Jérémy Faupin

Second Order Perturbation Theory for Embedded Eigenvalues

We study second order perturbation theory for embedded eigenvalues of an abstract class of self-adjoint operators. Using an extension of the Mourre theory, under assumptions on the regularity of bound states with respect to a conjugate operator, we prove upper semicontinuity of the point spectrum and establish the Fermi Golden Rule criterion. Our results apply to massless Pauli-Fierz Hamiltonians for arbitrary coupling.

REGULARITY OF BOUND STATES

We study regularity of bound states pertaining to embedded eigenvalues of a self-adjoint operator H, with respect to an auxiliary operator A that is conjugate to H in the sense of Mourre. We work within the framework of singular Mourre theory which enables us to deal with confined massless Pauli–Fierz models, our primary example, and many-body AC-Stark Hamiltonians. In the simpler context of regular Mourre theory, our results boil down to an improvement of results obtained recently in [8, 9].

Effective Dynamics of an Electron Coupled to an External Potential in Non-relativistic QED

In the framework of non-relativistic QED, we show that the renormalized mass of the electron (after having taken into account radiative corrections) appears as the kinematic mass in its response to an external potential force. Specifically, we study the dynamics of an electron in a slowly varying external potential and with slowly varying initial conditions and prove that, for a long time, it is accurately described by an associated effective dynamics of a Schrödinger electron in the same external potential and for the same initial data, with a kinetic energy operator determined by the renormalized dispersion law of the translation-invariant QED model.

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.

Scattering theory for Lindblad master equations

We study scattering theory for a quantum-mechanical system consisting of a particle scattered off a dynamical target that occupies a compact region in position space. After taking a trace over the degrees of freedom of the target, the dynamics of the particle is generated by a Lindbladian acting on the space of trace-class operators. We study scattering theory for a general class of Lindbladians with bounded interaction terms. First, we consider models where a particle approaching the target is always re-emitted by the target. Then we study models where the particle may be captured by the target. An important ingredient of our analysis is a scattering theory for dissipative operators on Hilbert space.

Spectral Theory near Thresholds for Weak Interactions with Massive Particles

We consider a Hamiltonian describing the weak decay of the massive vector boson Z0 into electrons and positrons. We show that the spectrum of the Hamiltonian is composed of a unique isolated ground state and a semi-axis of essential spectrum. Using a suitable extension of Mourres theory, we prove that the essential spectrum below the boson mass is purely absolutely continuous.

Quantum Electrodynamics of Atomic Resonances

A simple model of an atom interacting with the quantized electromagnetic field is studied. The atom has a finite mass m, finitely many excited states and an electric dipole moment,

Hyperfine Splitting of the Dressed Hydrogen Atom Ground State in Non-relativistic QED

{\vec{d}_0 = -\lambda_{0} \vec{d}}