Dirichlet Problem for Schrödinger Operators on Heisenberg Groups

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