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Séminaire de Physique Mathématique - EDP
Responsables : Jean-Baptiste Burie, Franck Sueur
Le 2 avril 2024
à 11:00
Séminaire de Physique Mathématique - EDP
Salle de conférences + zoom (BBT in Bordeaux)
San Vu-Ngoc IRMAR
Microlocal analysis of strong magnetic fields, from magnetic bottles to edge states
I will talk about recent work with Rayan Fahs, Loïc Le Treust, Léo Morin, and Nicolas Raymond.
It concerns the spectral study of purely magnetic Schrödinger operators in dimension 2, in the limit of large fields, which is transformed into a semiclassical limit.
A precise geometric and microlocal analysis (of "normal forms" ) gives a very useful heuristic to reduce the problem to an effective 1D operator.
I will present the case of the confinement of classical and quantum particles by a variable magnetic field, as well as more recent work on the appearance of edge states on bounded domains in the plane, with constant magnetic field.
In both cases we obtain spectral asymptotics with 2 or more terms, for Weyl formulas but also for the precise individual descriptions of a large number of eigenvalues, and their relation with the Landau levels.
Le 14 mai 2024
à 11:00
Séminaire de Physique Mathématique - EDP
Salle de conférences
David Krejcirik Czech Technical University in Prague\,
Is the optimal rectangle a square?
We give a light talk on optimality of shapes in geometry and physics. First, we recollect classical geometric results that the disk has the largest area (respectively, the smallest perimeter) among all domains of a given perimeter (respectively, area). Second, we recall that the circular drum has the lowest fundamental tone among all drums of a given area or perimeter and reinterpret the result in a quantum-mechanical language of nanostructures. In parallel, we discuss the analogous optimality of square among all rectangles in geometry and physics. As the main body of the talk, we present our recent attempts to prove the same spectral-geometric properties in relativistic quantum mechanics, where the mathematical model is a matrix-differential (Dirac) operator with complex (infinite-mass) boundary conditions. It is frustrating that such an illusively simple and expected result remains unproved and apparently out of the reach of current mathematical tools.
Les séminaires depuis 2013