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 $\widetilde{\mu}_{g}$ 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 ${\mathcal H}L^2({\mathbb C}, \widetilde{\mu}_{g})$ of analytic $L^2$-functions on ${\mathbb C}$ with respect to $\widetilde{\mu}_{g}$. 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 ${\mathcal H}L^2({\mathbb C}, \widetilde{\mu}_{g})$ 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 $L^2({\mathbb R}, \mu_p)$ act quite differently on ${\mathcal H}L^2({\mathbb C}, \widetilde{\mu}_{g})$.