On the Schr\'odinger equations with $B_\infty$ potentials in the region above a Lipschitz graph

In this paper we investigate the $L^p$ regularity, $L^p$ Neumann and $W^{1,p}$ problems for generalized Schr\"odinger operator $-\text{div}(A\nabla )+ V $ in the region above a Lipschitz graph under the assumption that $A$ is elliptic, symmetric and $x_d-$independent. Specifically, we prove that the $L^p$ regularity problem is uniquely solvable for $$1<p<2+\varepsilon.$$ Moreover, we also establish the $W^{1,p}$ estimate for Neumann problem for $$\frac{3}{2}-\varepsilon<p<3+\varepsilon.$$ As a by-product, we also obtain that the $L^p$ Neumann problem is uniquely solvable for $1<p<2+\varepsilon.$ The only previously known estimates of this type pertain to the classical Schr\"odinger equation $-\Delta u+ Vu=0$ in $\Omega$ and $\frac{\partial u}{\partial n}=g$ on $\partial\Omega$ which was obtained by Shen [Z. Shen, On the Neumann problem for Schr\"odinger operators in Lipschitz domains, Indiana Univ. Math. J. 43 (1994)] for ranges $1<p\leq 2$. All the ranges of $p$ are sharp.