Jean-Francois Bony

Counting Function of Characteristic Values and Magnetic Resonances

We consider the meromorphic operator-valued function I − K(z) = I − A(z)/z where A is holomorphic on the domain 𝒟 ⊂ ℂ, 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 I − K, i.e. the complex numbers w ≠ 0 for which the operator I − K(w) is not invertible, and we show that generically the characteristic values of I − K 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: ℝ3 → ℝ is the electric potential which admits a suitable decay at infinity. It is well known that the spectrum σ(H 0) of H 0 is purely absolutely continuous, coincides with [0, + ∞[, and the so-called Landau levels 2bq with integer q ≥ 0, play the role of thresholds in σ(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.

Semiclassical scattering amplitude at the maximum of the potential

We study the scattering amplitude for Schrodinger operators at a critical energy level, which is a unique non- degenerate maximum of the potential. We do not assume that the maximum point is non-resonant and use results by Bony, Fujiie, Ramond and Zerzeri to analyze the contributions of the trapped trajectories. We prove a semiclassical expansion of the scattering amplitude and compute its leading term. We show that it has different orders of magnitude in specific regions of phase space. We also prove upper and lower bounds for the resolvent in this setting.

Resonances for non-trapping time-periodic perturbations

For time-periodic perturbations of the wave equation in given by a potential q(t, x), we obtain an upper bound of the number of the resonances . We establish for m N large enough a trace formula relating the iterations of the monodromy operator U(mT, 0), T > 0, and the sum ∑jzmj of all resonances counted with their multiplicities.

Scattering theory for the Schrödinger equation with repulsive potential

Abstract We consider the scattering theory for the Schrodinger equation with − Δ − | x | α as a reference Hamiltonian, for 0 α ⩽ 2 , in any space dimension. We prove that, when this Hamiltonian is perturbed by a potential, the usual short range/long range condition is weakened: the limiting decay for the potential depends on the value of α, and is related to the growth of classical trajectories in the unperturbed case. The existence of wave operators and their asymptotic completeness are established thanks to Mourre estimates relying on new conjugate operators. We construct the asymptotic velocity and describe its spectrum. Some results are generalized to the case where − | x | α is replaced by a general second order polynomial.

Large deviations for Gaussian stationary processes and semi-classical analysis

In the present article we provide existence, uniqueness and stability results under an exponential moments condition for quadratic semimartingale backward stochastic differential equations (BSDEs) having convex generators. We show that the martingale part of the BSDE solution defines a true change of measure and provide an example which demonstrates that pointwise convergence of the drivers is not sufficient to guarantee a stability result within our framework.In this paper, we obtain a large deviation principle for quadratic forms of Gaussian stationary processes. It is established by the conjunction of a result of Roch and Silbermann on the spectrum of products of Toeplitz matrices together with the analysis of large deviations carried out by Gamboa, Rouault and the rst author. An alternative proof of the needed result on Toeplitz matrices, based on semi-classical analysis, is also provided.Context trees, variable length Markov chains and dynamical sources.- Martingale property of generalized stochastic exponentials.- Some classes of proper integrals and generalized Ornstein-Uhlenbeck processes.- Martingale representations for diffusion processes and backward stochastic differential equations.- Quadratic Semimartingale BSDEs Under an Exponential Moments Condition.- The derivative of the intersection local time of Brownian motion through Wiener chaos.- On the occupation times of Brownian excursions and Brownian loops.- Discrete approximation to solution flows of Tanakas SDE related to Walsh Brownian motion.- Spectral Distribution of the Free unitary Brownian motion: another approach.- Another failure in the analogy between Gaussian and semicircle laws.- Global solutions to rough differential equations with unbounded vector fields.- Asymptotic behavior of oscillatory fractional processes.- Time inversion property for rotation invariant self-similar diffusion processes.- On Peacocks: a general introduction to two articles.- Some examples of peacocks in a Markovian set-up.- Peacocks obtained by normalisation strong and very strong peacocks.- Branching Brownian motion: Almost sure growth along scaled paths.- On the delocalized phase of the random pinning model.- Large deviations for Gaussian stationary processes and semi-classical analysis.- Girsanov theory under a finite entropy condition.

Tunnel effect for semiclassical random walks

We study a semiclassical random walk with respect to a probability measure with a finite number n_0 of wells. We show that the associated operator has exactly n_0 exponentially close to 1 eigenvalues (in the semiclassical sense), and that the other are O(h) away from 1. We also give an asymptotic of these small eigenvalues. The key ingredient in our approach is a general factorization result of pseudodifferential operators, which allows us to use recent results on the Witten Laplacian.

Tunnel effect for semiclassical random walk

In this note we describe recent results on semiclassical random walk associated to a probability density which may also concentrate as the semiclassical parameter goes to zero. The main result gives a spectral asymptotics of the close to 1 eigenvalues. This problem was studied in [1] and relies on a general factorization result for pseudo-differential operators. In this note we just sketch the proof of this second theorem. At the end of the note, using the factorization, we give a new proof of the spectral asymptotics based on some comparison argument.