V \geq 0$, $V$ admitting a non-degenerate minimum at $x = 0$ and $α'(0) = 0$. We study its eigenfunctions associated to the smallest eigenvalues in magnitude in the semiclassical limit $h \to 0$. Elementary variational estimates show that these eigenfunctions are localized near the point $(0,0)$ at the scales $O(h^{1/3})$ in $x$ and $O(h^{2/3})$ in $y$. In this paper, we show that the $O(h^{1/3})$ localization in $x$ is not optimal; more precisely, we establish that the eigenfunctions are concentrated in a neighborhood of size $O(h^{1/2})$ of the axis $\{x = 0\}$, and this scale is shown to be sharp. The proof relies on the symbolic calculus of operator-valued pseudodifferential operators.

math-ph↗
arXiv2024

Special macroscopic modes and hypocoercivity

We study linear inhomogeneous kinetic equations with an external confining potential and a collision operator admitting several local conservation laws (local density, momentum and energy). We classify all special macroscopic modes (stationary solutions and time-periodic solutions). We also prove the convergence of all solutions of the evolution equation to such non-trivial modes, with a quantitative exponential rate. This is the first hypocoercivity result with multiple special macroscopic modes with constructive estimates depending on the geometry of the potential.

math.AP↗
arXiv2024

Semiclassical asymptotics of the Bloch--Torrey operator in two dimensions

The Bloch--Torrey operator $-h^2Δ+e^{iα}x_1$ on a bounded smooth planar domain, subject to Dirichlet boundary conditions, is analyzed. Assuming $α\in\left[0,\frac{3π}{5}\right)$ and a non-degeneracy assumption on the left-hand side of the domain, asymptotics of the eigenvalues with the smallest real part in the limit $h \to 0$ are derived. The strategy is a backward complex scaling and the reduction to a tensorized operator involving a real Airy operator and a complex harmonic oscillator.

math.SP↗
arXiv2023

Spectral asymptotics of the Neumann Laplacian with variable magnetic field on a smooth bounded domain in three dimensions

This article is devoted the semiclassical spectral analysis of the Neumann magnetic Laplacian on a smooth bounded domain in three dimensions. Under a generic assumption on the variable magnetic field (involving a localization of the eigenfunctions near the boundary), we establish a semiclassical expansion of the lowest eigenvalues. In particular, we prove that the eigenvalues become simple in the semiclassical limit.

math.SP↗
arXiv2023

Quantitative Geometric Control in Linear Kinetic Theory

We consider general linear kinetic equations combining transport and a linear collision on the kinetic variable with a spatial weight that can vanish on part of the domain. The considered transport operators include external potential forces and boundary conditions, e.g. specular, diffusive and Maxwell conditions. The considered collision operators include the linear relaxation (scattering) and the Fokker-Planck operators and the boundary conditions include specular, diffusive and Maxwell conditions. We prove quantitative estimates of exponential stabilisation (spectral gap) under a geometric control condition. The argument is new and relies entirely on trajectories and weighted functional inequalities on the divergence operators. The latter functional inequalities are of independent interest and imply quantitatively weighted Stokes and Korn inequalities. We finally show that uniform control conditions are not always necessary for the existence of a spectral gap when the equation is hypoelliptic, and prove weaker control conditions in this case.

math.AP↗
arXiv2022

Trajectorial hypocoercivity and application to control theory

We present the quantitative method of the recent work arXiv:2209.09340 in a simple setting, together with a compactness argument that was not included in arXiv:2209.09340 and has interest per se. We are concerned with the exponential stabilization (spectral gap) for linear kinetic equations with degenerate thermalization, i.e. when the collision operator vanishes on parts of the spatial domain. The method in arXiv:2209.09340 covers both scattering and Fokker-Planck type operators, and deals with external potential and boundary conditions, but in these notes we present only its core argument and restrict ourselves to the kinetic Fokker-Planck in the periodic torus with unit velocities and a thermalization degeneracy. This equation is not covered by the previous results of Bernard and Salvarani (2013), Han-Kwan and Léautaud (2015), Evans and Moyano (arXiv:1907.12836).

math.AP↗
arXiv2022

Semiclassical spectral gaps of the 3D Neumann Laplacian with constant magnetic field

This article deals with the spectral analysis of the semiclassical Neumann magnetic Laplacian on a smooth bounded domain in dimension three. When the magnetic field is constant and in the semiclassical limit, we establish a four-term asymptotic expansion of the low-lying eigenvalues, involving a geometric quantity along the apparent contour of the domain in the direction of the field. In particular, we prove that they are simple.

