  We prove uniform resolvent estimates in weighted $L^2$-spaces for radial
solutions of the sublaplacian $\mathcal{L}$ on the Heisenberg group
$\mathbb{H}^d$. The proofs are based on the multipliers methods, and strongly
rely on the use of suitable multipliers and of the associated Hardy
inequalities. The constants in our inequalities are explicit and depend only on
the dimension $d$. As application of the method, we obtain some suitable
smallness and repulsivity conditions on a complex radial potential $V$ on
$\mathbb{H}^d$ such that $\mathcal{L}+V$ has no radial eigenfunctions.
