  Let $\mathcal L=-\Delta_{\mathbb H^n}+V$ be a Schr\"odinger operator on the
Heisenberg group $\mathbb H^n$, where $\Delta_{\mathbb H^n}$ is the
sub-Laplacian on $\mathbb H^n$ and the nonnegative potential $V$ belongs to the
reverse H\"older class $RH_s$ with $s\geq Q/2$. Here $Q=2n+2$ is the
homogeneous dimension of $\mathbb H^n$. For given $\alpha\in(0,Q)$, the
fractional integrals associated to the Schr\"odinger operator $\mathcal L$ is
defined by $\mathcal I_{\alpha}={\mathcal L}^{-{\alpha}/2}$. In this article,
we first introduce the Morrey space $L^{p,\kappa}_{\rho,\infty}(\mathbb H^n)$
and weak Morrey space $WL^{p,\kappa}_{\rho,\infty}(\mathbb H^n)$ related to the
nonnegative potential $V$. Then we establish the boundedness of fractional
integrals ${\mathcal L}^{-{\alpha}/2}$ on these new spaces. Furthermore, in
order to deal with certain extreme cases, we also introduce the spaces
$\mathrm{BMO}_{\rho,\infty}(\mathbb H^n)$ and
$\mathcal{C}^{\beta}_{\rho,\infty}(\mathbb H^n)$ with exponent $\beta\in(0,1]$.
