Résumé

In this talk I will present the problem of finding extremal potentials for the functional determinant of a one-dimensional Schrödinger operator defined on a bounded interval with Dirichlet boundary conditions. We consider potentials in a fixed $L^q$ space with $qgeq 1$. Functional determinants of Sturm-Liouville operators with smooth potentials or with potentials with prescribed singularities have been widely studied, I will present a short review of these results and will explain how to extend the definition of the functional determinant to potentials in $L^q$. The maximization problem turns out to be equivalent to a problem in optimal control. I will explain how we obtain existence and uniqueness of the maximizers. The results presented in the talk are join work with J-B. Caillau (UCDA, CNRS, Inria, LJAD) and P. Freitas (Lisboa).

Résumé

Résumé

For an arbitrary morphism of (super)manifolds, the pull-back is a linear map of the space of functions. In 2014 Th.Voronov have introduced thick morphisms of (super)manifolds which define a generally non-linear pull-back of functions. This construction was introduced as an adequate tool to describe L-infinity morphisms of algebras of functions provided with the structure of a homotopy Poisson algebra. It turns out that if you go down from `heaven to earth’, and consider usual (not super!) manifolds, then we come to constructions which have natural interpretation in classical and quantum mechanics. In particular in this case the geometrical object which defines the thick diffeomorphism becomes an action of classsical mecanics, and pull-back of the thick diffeomorphism with a quadratic action give a spinor representation. I will define a thick morphism and will tell shortly about their application in homotopy Poisson algebras. Then I will discuss the relation of thick morphisms with notions such as the action in classical mechanics and spinors. The talk is based on the work: “Thick morphisms of supermanifolds, quantum mechanics and spinor representation’, J.Geom. and Phys., 2019, article Number: 103540, https://doi.org/10.1016/ j.geomphys.2019.103540, arXiv:1909.00290 Authors: Hovhannes Khudaverdian, Theodore Voronov

https://dev-iecl.univ-lorraine.fr/Les-Seminaires/Theorie-Des-Nombres/wolfcms/seminaire.html

Résumé

We study a specific Poincaré-Sobolev inequality in bounded domains, that has recently been used to prove a semi-classical bound on the kinetic energy of fermionic many-body states. The corresponding inequality in the entire space is precisely scale invariant and this gives rise to an interesting phenomenon. Optimizers exist for some (most ?) domains and do not exist for some other domains, at least for the isosceles triangle in two dimensions. In this talk, I will discuss bounds on the constant in the inequality and the proofs of existence and non-existence. This is joint work with Rafael Benguria and Cristà³bal Vallejos (PUC, Chile).

Résumé

Le théorème de Hohenberg-Kohn est un résultat fondamental de la mécanique quantique à  N corps, il prouve l’injectivité de l’application allant des champs électriques vers l’état du système quantique électronique ou nucléaire à  l’équilibre. Il justifie la Density functional theory, qui est la méthode la plus utilisée pour étudier pratiquement les systèmes électroniques à  l’échelle microscopique. Nous introduirons le contexte mathématique, puis le théorème, et nous montrerons en quoi sa preuve repose sur un résultat de continuation unique forte. Enfin, nous essaierons d’étendre le théorème à  des champs magnétiques, des potentiels non locaux, des interactions et des températures positives, ce qui soulève des questions mathématiques nouvelles d’unicité.