Title: Characterization and computation of eigenvalues of operators with gaps. Applications in relativistic Quantum Mechanics
Professor Maria J. Esteban is a Basque-French mathematician. In her research, she studies nonlinear partial differential equations, mainly by the use of variational methods, with applications to physics and quantum chemistry. She has also worked on fluid-structure interaction. She did her PhD thesis at the Pierre and Marie Curie University (Paris), under the direction of Pierre-Louis Lions. After graduation, she became full-time researcher at CNRS, where she holds now a position of director of research. From 2015 to 2019, she is president of International Council for Industrial and Applied Mathematics (ICIAM). She was president of the Société de Mathématiques Appliquées et Industrielles from 2009 to 2012 and chair of the Applied Mathematics Committee of the European Mathematical Society in 2012 and 2013. She participated in the Forward Look on "Mathematics and Industry" funded by the European Science Foundation and is one of the launchers of the EU-MATHS-IN European network for industrial mathematics.
Abstract: In this talk I will present variational methods devised to compute the eigenvalues of operators with gaps, inside those gaps. This is a nontrivial problem, since those eigenvalues have infinite Morse index and therefore computing them is a very unstable matter. But we have been able to find a variational characterization which is easy to implement and which avoids all those instabilities. Our main application is for the computation of energy levels for Dirac Hamiltonians in relativistic Quantum Mechanics. The variational characterization allows to construct easy to implement algorithms which are efficient and very accurate. The presentation will contain 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 has been done in collaboration with J. Dolbeault, M. Lewin and E. Séré.|