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Lundi 15 octobre 11:00-12:00 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
Lundi 15 octobre 09:30-10:30 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