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

Abstract

Let $μ_{g}$ and $μ_{p}$ denote the Gaussian and Poisson measures on ${\Bbb R}$, respectively. We show that there exists a unique measure $\widetildeμ_{g}$ on ${\Bbb C}$ such that under the Segal-Bargmann transform $S_{μ_g}$ the space $L^2({\Bbb R},μ_g)$ is isomorphic to the space ${\cal H}L^2({\Bbb C}, \widetildeμ_{g})$ of analytic $L^2$-functions on ${\Bbb C}$ with respect to $\widetildeμ_{g}$. We also introduce the Segal-Bargmann transform $S_{μ_p}$ for the Poisson measure $μ_{p}$ and prove the corresponding result. As a consequence, when $μ_{g}$ and $μ_{p}$ have the same variance, $L^2({\Bbb R},μ_g)$ and $L^2({\Bbb R},μ_p)$ are isomorphic to the same space ${\cal H}L^2({\Bbb C}, \widetildeμ_{g})$ under the $S_{μ_g}$ and $S_{μ_p}$-transforms, respectively. However, we show that the multiplication operators by $x$ on $L^2({\Bbb R}, μ_g)$ and on $L^2({\Bbb R}, μ_p)$ act quite differently on ${\cal H}L^2({\Bbb C}, \widetildeμ_{g})$.