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

Study level:
Advanced studies
Grading scale:
0-5
Language:
English
Responsible organisation:
Department of Mathematics and Statistics
Curriculum periods:
2017-2018, 2018-2019, 2019-2020

Description

Content

1st part: Ground state problem for Many-body Schödinger 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 finite 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.

Completion methods

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

Assessment details

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

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 field are also considered.

Description of prerequisites

This is a master level course in mathematics. No background in physics will be necessary to follow this course.

Study materials

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

Completion methods

Method 1

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