AbstractWe prove the Hopf boundary point lemma for solutions of the Dirichlet problem involving the Schrödinger operator {-\Delta+V} with a nonnegative potential V which merely belongs to {L_{\mathrm{loc}}^{1}(\Omega)}.
More precisely, if {u\in W_{0}^{1,2}(\Omega)\cap L^{2}(\Omega;V\mathop{}\!\mathrm{d}{x})} satisfies {-\Delta u+Vu=f} on Ω for some nonnegative datum {f\in L^{\infty}(\Omega)}, {f\not\equiv 0}, then we show that at every point {a\in\partial\Omega} where the classical normal derivative {\frac{\partial u(a)}{\partial n}} exists and satisfies the Poisson representation formula, one has {\frac{\partial u(a)}{\partial n}>0} if and only if the boundary value problem\begin{dcases}\begin{aligned} \displaystyle-\Delta v+Vv&\displaystyle=0&&%
\displaystyle\phantom{}\text{in ${\Omega}$,}\\
\displaystyle v&\displaystyle=\nu&&\displaystyle\phantom{}\text{on ${\partial%
\Omega}$,}\end{aligned}\end{dcases}involving the Dirac measure {\nu=\delta_{a}} has a solution. More generally, we characterize the nonnegative finite Borel measures ν on {\partial\Omega} for which the boundary value problem above has a solution in terms of the set where the Hopf lemma fails.
On the Riesz summability of the eigenfunction expansions on a closed domain