math.SP↗
arXiv2021

Purely magnetic tunneling effect in two dimensions

The semiclassical magnetic Neumann Schrödinger operator on a smooth, bounded, and simply connected domain $Ω$ of the Euclidean plane is considered. When $Ω$ has a symmetry axis, the semiclassical splitting of the first two eigenvalues is analyzed. The first explicit tunneling formula in a pure magnetic field is established. The analysis is based on a pseudo-differential reduction to the boundary and the proof of the first known optimal purely magnetic Agmon estimates.

math.SP↗
arXiv2020

Weighted Korn and Poincaré-Korn inequalities in the Euclidean space and associated operators

We prove functional inequalities on vector fields on the Euclidean space when it is equipped with a bounded measure that satisfies a Poincaré inequality, and study associated self-adjoint operators. The weighted Korn inequality compares the differential matrix, once projected orthogonally to certain finite-dimensional spaces, with its symmetric part and, in an improved form of the inequality, an additional term. We also consider Poincaré-Korn inequalities for estimating a projection of the vector field by the symmetric part of the differential matrix and zeroth-order versions of these inequalities obtained using the Witten-Laplace operator. The constants depend on geometric properties of the potential and the estimates are quantitative and constructive. These inequalities are motivated by kinetic theory and related with the Korn inequality (1906) in mechanics, on a bounded domain.

math.AP↗
arXiv2020

Regularization estimates and Cauchy theory for inhomogeneous Boltzmann equation for hard potentials without cut-off

In this paper, we investigate the problems of Cauchy theory and exponential stability for the inhomogeneous Boltzmann equation without angular cut-off. We only deal with the physical case of hard potentials type interactions (with a moderate angular singularity). We prove a result of existence and uniqueness of solutions in a close-to-equilibrium regime for this equation in weighted Sobolev spaces with a polynomial weight, contrary to previous works on the subject, all developed with a weight prescribed by the equilibrium. It is the first result in this more physically relevant frameworkfor this equation. Moreover, we prove an exponential stability for such a solution, with a rate as close as we want to the optimal rate given by the semigroup decay of the linearized equation. Let us highlight the fact that a key point of the development of our Cauchy theory is the proof of new regularization estimates in short time for the linearized operator thanks to pseudo-differential tools.

math.AP↗
arXiv2018

Global hypoelliptic and symbolic estimates for the linearized Boltzmann operator without angular cutoff

In this article we provide global subelliptic estimates for the linearized inhomogeneous Boltzmann equation without angular cutoff, and show that some global gain in the spatial direction is available although the corresponding operator is not elliptic in this direction. The proof is based on a multiplier method and the so-called Wick quantization, together with a careful analysis of the symbolic properties of the Weyl symbol of the Boltzmann collision operator.

math.AP↗
arXiv2018

Coercivity, hypocoercivity, exponential time decay and simulations for discrete Fokker-Planck equations

In this article, we propose and study several discrete versions of homogeneous and inhomogeneous one-dimensional Fokker-Planck equations. In particular, for these discretizations of velocity and space, we prove the exponential convergence to the equilibrium of the solutions, for time-continuous equations as well as for time-discrete equations. Our method uses new types of discrete Poincaré inequalities for a "two-direction" discretization of the derivative in velocity. For the inhomogeneous problem, we adapt hypocoercive methods to the discrete cases.

math.NA↗
arXiv2016

On global existence and trend to the equilibrium for the Vlasov-Poisson-Fokker-Planck system with exterior confining potential

We prove a global existence result with initial data of low regularity, and prove the trend to the equilibrium for the Vlasov-Poisson-Fokker-Planck system with small non linear term but with a possibly large exterior confining potential in dimension $d=2$ and $d=3$. The proof relies on a fixed point argument using sharp estimates (at short and long time scales) of the semi-group associated to the Fokker-Planck operator, which were obtained by the first author.

math.AP↗
arXiv2016

Magnetic WKB Constructions

This paper is devoted to the semiclassical magnetic Laplacian. Until now WKB expansions for the eigenfunctions were only established in presence of a non-zero electric potential. Here we tackle the pure magnetic case. Thanks to Feynman-Hellmann type formulas and coherent states decomposition, we develop here a magnetic Born-Oppenheimer theory. Exploiting the multiple scales of the problem, we are led to solve an effective eikonal equation in pure magnetic cases and to obtain WKB expansions. We also investigate explicit examples for which we can improve our general theorem: global WKB expansions, quasi-optimal estimates of Agmon and upper bound of the tunelling effect (in symmetric cases). We also apply our strategy to get more accurate descriptions of the eigenvalues and eigenfunctions in a wide range of situations analyzed in the last two decades.

math.AP↗