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Let $\mathcal L=-\Delta_{\mathbb H^n}+V$ be a Schr\"odinger operator on theHeisenberg group $\mathbb H^n$, where $\Delta_{\mathbb H^n}$ is thesublaplacian on $\mathbb H^n$ and the nonnegative potential $V$ belongs to thereverse H\"older class $RH_q$ with $q\geq Q/2$. Here $Q=2n+2$ is thehomogeneous dimension of $\mathbb H^n$. Assume that $\{e^{-s\mathcalL}\}_{s>0}$ is the heat semigroup generated by $\mathcal L$. TheLittlewood-Paley function $\mathfrak{g}_{\mathcal L}$ and the Lusin areaintegral $\mathcal{S}_{\mathcal L}$ associated with the Schr\"odinger operator$\mathcal L$ are defined, respectively, by \begin{equation*}\mathfrak{g}_{\mathcal L}(f)(u) := \bigg(\int_0^{\infty}\bigg|s\frac{d}{ds}e^{-s\mathcal L}f(u) \bigg|^2\frac{ds}{s}\bigg)^{1/2} \end{equation*} and\begin{equation*} \mathcal{S}_{\mathcal L}(f)(u) := \bigg(\iint_{\Gamma(u)}\bigg|s\frac{d}{ds} e^{-s\mathcal L}f(v) \bigg|^2\frac{dvds}{s^{Q/2+1}}\bigg)^{1/2}, \end{equation*} where \begin{equation*}\Gamma(u) := \big\{(v,s)\in\mathbb H^n\times(0,\infty): |u^{-1}v| <\sqrt{s\,}\big\}. \end{equation*} In this paper the author first introduces aclass of Morrey spaces associated with the Schr\"odinger operator $\mathcal L$on $\mathbb H^n$. Then by using some pointwise estimates of the kernels relatedto the nonnegative potential $V$, the author establishes the boundednessproperties of these two operators $\mathfrak{g}_{\mathcal L}$ and$\mathcal{S}_{\mathcal L}$ acting on the Morrey spaces. It can be shown thatthe same conclusions also hold for the operators $\mathfrak{g}_{\sqrt{\mathcalL}}$ and $\mathcal{S}_{\sqrt{\mathcal L}}$ with respect to the Poissonsemigroup $\{e^{-s\sqrt{\mathcal L}}\}_{s>0}$.

Morrey spaces for Schr\"odinger operators with certain nonnegative potentials, Littlewood-Paley and Lusin functions on the Heisenberg groups