# Hopf potentials for the Schr\"odinger operator

## TL;DR

This paper proves a boundary point lemma for the Schrödinger operator with potentials in L^1_loc, showing that if one supersolution has a negative normal derivative at the boundary, then all do, under certain conditions.

## Contribution

It extends the Hopf boundary point lemma to Schrödinger operators with minimal regularity potentials, using solutions of nonhomogeneous Dirichlet problems.

## Key findings

- Established the Hopf boundary point lemma for Schrödinger operators with L^1_loc potentials.
- Proved that the boundary derivative property for one supersolution extends to all supersolutions.
- Utilized existence results for nonhomogeneous Dirichlet problems with boundary data in L^∞.

## Abstract

We establish the Hopf boundary point lemma for the Schr\"odinger operator $-\Delta + V$ involving potentials $V$ that merely belong to the space $L^{1}_{loc}(\Omega)$. More precisely, we prove that among all supersolutions $u$ of $-\Delta + V$ which vanish on the boundary $\partial\Omega$ and are such that $V u \in L^{1}(\Omega)$, if there exists one supersolution which satisfies $\partial u/\partial n < 0$ almost everywhere on $\partial\Omega$ with respect to the outward unit vector $n$, then such a property holds for every nontrivial supersolution in the same class. We rely on the existence of nontrivial solutions of the nonhomogeneous Dirichlet problem with boundary datum in $L^{\infty}(\partial\Omega)$.