15 octobre 2018

 
Relâche (GT Théorie spectrale et physique mathématique et Colloquium)

Matej Tusek (CTU Prague)
Dispersionless states in graphene and linear bands of Dirac operators

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Lieu : Salle 3L15, IMO (Bâtiment 307)

Résumé : In the first part of my talk, I will show that a quantum system with translational invariance may host dispersionless wave packets and that a sufficient condition for their existence is presence of a linear energy band in the spectrum of the corresponding Hamiltonian. This condition is, of course, very restrictive, so we will also briefly study what happens if it is slightly violated. Still, there are some rare examples of systems with exactly linear bands – those I am aware of are always governed by 2D Dirac Hamiltonian. In the second part of my talk, I will show how to construct a new example of such a system employing some basic self-adjoint extension techniques.

Dispersionless states in graphene and linear bands of Dirac operators  Version PDF

Davi Obata (Orsay)
A proof of stable ergodicity without accessibility

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Lieu : salle 3L8

Résumé : In the last two decades several works have been done on stable ergodicity
of volume-preserving diffeomorphisms. In the partially hyperbolic world,
all the works on stable ergodicity use a notion called accessibility. In
this talk, I will present the main points in the proof of the stable
ergodicity of an example of a volume-preserving, partially hyperbolic
diffeomorphism introduced by Pierre Berger and Pablo Carrasco. The main
novelty of this proof is that it does not use accessibility.

A proof of stable ergodicity without accessibility  Version PDF

David Krejcirik (CTU Prague)
Spectral analysis of the diffusion operator with random jumps from the boundary

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Lieu : Salle 3L15, IMO (Bâtiment 307)

Résumé : Using an operator-theoretic framework in a Hilbert-space setting, we perform a detailed spectral analysis of the one-dimensional Laplacian in a bounded interval, subject to specific non-self-adjoint connected boundary conditions modelling a random jump from the boundary to a point inside the interval. In accordance with previous works, we find that all the eigenvalues are real. As the new results, we derive and analyse the adjoint operator, determine the geometric and algebraic multiplicities of the eigenvalues, write down formulae for the eigenfunctions together with the generalised eigenfunctions and study their basis properties. It turns out that the latter heavily depend on whether the distance of the interior point to the centre of the interval divided by the length of the interval is rational or irrational. Finally, we find a closed formula for the metric operator that provides a similarity transform of the problem to a self-adjoint operator. This is joint work with Martin Kolb.

Spectral analysis of the diffusion operator with random jumps from the boundary  Version PDF

Nalini Anantharaman (IRMA - Université de Strasbourg)
Délocalisation des fonctions propres de Schrödinger

Résumé : Il y a 100 ans, Einstein se posait la question de trouver des « conditions de quantification » pour les systèmes ergodiques. Alors que nous sommes encore loin d’avoir une bonne description du spectre d’opérateurs de Schrödinger associés à une dynamique classique ergodique, des progrès ont été faits récemment concernant la délocalisation des fonctions propres associées.

Délocalisation des fonctions propres de Schrödinger  Version PDF