Generalized quaternionic Bargmann-Fock spaces and associated Segal-Bargmann transforms

TL;DR

This paper introduces a new family of quaternionic Bargmann-Fock spaces, generalizing existing models, and explores their properties, kernels, and associated Segal-Bargmann transforms, linking to quaternionic Fourier-Wigner transforms.

Contribution

It defines and analyzes generalized quaternionic Bargmann-Fock spaces labeled by an integer m, extending prior work and providing explicit kernels and transform connections.

Findings

Defined new quaternionic Hilbert spaces of Bargmann-Fock type.

Derived explicit formulas for reproducing kernels of these spaces.

Established connections between Segal-Bargmann transforms and quaternionic Fourier-Wigner transform.

Abstract

We introduce new classes of right quaternionic Hilbert spaces of Bargmann-Fock type $\mathcal{GB}_{m}^{2}(\mathbb{H})$, labeled by nonnegative integer $m$, generalizing the so-called slice hyperholomorphic Bargmann-Fock space introduced recently by Alpay, Colombo, Sabadini and Salomon (2014). They are realized as $L^2$-eigenspaces of a sliced second order differential operator. The concrete description of these spaces is investigated and involves the so-called quaternionic Hermite polynomials. Their basic properties are discussed and the explicit formulae of their reproducing kernels are given. Associated Segal-Bargmann transforms, generalizing the one considered quite recently by Diki and Ghanmi (2017), are also introduced and studied. Connection to the quaternionic Fourier-Wigner transform is established.