Enhanced binding revisited for a spinless particle in non-relativistic QED

We consider a spinless particle coupled to a quantized Bose field and show that such a system has a ground state for two classes of short-range potentials which are alone too weak to have a zero-energy resonance

Properties of periodic Dirac--Fock functional and minimizers

Existence of minimizers for the Dirac--Fock model in crystals was recently proved by Paturel and S\'er\'e and the authors \cite{crystals} by a retraction technique due to S\'er\'e \cite{Ser09}. In this paper, inspired by Ghimenti and Lewin's result \cite{ghimenti2009properties} for the periodic Hartree--Fock model, we prove that the Fermi level of any periodic Dirac--Fock minimizer is either empty or totally filled when $\frac{\alpha}{c}\leq C_{\rm cri}$ and $\alpha>0$. Here $c$ is the speed of light, $\alpha$ is the fine structure constant, and $C_{\rm cri}$ is a constant only depending on the number of electrons and on the charge of nuclei per cell. More importantly, we provide an explicit upper bound for $C_{\rm cri}$. Our result implies that any minimizer of the periodic Dirac--Fock model is a projector when $\frac{\alpha}{c}\leq C_{\rm cri}$ and $\alpha>0$. In particular, the non-relativistic regime (i.e., $c\gg1$) and the weak coupling regime (i.e., $0<\alpha\ll1$) are covered. The proof is based on a delicate study of a second-order expansion of the periodic Dirac--Fock functional composed with the retraction used in \cite{crystals}

Self-energy of one electron in non-relativistic QED

AbstractWe investigate the self-energy of one electron coupled to a quantized radiation field by extending the ideas developed in Hainzl (Ann. H. Poincaré, in press). We fix an arbitrary cut-off parameter Λ and recover the α2-term of the self-energy, where α is the coupling parameter representing the fine structure constant. Thereby we develop a method which allows to expand the self-energy up to any power of α. This implies that perturbation theory in α is correct if Λ is fix. As an immediate consequence we obtain enhanced binding for electrons

Properties of periodic Dirac-Fock functional and minimizers

Existence of minimizers for the Dirac-Fock model for crystals was recently proved by Paturel and Séré and the authors [9]. In this paper, inspired by Ghimenti and Lewin's result [12] for the periodic Hartree-Fock model, we prove that the Fermi level of any periodic Dirac-Fock minimizer is either empty or totally filled when α/c\leq C_{cri} and α &gt; 0. Here c is the speed of light, α is the fine structure constant, and C cri is a constant only depending on the number of electrons and on the charge of nuclei per cell. More importantly, we provide an explicit upper bound for C_{cri} .Our result implies that any minimizer of the periodic Dirac-Fock model is a projector when α/c\leq C_{cri} and α &gt; 0 In particular, the non-relativistic regime (i.e., c &gt;&gt;1) and the weak coupling regime (i.e., 0&lt; α &lt;&lt;1) are covered.The proof is based on a delicate study of a second-order expansion of the periodic Dirac-Fock functional composed with a retraction that was introduced by Séré in [23] for atoms and molecules and later extended to the case of crystals in

Mathematical analysis of a nonlinear parabolic equation arising in the modelling of non-newtonian flows

add proof of uniqueness of steady statesThe mathematical properties of a nonlinear parabolic equation arising in the modelling of non-newtonian flows are investigated. The peculiarity of this equation is that it may degenerate into a hyperbolic equation (in fact a linear advection equation). Depending on the initial data, at least two situations can be encountered: the equation may have a unique solution in a convenient class, or it may have infinitely many solutions