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Let n ≥ 3 , Ω be a bounded (semi-)convex domain in ℝ n and the non-negative potential V belong to the reverse Hölder class RH n (ℝ n). Assume that p ∈ (1 , ∞) and ω ∈ A p (∂ Ω) , where A p (∂ Ω) denotes the Muckenhoupt weight class on ∂ Ω , the boundary of Ω. In this paper, the authors show that, for any p ∈ (1 , ∞) , the Neumann problem for the Schrödinger equation − Δ u + V u = 0 in Ω with boundary data in (weighted) L p is uniquely solvable. The obtained results in this paper essentially improve the known results which are special cases of the results obtained by Shen [Indiana Univ. Math. J.43 (1994) 143–176] and Tao and Wang [Canad. J. Math.56 (2004) 655–672], via extending the range p ∈ (1 , 2 ] of p into p ∈ (1 , ∞). [ABSTRACT FROM AUTHOR]