Journal of the
Korean Mathematical Society JKMS

Noncommutative extensions of the Gross and Beltrami Laplacians, called the quantum Gross Laplacian and the quantum Beltrami Laplacian, resp., are introduced and their basic properties are studied. As noncommutative extensions of the Fourier-Gauss and Fourier-Mehler transforms, we introduce the quantum Fourier-Gauss and quantum Fourier-Mehler transforms. The infinitesimal generators of all differentiable one parameter groups induced by the quantum Fourier-Gauss transform are linear combinations of the quantum Gross Laplacian and quantum Beltrami Laplacian. A characterization of the quantum Fourier-Mehler transform is studied.

Keywords: Fock space operator, quantum Gross Laplacian, quantum Beltrami Laplacian, quantum Fourier-Gauss transform, quantum generalized Fourier-Mehler transform, infinitesimal generator

MSC numbers: Primary 46F25; Secondary 60H40