Master's Degree Programme in Mathematics

MATS4400 Density Functional Theory for Strong Correlated Systems and Optimal Transport (5 cr)

Grading scale

0-5

Teaching languages

English

Responsible person(s)

Tero Kilpel盲inen

Learning outcomes

The students will be familiarized with and combine techniques on Calculus of Variations, Functional Analysis, Convex Analysis and Measure Theory. The course has a good environment to introduce master students to applied analysis. In particular, in putting in mathematical grounds a problem that comes from physics. Numerical aspects and open problems in the 铿乪ld are also considered.

Study methods

Homework assignments and a course exam. Grades will be based on the homeworks (1/3) and the 铿乶al exam (2/3).

Content

1st part: Ground state problem for Many-body Sch枚dinger Equation; A brief introduction of Density Functional Theory: Hohenberg-Kohn functional, Kohm Sham Equations; Density Functional Theory for strongly correlated systems (adiabatic limit). Co-motion functions for spherically symmetric systems.

2nd part: Duality between the space of 铿乶ite measures and continuous bounded functions; Monge and Monge-Kantorovich problems, existence of optimal plans, Kantorovich duality and existence of Kantorovich potentials. Monge problem (two marginal case) and Wasserstein distances. Multi-marginal Optimal Transport for the attractive harmonic case (existence of Monge minimizers). Study the two electrons (marginals) case for Coulomb costs and the N electrons (multi-marginal) case for radially symmetric densities.

Depending of the interests of the students the following topics (not limited of) can be covered as well: Regularity of Kantorovich potentials for Coulomb costs. Entropic Transport. Optimal Transport for Repulsive harmonic costs. Semi-classical limit of the Hohenberg-Kohn functional.

Materials

Lecture notes will be posted online or send by e-mail every week. Other references will be provided
during the lectures.

Assessment criteria