  We investigate the Dirichlet problem associated to the Schr\"odinger operator
$\mathcal L=-\Delta_{\mathbb{H}^n}+V$ on Heisenberg group $\mathbb H^n$:
\begin{align*} \begin{cases} \partial_{ss}u(g,s)-\mathcal L u(g,s)=0\,,\quad
&{\rm in \,\ } \mathbb{H}^n\times\mathbb{R}^+,\\ u(g,0)=f \,,\quad &{\rm on \,\
} \mathbb{H}^n \end{cases} \end{align*} with $f$ in $L^p(\mathbb{H}^n)$ ($1<
p<\infty$) and in $H^1_{\mathcal L}(\mathbb{H}^n)$, i.e., the Hardy space
associated with $\mathcal L$. Here $\Delta_{\mathbb{H}^n}$ is the sub-Laplacian
on $\mathbb H^n$ and the nonnegative potential $V$ belongs to the reverse
H\"older class $B_{Q/2}$ with $Q$ the homogeneous dimension of $\mathbb{H}^n$.
The new approach is to establish a suitable weak maximum principle, which is
the key to solve this problem under the condition $V\in B_{Q/2}$. This result
is new even back to $\mathbb R^n$ (the condition will become $V\in B_{n/2}$)
since the previous known result requires $V\in B_{(n+1)/2}$ which went through
a Liouville type theorem.
