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Let $L= -\Delta_{\mathbb{H}^n}+V$ be a Schr\"odinger operator on theHeisenberg group $\mathbb{H}^n$, where $\Delta_{\mathbb{H}^n}$ is thesub-Laplacian and the nonnegative potential $V$ belongs to the reverse H\"olderclass $B_{\frac{Q}{2}}$ and $Q$ is the homogeneous dimension of $\mathbb{H}^n$.The Riesz transforms associated with the Schr\"odinger operator $L$ are boundedfrom $L^1(\mathbb{H}^n)$ to $L^{1,\infty}(\mathbb{H}^n)$. The $L^1$integrability of the Riesz transforms associated with $L$ characterizes acertain Hardy type space denoted by $H^1_L(\mathbb{H}^n)$ which is larger thanthe usual Hardy space $H^1(\mathbb{H}^n)$. We define $H^1_L(\mathbb{H}^n)$ interms of the maximal function with respect to the semigroup $\big \{e^{-s L}:\;s>0 \big\}$, and give the atomic decomposition of $H^1_L(\mathbb{H}^n)$. As anapplication of the atomic decomposition theorem, we prove that$H^1_L(\mathbb{H}^n)$ can be characterized by the Riesz transforms associatedwith $L$. All results hold for stratified groups as well.

Hardy spaces associated with Schrodinger operators on the Heisenberg group