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 R, respectively. We show that there exists a unique measure (μ) over tilde (g) on C such that under the Segal- Bargmann transform S-mug the space L-2(R, mu(g)) is isomorphic to the space HL2(C, (μ) over tilde (g)) of analytic L-2- functions on C with respect to (μ) over tilde (g). We also introduce the Segal- Bargmann transform S-mup 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( R, mu(g)) and L-2( R, mu(p)) are isomorphic to the same space HL2(C, (μ) over tilde (g)) under the S-mug - and S-mup - transforms, respectively. However, we show that the multiplication operators by x on L-2( R, mu(g)) and on L-2( R, mu(p)) act quite differently on HL2(C, (μ) over tilde (g)).