By decomposing the Laplacian on the Heisenberg group into a family of parametrized partial differential operators Lτ,τ ∈ ℝ \ {0}, and using parametrized Fourier-Wigner transforms, we give formulas and estimates for the strongly continuous one-parameter semigroup generated by Lτ, and the inverse of Lτ. Using these formulas and estimates, we obtain Sobolev estimates for the one-parameter semigroup and the inverse of the Laplacian.