On the Neumann Problem for the SchrÃ¶dinger Equations with Singular Potentials in Lipschitz Domains We consider the Neumann problem for the Schr\"odinger equations $-\Delta u+Vu=0$, with singular nonnegative potentials $V$ belonging to the reverse H\"older class $\B_n$, in a connected Lipschitz domain $\Omega\subset\mathbf{R}^n$. Given boundary data $g$ in $H^p$ or $L^p$ for $1-\epsilon Keywords:Neumann problem, SchrÃ¶dinger equation, Lipschitz, domain, reverse HÃ¶lder class,$H^p$spaceCategories:42B20, 35J10 2. CJM 1998 (vol 50 pp. 487) Barlow, Martin T.  On the Liouville property for divergence form operators In this paper we construct a bounded strictly positive function$\sigma$such that the Liouville property fails for the divergence form operator$L=\nabla (\sigma^2 \nabla)$. Since in addition$\Delta \sigma/\sigma\$ is bounded, this example also gives a negative answer to a problem of Berestycki, Caffarelli and Nirenberg concerning linear Schr\"odinger operators. Categories:31C05, 60H10, 35J10