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Jean-Francois Bony

Publications and source records attributed to Jean-Francois Bony.

At least 19 recordsLinked to original sources
arXiv2026

Semiclassical Schr\"odinger operators with purely imaginary potential

We consider Schr\"odinger operators with purely imaginary potential $P = - h^{2} \Delta + i V ( x )$ on a bounded domain. Assuming that near its critical points the potential $V$ can be approximated by an homogeneous polynomial, we show that in the limit $h \to 0$ the leftmost eigenvalues of $P$ are asymptotically given by the local model associated to the most degenerated critical points of $V$. We give applications of this result to the associated evolution problem including shear flows in fluid mechanics.

math.AP↗
arXiv2025

Counter-examples to the fractal Weyl law for semiclassical resonances

Under general assumptions, the numbers of semiclassical resonances is known to be bounded from above by a negative power of $h$ which is given by the fractal dimension of the trapped set. In this paper we provide examples of operators with much less resonances, showing that these upper bounds are not always sharp.

math.AP↗
arXiv2023

Real diffusion with complex spectral gap

The low-lying eigenvalues of the generator of a Langevin process are known to satisfy the Eyring-Kramers law in the low temperature regime under suitable assumptions. These eigenvalues are generically real. We construct generators whose spectral gap is given by non-real eigenvalues or by a real eigenvalue having a Jordan block.

math.AP↗
arXiv2022

Metastable diffusions with degenerate drifts

We study the spectrum of the semiclassical Witten Laplacian $Δ_{f}$ associated to a smooth function $f$ on ${\mathbb R}^d$. We assume that $f$ is a confining Morse--Bott function. Under this assumption we show that $Δ_{f}$ admits exponentially small eigenvalues separated from the rest of the spectrum. Moreover, we establish Eyring-Kramers formula for these eigenvalues. Our approach is based on microlocal constructions of quasimodes near the critical submanifolds.

math.AP↗
arXiv2022

Eyring-Kramers law for Fokker-Planck type differential operators

We consider Fokker-Planck type differential operators associated with general Langevin processes admitting a Gibbs stationary distribution. Under assumptions insuring suitable resolvent estimates, we prove Eyring-Kramers formulas for the bottom of the spectrum of these operators in the low temperature regime. Our approach is based on the construction of sharp Gaussian quasimodes which avoids supersymmetry or PT-symmetry assumptions.

math.AP↗
arXiv2021

Spectral properties of 2D Pauli operators with almost periodic electromagnetic fields

We consider a 2D Pauli operator with almost periodic field $b$ and electric potential $V$. First, we study the ergodic properties of $H$ and show, in particular, that its discrete spectrum is empty if there exists an almost periodic magnetic potential which generates the magnetic field $b - b_{0}$, $b_{0}$ being the mean value of $b$. Next, we assume that $V = 0$, and investigate the zero modes of $H$. As expected, if $b_{0} \neq 0$, then generically $\operatorname{dim} \operatorname{Ker} H = \infty$. If $b_{0} = 0$, then for each $m \in {\mathbb N} \cup \{ \infty \}$, we construct almost periodic $b$ such that $\operatorname{dim} \operatorname{Ker} H = m$. This construction depends strongly on results concerning the asymptotic behavior of Dirichlet series, also obtained in the present article.

math.SP↗
arXiv2020

An example of resonance instability

We construct a semiclassical Schrödinger operator such that the imaginary part of its resonances closest to the real axis changes by a term of size $h$ when a real compactly supported potential of size $o ( h )$ is added.

math.SP↗
arXiv2019

Applications of resonance theory without analyticity assumption

We prove that the results in scattering theory that involve resonances are still valid for non-analytic potentials, even if the notion of resonance is not defined in this setting. More precisely, we show that if the potential of a semiclassical Schrödinger operator is supposed to be smooth and to decrease at infinity, the usual formulas relating scattering quantities and resonances still hold. The main ingredient for the proofs is a resolvent estimate of a new type, relating the resolvent of an operator with the resolvent of its cut-off counterpart.

math.AP↗
arXiv2018

Low-lying eigenvalues of semiclassical Schrödinger operator with degenerate wells

In this article, we consider the semiclassical Schrödinger operator $P = - h^{2} Δ+ V$ in $\mathbb{R}^{d}$ with confining non-negative potential $V$ which vanishes, and study its low-lying eigenvalues $λ_{k} ( P )$ as $h \to 0$. First, we give a necessary and sufficient criterion upon $V^{-1} ( 0 )$ for $λ_{1} ( P ) h^{- 2}$ to be bounded. When $d = 1$ and $V^{-1} ( 0 ) = \{ 0 \}$, we are able to control the eigenvalues $λ_{k} ( P )$ for monotonous potentials by a quantity linked to an interval $I_{h}$, determined by an implicit relation involving $V$ and $h$. Next, we consider the case where $V$ has a flat minimum, in the sense that it vanishes to infinite order. We give the asymptotic of the eigenvalues: they behave as the eigenvalues of the Dirichlet Laplacian on $I_{h}$. Our analysis includes an asymptotic of the associated eigenvectors and extends in particular cases to higher dimensions.

