Lundi 30 janvier 10:00-11:00 -
Zouhair Mouayn - Maroc et IHES
Coherent states and Berezin transforms attached to Landau levels
Résumé : In general, coherent states (|x⟩)x∈X are a specific overcomplete family of normalized vectors in the Hilbert space H of the problem that describes the quantum phenomena and solves the identity of H as 1H = ∫ X|x⟩ ⟨x|dμ(x). These states have long been known for the harmonic oscillator and their properties have frequently been taken as models for defining this notion for other models. We review the definition and properties of coherent states with examples. We construct coherent states attached to Landau levels (discrete energies of a uniform magnetic field) on three known examples of Kähler manifolds X : the Poincare disk D, the Euclidean plane ℂ and the Riemann sphere Cℙ1. After defining their corresponding integral transforms, we obtain characterization theorems for spaces of bound states of the particle. Generalization to ℂn and to the complex unit ball Bn and Cℙn are also discussed. In these cases, we apply a coherent states quantization method to recover the corresponding Berezin transforms and we give formulae representing these transforms as functions of Laplace-Beltrami operators.
Lundi 30 janvier 11:15-12:15 -
Anton Baranov - Saint Petersburg
Hypercyclic Toeplitz operators
Résumé : We study hypercyclicity of the Toeplitz operators in the Hardy space H2(D) with symbols of the form p(z) + φ(z), where p is a polynomial and φ ∈ H∞(D). We find both necessary and sufficient conditions for hypercyclicity which almost coincide in the case when deg p = 1.
Lundi 6 mars 10:00-11:00 -
Alexander Logunov - Saint Petersburg et Tel Aviv
Zero set of a non-constant harmonic function in R^3 has infinite area
Résumé : Nadirashvili conjectured that for any non-constant harmonic function in R^3 its zero set has infinite surface area. This question was motivated by the Yau conjecture on zero sets of Laplace eigenfunctions. We will give a sketch of the proof of Nadirashvili’s conjecture.