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Flensted-Jensen's functions attached to the Landau problem on the hyperbolic disc

Zouhaïr Mouayn (2007)

Applications of Mathematics

We give an explicit expression of a two-parameter family of Flensted-Jensen’s functions Ψ μ , α on a concrete realization of the universal covering group of U ( 1 , 1 ) . We prove that these functions are, up to a phase factor, radial eigenfunctions of the Landau Hamiltonian on the hyperbolic disc with a magnetic field strength proportional to μ , and corresponding to the eigenvalue 4 α ( α - 1 ) .

Harmonic analysis for spinors on real hyperbolic spaces

Roberto Camporesi, Emmanuel Pedon (2001)

Colloquium Mathematicae

We develop the L² harmonic analysis for (Dirac) spinors on the real hyperbolic space Hⁿ(ℝ) and give the analogue of the classical notions and results known for functions and differential forms: we investigate the Poisson transform, spherical function theory, spherical Fourier transform and Fourier transform. Very explicit expressions and statements are obtained by reduction to Jacobi analysis on L²(ℝ). As applications, we describe the exact spectrum of the Dirac operator, study the Abel transform...

Harmonic analysis of spherical functions on S U ( 1 , 1 )

Y. Benyamini, Yitzhak Weit (1992)

Annales de l'institut Fourier

Denote by L 1 ( K G / K ) the algebra of spherical integrable functions on S U ( 1 , 1 ) , with convolution as multiplication. This is a commutative semi-simple algebra, and we use its Gelfand transform to study the ideals in L 1 ( K G / K ) . In particular, we are interested in conditions on an ideal that ensure that it is all of L 1 ( K G / K ) , or that it is L 0 1 ( K G / K ) . Spherical functions on S U ( 1 , 1 ) are naturally represented as radial functions on the unit disk D in the complex plane. Using this representation, these results are applied to characterize harmonic...

Macdonald formula for spherical functions on affine buildings

A. M. Mantero, A. Zappa (2011)

Annales de la faculté des sciences de Toulouse Mathématiques

In this paper we explicitly determine the Macdonald formula for spherical functions on any locally finite, regular and affine Bruhat-Tits building, by constructing the finite difference equations that must be satisfied and explaining how they arise, by only using the geometric properties of the building.

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