  Given $\alpha > -1$, consider the second order differential operator in
$(0,\infty)$,
  $$L_\alpha f \equiv (x^2 \frac{d^2}{dx^2} + (2\alpha+3)x \frac{d}{dx} + x^2 +
(\alpha+1)^2)(f), $$
  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.
