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Jeudi 30 novembre 10:45-11:45 Andrei Shafarevich  (Moscow State University)
Laplacians and wave equations on polyhedral surfaces

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Lieu : Salle 121-123, bâtiment 425

Résumé : Differential operators on polyhedral surfaces are intensively studied during last decades. Many papers are devoted to such topics as spectral theory, determinants, trace formulas etc. Nice properties of such operators are due to the fact that polyhedra are almost everywhere flat ; on the other hand, there appear interesting effects caused by the singularities (vertices). In the talk, we present some results concerning various properties of Laplacians and the behavior of solutions to wave equations on polyhedral surfaces.

Laplacians and wave equations on polyhedral surfaces  Version PDF

Mercredi 29 novembre 16:00-17:00 Markus Holzmann  (TU Graz)
Self-adjoint Dirac operators with boundary conditions on domains

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Lieu : salle 229, bâtiment 440

Résumé : Let \Omega \subset\mathbb{R}^3 be a domain with compact C^2-smooth boundary.
In this talk we discuss Dirac operators on \Omega acting on functions which satisfy
suitable boundary conditions which yield self-adjoint operators in L^2(\Omega; \mathbb{C}^4).
Such operators are the relativistic counterparts of Laplacians on \Omega with Robin-type boundary conditions. Using a boundary triple approach the self-adjointness of the operators can be shown.
It turns out that there exist critical boundary values for which functions in the domains of the corresponding operators have less Sobolev-regularity.
Furthermore, several basic spectral properties of the operators are obtained,
which can be analyzed and formulated in terms of well-studied integral operators for the Dirac equation.
This talk is based on a joint work with J. Behrndt and A. Mas.

Self-adjoint Dirac operators with boundary conditions on domains  Version PDF

Mercredi 29 novembre 15:00-16:00 Anna Allilueva  (MIPT, Moscow)
Localized solutions of wave equations on simplest geometric graphs

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Lieu : salle 229, bâtiment 440

Résumé : Behavior of solutions for evolution equations on geometric graphs was studied by various authors. In particular, a number of papers were devoted to the distribution of the number of localized (Gaussian) wave packets after multiple scattering on vertices ; this distribution is connected with well-known problems of the analytic number theory. However, a description of the distribution of energy appeared to be much more complicated. In the talk we discuss a construction of localized asymptotic solutions for wave equations and the distribution of their energy for simplest graphs.

Notes de dernières minutes : ATTENTION DOUBLE SEANCE

Localized solutions of wave equations on simplest geometric graphs  Version PDF

Mercredi 15 novembre 16:00-17:00 Jan Dereziński  (Warsaw university)
Almost homogeneous Schroedinger operator

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Lieu : salle 228, bâtiment 440

Résumé : First I will describe a certain natural holomorphic family of closed operators with interesting spectral properties. These operators can be fully analyzed using just trigonometric functions.
Then I will discuss 1-dimensional Schroedinger operators with a 1/x^2 potential with general boundary conditions, which I studied recently with S.Richard. Even though their description involves Bessel and Gamma functions, they turn out to be equivalent to the previous family.
Some operators that I will describe are homogeneous–they get multiplied by a constant after a change of the scale. In general, their homogeneity is weakly broken–scaling induces a simple but nontrivial flow in the parameter space. One can say (with some exaggeration) that they can be viewed as « toy models of the renormalization group ».
Based on
J.D. Laurent Bruneau and Vladimir Georgescu : Homogeneous Schrödinger operators on half-line, Annales Henri Poincare 12 (2011), 547-590
J.D., Serge Richard : On Schrödinger operators with inverse square potentials on the half-line, Annales Henri Poincare 18 (2017) 869-928
J.D. : Homogeneous rank one perturbations, to appear in Annales Henri Poincare

Almost homogeneous Schroedinger operator  Version PDF