Prof. Maria J. Esteban, gave a lecture titled "Characterization and computation of eigenvalues of operators with gaps. Applications in relativistic Quantum Mechanics" at AMSS on 1, November 2017.
In her talk She presented variational methods devised to compute the eigenvalues of operators with gaps, inside those gaps. This was a nontrivial problem, since those eigenvalues had infinite Morse index and therefore computing them was a very unstable matter. But they would have been able to find a variational characterization which was easy to implement and which avoided all those instabilities. Their main application was for the computation of energy levels for Dirac Hamiltonians in relativistic Quantum Mechanics. The variational characterization allowed to construct easy to implement algorithms which were efficient and very accurate. The presentation contained the theoretical description of the variational methods and also a description of the computing algorithms and of the results obtained for some atomic and molecular relativistic models. This work had been done in collaboration with J. Dolbeault, M. Lewin and E. Séré.
Maria J. Esteban is a CNRS senior researcher in mathematics working at the University Paris-Dauphine. She is currently the president of the International Council for Industrial and Applied Mathematics (ICIAM) and member of Jakiunde, the Basque Academy of Sciences and Letters. Her research is mostly concerned with the study of nonlinear partial differential equations and the use of variational methods to study problems in relativistic quantum mechanics and quantum chemistry. She has also made contributions to the study of fluid-structure interaction problems and is recently interested in the understanding of symmetry and symmetry breaking phenomena in the framework of functional inequalities. She is co-editor in chief of the Annales de l′Institut Henri Poincaré (Analyse non linéaire) and editor of several other mathematical journals. She was plenary speaker at the International Congress of Mathematical Physics in 2000, invited speaker at the European Congress of Mathematics in 2008 and at the International Congress of Mathematics in 2018. She has got a Honorary doctorate at the Universities of the Basque Country and of Valencia.