Segal-Bargmann Transforms of One-mode Interacting Fock Spaces Associated with Gaussian and Poisson Measures

Let ${\mu }_g$ and ${\mu }_p$ denote the Gaussian and Poisson measures on $\mathbb{R}$, respectively. We show that there exists a unique measure ${\tilde{\mu }}_g$ on $\mathbb{C}$ such that under the Segal-Bargmann transform $S_{{\mu }_g}$ the space $L^2(\mathbb{R},{\mu }_g)$ is isomorphic to the space $\mathcal{H}{L}^2(\mathbb{C},{\tilde{\mu }}_g)$ of analytic $L^2$-functions on $\mathbb{C}$ with respect to ${\tilde{\mu }}_g$. We also introduce the Segal-Bargmann transform $S_{{\mu }_p}$ for the Poisson measure ${\mu }_p$ and prove the corresponding result. As a consequence, when ${\mu }_g$ and ${\mu }_p$ have the same variance, $L^2(\mathbb{R},{\mu }_g)$ and $L^2(\mathbb{R},{\mu }_p)$ are isomorphic to the same space $\mathcal{H}{L}^2(\mathbb{C},{\tilde{\mu }}_g)$ under the $S_{{\mu }_g}$ and $S_{{\mu }_p}$-transforms, respectively. However, we show that the multiplication operators by $x$ on $L^2(\mathbb{R},{\mu }_g)$ and on $L^2({\Bbb R}, \mu_p)$actquitedifferentlyon${\cal H}L^2({\Bbb C}, \widetilde{\mu}_{g})$.