77,711 research outputs found
Fock model and Segal-Bargmann transform for minimal representations of Hermitian Lie groups
For any Hermitian Lie group G of tube type we construct a Fock model of its
minimal representation. The Fock space is defined on the minimal nilpotent
K_C-orbit X in p_C and the L^2-inner product involves a K-Bessel function as
density. Here K is a maximal compact subgroup of G, and g_C=k_C+p_C is a
complexified Cartan decomposition. In this realization the space of k-finite
vectors consists of holomorphic polynomials on X. The reproducing kernel of the
Fock space is calculated explicitly in terms of an I-Bessel function. We
further find an explicit formula of a generalized Segal-Bargmann transform
which intertwines the Schroedinger and Fock model. Its kernel involves the same
I-Bessel function. Using the Segal--Bargmann transform we also determine the
integral kernel of the unitary inversion operator in the Schroedinger model
which is given by a J-Bessel function.Comment: 77page
Models of q-algebra representations: q-integral transforms and "addition theorems''
In his classic book on group representations and special functions Vilenkin studied the matrix elements of irreducible representations of the Euclidean and oscillator Lie algebras with respect to countable bases of eigenfunctions of the Cartan subalgebras, and he computed the summation identities for Bessel functions and Laguerre polynomials associated with the addition theorems for these matrix elements. He also studied matrix elements of the pseudo-Euclidean and pseudo-oscillator algebras with respect to the continuum bases of generalized eigenfunctions of the Cartan subalgebras of these Lie algebras and this resulted in realizations of the addition theorems for the matrix elements as integral transform identities for Bessel functions and for confluent hypergeometric functions. Here we work out q analogs of these results in which the usual exponential function mapping from the Lie algebra to the Lie group is replaced by the q-exponential mappings Eq and eq. This study of representations of the Euclidean quantum algebra and the q-oscillator algebra (not a quantum algebra) leads to summation, integral transform, and q-integral transform identities for q analogs of the Bessel and confluent hypergeometric functions, extending the results of Vilenkin for the q=1 case
An integral containing a Bessel Function and a Modified Bessel Function of the First Kind
Here we discuss the calculation of an integral containing the Bessel function J_0(r) and the modified Bessel function of the first kind I_1(r). The calculus is based on a function of J_0(r), I_1(r) and of their derivatives, having a Wronskian form. The method here described could be useful for training the students in the manipulation of such integrals
The inversion formula and holomorphic extension of the minimal representation of the conformal group
The minimal representation of the indefinite orthogonal group
is realized on the Hilbert space of square integrable functions on
with respect to the measure .
This article gives an explicit integral formula for the holomorphic extension
of to a holomorphic semigroup of by means of the
Bessel function.
Taking its `boundary value', we also find the integral kernel of the
`inversion operator' corresponding to the inversion element on the Minkowski
space
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