Laurent Amour

Inverse spectral theory for the AKNS system with separated boundary conditions

Results in inverse spectral theory are given for the AKNS systems with separated self-adjoint boundary conditions. More precisely we construct a coordinate system well adapted to the isospectral sets. The isospectral sets are analytic, unbounded submanifolds. We give explicit formulae for the solutions to some isospectral flows. Finally we characterize all possible spectra associated with separated self-adjoint boundary conditions.

Inverse spectral results for Schrödinger operators on the unit interval with partial information given on the potentials

We pursue the analysis of the Schrodinger operator on the unit interval in inverse spectral theory initiated in the work of Amour and Raoux [“Inverse spectral results for Schrodinger operators on the unit interval with potentials in Lp spaces,” Inverse Probl. 23, 2367 (2007)]. While the potentials in the work of Amour and Raoux belong to L1 with their difference in Lp (1≤p<∞), we consider here potentials in Wk,1 spaces having their difference in Wk,p, where 1≤p≤+∞, k∊{0,1,2}. It is proved that two potentials in Wk,1([0,1]) being equal on [a,1] are also equal on [0,1] if their difference belongs to Wk,p([0,a]) and if the number of their common eigenvalues is sufficiently high. Naturally, this number decreases as the parameter a decreases and as the parameters k and p increase.

Inverse spectral results for Schrödinger operators on the unit interval with potentials in Lp spaces

We consider the Schr?dinger operator on [0, 1] with potential in L1. We prove that two potentials already known on [a, 1] and having their difference in Lp are equal if the number of their common eigenvalues is sufficiently large. The result here is to write down explicitly this number in terms of p (and a) showing the role of p.

Isospectral sets for akns systems on the unit interval with generalized periodic boundary conditions

We analyze isospectral sets of potentials associated with generalized periodic boundary conditionsB inSL (2, ℝ) for the 2 ×2AKNS systems on the unit interval. WhenB is a rotation we get the usual periodic case. WhenB is not a rotation isospectral sets are cylindrical real analytic submanifolds ofLℝ2([0, 1])2 ×SL(2, ℝ) and their sections for fixed boundary conditions are real analytic submanifolds ofLℝ2([0, 1])2.

On bounded pseudodifferential operators in a high-dimensional setting

This work is concerned with extending the results of Calderon and Vaillancourt proving the boundedness of Weyl pseudodifferential operators Op W eyl h (F) in L 2 (R n). We state conditions under which the norm of such operators has an upper bound independent of n. To this aim, we apply a decomposition of the identity to the symbol F , thus obtaining a sum of operators of a hybrid type, each of them behaving as a Weyl operator with respect to some of the variables and as an anti-Wick operator with respect to the other ones. Then we establish upper bounds for these auxiliary operators, using suitably adapted classical methods like coherent states.

The semiclassical limit of the time dependent Hartree–Fock equation: The Weyl symbol of the solution

We study the Wick symbol of a solution of the time dependent Hartree Fock equation, under weaker hypotheses than those needed for the Weyl symbol in the first paper with thesame title. With similar, we prove some kind of Ehrenfest theorem for observables that are not pseudo-differential operators.

Weyl calculus in QED I. The unitary group

In this work, we consider fixed

Resonances of the Dirac Hamiltonian in the Non Relativistic Limit

1/2

The Cauchy problem for a coupled semilinear parabolic system

spin particles interacting with the quantized radiation field in the context of quantum electrodynamics (QED). We investigate the time evolution operator in studying the reduced propagator (interaction picture). We first prove that this propagator belongs to the class of infinite dimensional Weyl pseudodifferential operators recently introduced in \cite {A-J-N} on Wiener spaces. We give a semiclassical expansion of the symbol of the reduced propagator up to any order with estimates on the remainder terms. Next, taking into account analyticity properties for the Weyl symbol of the reduced propagator, we derive estimates concerning transition probabilities between coherent states.

L1 decay properties for a semilinear parabolic system

Abstract. For a Dirac operator in