Heat and Poisson semigroups for Fourier-Neumann expansions

Given $\alpha >-1$, consider the second order differential operator in $(0,\mathrm{\infty })$, $$L_{\alpha }f\equiv (x^2\frac{d^2}{d{x}^2}+(2\alpha +3)x\frac{d}{dx}+x^2+(\alpha +1{)}^2)(f),$$ which appears in the theory of Bessel functions. The purpose of this paper is to develop the corresponding harmonic analysis taking $L_{\alpha }$ as the analogue to the classical Laplacian. Namely we study the boundedness properties of the heat and Poisson semigroups. These boundedness properties allow us to obtain some convergence results that can be used to solve the Cauchy problem for the corresponding heat and Poisson equations.