In this paper we investigate the \(L^p\) regularity, \(L^p\) Neumann and \(W^{1,p}\) problems for generalized Schrödinger 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 \(\begin{aligned}1<p<2+\varepsilon .\end{aligned}\) Moreover, we also establish the \(W^{1,p}\) estimate for Neumann problem for \(\begin{aligned}\frac{3}{2}-\varepsilon<p<3+\varepsilon .\end{aligned}\) 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 Schrödinger equation \(-\Delta u+ Vu=0\) in \(\Omega \) and \(\frac{\partial u}{\partial n}=g\) on \(\partial \Omega \) which was obtained by Shen (Indiana Univ Math J 43(1):143–176, 1994) for ranges \(1<p\le 2\) . All the ranges of p are sharp.