math.SP↗
arXiv2017

Propagation des singularités et résonances

In the framework of semiclassical resonances, we make more precise the link between polynomial estimates of the extension of the resolvent and propagation of the singularities through the trapped set. This approach makes it possible to eliminate infinity and to concentrate the study near the trapped set. It has allowed us in previous papers to obtain the asymptotic of resonances in various geometric situations.

math.AP↗
arXiv2016

Barrier-top resonances for non globally analytic potentials

We give the semiclassical asymptotic of barrier-top resonances for Schrödinger operators on ${\mathbb R}^{n}$, $n \geq 1$, whose potential is $C^{\infty}$ everywhere and analytic at infinity. In the globally analytic setting, this has already been obtained. Our proof is based on a propagation of singularities theorem at a hyperbolic fixed point that we establish here. This last result refines a theorem of the same authors, and its proof follows another approach.

math.AP↗
arXiv2016

Resonances for homoclinic trapped sets

We study semiclassical resonances generated by homoclinic trapped sets. First, under some general assumptions, we prove that there is no resonance in a region below the real axis. Then, we obtain a quantization rule and the asymptotic expansion of the resonances when there is a finite number of homoclinic trajectories. The same kind of results is proved for homoclinic sets of maximal dimension. Next, we generalize to the case of homoclinic/heteroclinic trajectories and we study the three bump case. In all these settings, the resonances may either accumulate on curves or form clouds. We also describe the corresponding resonant states.

math.AP↗
arXiv2012

Semiclassical estimates of the cut-off resolvent for trapping perturbations

This paper is devoted to the study of a semiclassical "black box" operator $P$. We estimate the norm of its resolvent truncated near the trapped set by the norm of its resolvent truncated on rings far away from the origin. For $z$ in the unphysical sheet with $- h |ln h| < Im z < 0$, we prove that this estimate holds with a constant $h |Im z|^{-1} e^{C|Im z|/h}$. We also obtain analogous bounds for the resonances states of $P$. These results hold without any assumption on the trapped set neither any assumption on the multiplicity of the resonances.

math-ph↗
arXiv2011

Maximal velocity of photons in non-relativistic QED

We consider the problem of propagation of photons in the quantum theory of non-relativistic matter coupled to electromagnetic radiation, which is, presently, the only consistent quantum theory of matter and radiation. Assuming that the matter system is in a localized state (i.e for energies below the ionization threshold), we show that the probability to find photons at time t at the distance greater than ct, where c is the speed of light, vanishes as t goes to infinity as an inverse power of t.

math-ph↗
arXiv2011

Counting function of characteristic values and magnetic resonances

We consider the meromorphic operator-valued function 1-K(z) = 1-A(z)/z where A(z) is holomorphic on the domain D, and has values in the class of compact operators acting in a given Hilbert space. Under the assumption that A(0) is a selfadjoint operator which can be of infinite rank, we study the distribution near the origin of the characteristic values of 1-K(z), i.e. the complex numbers w for which the operator 1-K(w) is not invertible, and we show that generically the characteristic values of 1-K(z) converge to 0 with the same rate as the eigenvalues of A(0). We apply our abstract results to the investigation of the resonances of the operator H = H_0 + V where H_0 is the shifted 3D Schrödinger operator with constant magnetic field of scalar intensity b>0, and V is a real electric potential which admits a suitable decay at infinity. It is well known that the spectrum of H_0 is purely absolutely continuous, coincides with [0,+\infty[, and the so-called Landau levels 2bq with integer q, play the role of thresholds in the spectrum of H_0. We study the asymptotic distribution of the resonances near any given Landau level, and under generic assumptions obtain the main asymptotic term of the corresponding resonance counting function, written explicitly in the terms of appropriate Toeplitz operators.

math.SP↗
arXiv2011

Resolvent smoothness and local decay at low energies for the standard model of non-relativistic QED

We consider an atom interacting with the quantized electromagnetic field in the standard model of non-relativistic QED. The nucleus is supposed to be fixed. We prove smoothness of the resolvent and local decay of the photon dynamics for quantum states in a spectral interval I just above the ground state energy. Our results are uniform with respect to I. Their proofs are based on abstract Mourre's theory, a Mourre inequality established in [FGS1], Hardy-type estimates in Fock space, and a low-energy dyadic decomposition.

math-ph↗
arXiv2010

Minoration de la resolvante dans le cas captif

In this note, we prove an optimal universal lower bound on the truncated resolvent for semiclassical Schroedinger operators near a trapping energy. In particular, this shows that known upper bounds for hyperbolic trapping are optimal. The proof rely on an idea of X. P. Wang, and on propagation of coherent states for Ehrenfest times.

math.AP↗