A Dirac field interacting with point nuclear dynamics

Abstract

The system describing a single Dirac electron field coupled with classically moving point nuclei is presented and studied. The model is a semi-relativistic extension of corresponding time-dependent onebody Hartree-Fock equation coupled with classical nuclear dynamics, already known and studied both in quantum chemistry and in rigorous mathematical literature. We prove local existence of solutions for data in H σ with σ ∈ [0, 3 2 [. In the course of the analysis a second new result of independent interest is discussed and proved, namely the construction of the propagator for the Dirac operator with several moving Coulomb singularities.

Large time well posedness for a Dirac--Klein-Gordon system

“…is Lipschitz continuous and thus to be able to apply Picard fixed point Theorem; therefore, it represents an unavoidable threshold. This fact has been already noticed and discussed in [8] (see Remark 1.5 there). On the other hand, the (additional) upper bound s ď 2 turns out to be necessary in view of providing suitable estimates on the function W (see e.g.…”

“…We now deal with the proof of Theorem 1.4, that is we prove local well posedness for system (10). To do this, we essentially follow the strategy developed in [8] (see also [1]). First of all, we need to deal with the classical dynamics driven by q.…”

“…The aim here is to pursue our investigation on the toy models for the Helium atom with relativistic corrections that the first and second author started in [8]. The map u represents the wave function of the electrons, the map W represents the electric field generated by a nucleus centered in the position qptq at time t and with a charge density given by χpx ´qptqq.…”

“…The map u represents the wave function of the electrons, the map W represents the electric field generated by a nucleus centered in the position qptq at time t and with a charge density given by χpx ´qptqq. In [8], the potential generated by the nucleus, instead of being a solution to the Klein-Gordon equation, was an "electrostatic" approximate field generated by a slowly moving charge, namely, it took the form of a moving Coulomb potential 1 |x ´qptq| where x is the space variable and qptq is the position of the nucleus. Unfortunately, the Coulomb potential in this setting is an approximation that does not help getting a global solution to the Dirac equation.…”

On the Cauchy problem for the Hartree approximation in quantum dynamics

“…Existence and uniqueness have been studied in the case where the interaction potential is of convolution type, i.e. for w(x, y) = W (x − y) and with one of the subsystems moving by classical mechanics (see [7,2,6] for example). A related investigation has targeted the time-dependent self-consistent field system [12] with coupling potentials of Schwartz class.…”