Dirichlet problem for Schr\"odinger operators on Heisenberg groups

TL;DR

This paper studies the Dirichlet problem for Schrödinger operators on Heisenberg groups, establishing existence and uniqueness results under minimal conditions on the potential by developing a new weak maximum principle.

Contribution

It introduces a novel approach using a weak maximum principle to solve the Dirichlet problem for Schrödinger operators on Heisenberg groups with potentials in the reverse Hölder class, improving previous results.

Findings

Established solvability of the Dirichlet problem under $V \,\in B_{Q/2}$.

Developed a new weak maximum principle for these operators.

Extended results to the Euclidean setting with less restrictive potential conditions.

Abstract

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.