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

A. El Hamyani; A. Ghanmi

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arXiv · 1707.01674

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

A. El Hamyani·A. Ghanmi

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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.

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A. El Hamyani, A. Ghanmi. 2017-07-06. Generalized quaternionic Bargmann-Fock spaces and associated Segal-Bargmann transforms. https://arxiv.org/abs/1707.01674

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