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

Original · EN

Let μg and μₚ denote the Gaussian and Poisson measures on R, respectively. We show that there exists a unique measure μg on C such that under the Segal-Bargmann transform Sμg the space L²(R,μg) is isomorphic to the space HL²(C, μg) of analytic L²-functions on C with respect to μg. We also introduce the Segal-Bargmann transform Sμₚ for the Poisson measure μₚ and prove the corresponding result. As a consequence, when μg and μₚ have the same variance, L²(R,μg) and L²(R,μₚ) are isomorphic to the same space HL²(C, μg) under the Sμg and Sμₚ-transforms, respectively. However, we show that the multiplication operators by x on L²(R, μg) and on L²(R, μₚ) act quite differently on HL²(C, μg).