Non-trivial 1d and 2d integral transforms of Segal-Bargmann type

Abstract

Generating functions for the univariate complex Hermite polynomials (UCHP) are employed to introduce some non-trivial one and two-dimensional integral transforms of Segal-Bargmann type in the framework of specific functional Hilbert spaces. The approach used is issued from the coherent states framework. Basic properties of these transforms are studied. Connection to some special known transforms like Fourier and Wigner transforms is established. The first transform is a two-dimensional Segal-Bargmann transform whose kernel is related the exponential generating function of the UCHP. The second one connects the so-called generalized Bargmann-Fock spaces (or also true-poly-Fock spaces in the terminology of Vasilevski) that are realized as the $L^2$-eigenspaces of a specific magnetic Schr\"odinger operator. Its kernel function is related to a Mehler formula of the UCHP.