Georgia Institute of Technology College of Sciences

This talk is about the Dirac equation. We consider an electron modeled by awave function and evolving in the Coulomb field generated by a nucleus. Ina very rough way, this should be an equation of the form$$i\partial_t u = -\Delta u + V( \cdot - q(t)) u$$where $u$ represents the electron while $q(t)$ is the position of thenucleus. When one considers relativitic corrections on the dynamics of anelectron, one should replace the Laplacian in the equation by the Diracoperator. Because of limiting processes in the chemistry model from whichthis is derived, there is also a cubic term in $u$ as a correction in theequation. What is more, the position of the nucleus is also influenced bythe dynamics of the electron. Therefore, this equation should be coupledwith an equation on $q$ depending on $u$.I will present this model and give the first properties of the equation.Then, I will explain why it is well-posed on $H^2$ with a time of existencedepending only on the $H^1$ norm of the initial datum for $u$ and on theinitial datum for $q$. The linear analysis, namely the properties of thepropagator of the equation $i\partial_t u = D u + V( \cdot - q(t))$ where$D$ is the Dirac operator is based on works by Kato, while the non linearanalysis is based on a work by Cancès and Lebris.It is possible to have more than one nucleus. I will explain why.(Joint work with F. Cacciafesta, D. Noja and E. Séré)