Annales Henri Poincaré

, Volume 16, Issue 2, pp 641–650 | Cite as

Discrete Bargmann Transforms Attached to Landau Levels on the Riemann Sphere

  • Zouhaïr MouaynEmail author


We construct a family of transforms labeled by (ν, m) and mapping isometrically square integrable functions on a finite subset of \({\mathbb{R}}\) onto L 2-eigenspaces associated with the discrete spectrum of a charged particle evolving in the Riemann sphere under influence of a uniform magnetic field with a strength proportional to \({2\nu \in \mathbb{Z}_{+}^{\ast}}\). These transforms are attached to spherical Landau levels \({\lambda _{m}^{\nu}=\left( 2m+1\right) \nu +m\left( m+1\right)}\) with \({m\in \mathbb{Z}_{+}}\) and are called discrete Bargmann transforms.


Coherent State Landau Level Bergman Space Riemann Sphere Hermitian Line Bundle 
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Transformations de Bargmann Discrètes Attachées aux Niveaux de Landau sur la Sphère de Riemann


On construit une famille de transformations indexées par (ν, m) qui appliquent isométriquement les fonctions de carré intégrables sur un sous-ensemble fini de \({\mathbb{R}}\) sur les espaces propres L 2 associés au spectre discret d’une particule chargée évoluant sur la sphère de Riemann sous l’influence d’un champ magnétique uniforme d’une intensité proportionnelle à \({2\nu \in \mathbb{Z}_{+}^{\ast}}\). Ces transformations sont attachées aux niveaux de Landau sphériques \({\lambda _{m}^{\nu}=\left( 2m+1\right) \nu +m\left( m+1\right)}\) avec \({m\in \mathbb{Z}_{+}}\) et sont appelées transformations de Bargmann discrètes.


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Authors and Affiliations

  1. 1.CMUC, Department of MathematicsUniversity of CoimbraCoimbraPortugal
  2. 2.Sultan Moulay Slimane University, Faculty of Sciences and Technics (M’Ghila)Béni MellalMorocco

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