This thesis, co-supervised by Eric Cancès (CERMICS, ENPC) and Mathieu Lewin (CEREMADE, Dauphine PSL), will be carried out as part of the CNRS MaQui project on 'New mathematical approaches for interacting quantum systems'. The PhD student will be based at CERMICS, the applied mathematics laboratory of Ecole des Ponts ParisTech. He/she will interact on a weekly basis with senior and junior mathematicians and theoretical chemists in the MaQui project, in particular Julien Toulouse (LCT, Sorbonne University).

Time-Dependent Density-Functional theory (RS-DFT) is one of the most popular computational electronic-structure methods for describing excited electronic states or out-of-equilibrium electronic processes (see e.g. [1]). From a theoretical viewpoint, it relies on the Runge-Gross Theorem. Despite significant successes, TDDFT suffers from several limitations. First, its theoretical foundations are somewhat shaky since the assumptions of the Runge-Gross theorem are too strong for it to be applied to real molecular systems with Coulombic interactions [2]. Second, the approximations commonly used to turn TDDFT into a practical numerical scheme are not accurate enough to model non-adiabatic processes and/or interactions with strong external fields [3].
The aim of this PhD thesis is to contribute to the development of new approximations and numerical methods to increase the potential of TDDFT, notably through
(1) an analysis of the theoretical foundations of TDDFT and a mathematical and numerical investigation of specific out-of-equilibrium states in order to better understand the non- adiabatic regime;
(2) the design, numerical analysis, and implementation in the Julia language of new numerical schemes to take continuous spectrum into account, in the spirit of e.g. [4].

[1] C.A. Ullrich, Time-Dependent Density-Functional Theory: Concepts and Applications, Oxford University Press, 2011.
[2] E. Runge and E. K. U. Gross, Density-functional theory for time-dependent systems, Phys. Rev. Lett., 52 (1984), pp. 997?1000.
[3] S. Fournais, J. Lampart, M. Lewin, and T. Østergaard Sørensen, Coulomb potentials and Taylor expansions in Time-Dependent Density Functional Theory, Phys. Rev. A, 93 (2016), p. 062510.
[4] K. Schwinn, F. Zapata, A. Levitt, E. Cancès, E. Luppi, and J. Toulouse, Photoionization and core resonances from range-separated density-functional theory : General formalism and example of the beryllium atom, J. Chem. Phys., 156 (2022), p. 224106.