Bernard Parisse

Équilibre instable en régime semi-classique: i—concentration microlocale

On s’intéresse tout d’abord aux fonctions propres de l’opérateur de Schrödinger en dimension 1: (− 2 2 ∂ ∂x2 + V (x))φ(x) = E(h)φ(x), (1) • où V est un potentiel C∞, V (x) = −x/2 + o(x) au voisinage de x = 0 et lim inf |x|→∞ V (x) > 0. • où h tend vers 0 (limite semi-classique) • et où E(h) est un niveau d’énergie qui tend vers 0 lorsque h tend vers 0. On montrera que les fonctions propres se concentrent au point x = 0 à une vitesse au plus logarithmique en h. On en déduira l’existence de fonctions propres du laplacien sur des surfaces de révolution de courbure -1 qui se concentrent sur une géodésique instable (en 1/ ln(λk)).

Giac and GeoGebra - Improved Grobner Basis Computations

GeoGebra is open source mathematics education software being used in thousands of schools worldwide. It already supports equation system solving, locus equation computation and automatic geometry theorem proving by using an embedded or outsourced CAS. GeoGebra recently changed its embedded CAS from Reduce to Giac because it fits better into the educational use. Also careful benchmarking of open source Grobner basis implementations showed that Giac is fast in algebraic computations, too, therefore it allows heavy Grobner basis calculations even in a web browser via Javascript.

Semi-classical study of the origin of quantized Hall conductance in periodic potentials

The semi-classical study of the integer quantum Hall conductivity is investigated for electrons in a biperiodic potential V(x,y). The Hall conductivity is due to the tunnelling effect and we concentrate our study on potentials having three wells in a periodic cell. We show that a nonzero topological conductivity requires special conditions for the positions and shapes of the wells. The results are derived by changing the potential, using the topological nature of Chern indices. Our numerical calculations show that these semi-classical results are still valid for small value of B.The semi-classical study of the integer Quantum Hall conductivity is investigated for electrons in a bi-periodic potential

Singular Bohr–Sommerfeld Rules

V(x,y)

Équilibre instable en régime semi-classique

. The Hall conductivity is due to the tunnelling effect and we concentrate our study to potentials having three wells in a periodic cell. A non-zero topological conductivity requires special conditions for the positions, and shapes of the wells. The results are derived analytically and well confirmed by numerical calculations.