Generalized Fock Spaces and Associated Operators

Abstract

A class of generalized Fock spaces associated with Bessel functions is studied. The generalized Fock space is a Hilbert space of even entire functions weighted by a modified Bessel function of the third kind, whereas ordinary Fock space is a Hilbert space of entire functions of several complex variables weighted by a Gaussian kernel. The generalized Fock space has a reproducing kernel which is a modified Bessel function of the first kind. Commutator relations between the Schrödinger radial kinetic energy operator and multiplication by $z^2 $ lead to a generalized class of Weyl relations for the Bessel functions.