In this paper, we establish the non-positivity of the second eigenvalue of the Schrödinger operator − div ( P r ∇ ⋅ ) − W r 2 -\textrm {div}\big ( P_r \nabla \cdot \big ) - W_r^2 on a closed hypersurface Σ n \Sigma ^n of R n + 1 \mathbb {R}^{n+1} , where W r W_r is a power of the ( r + 1 ) (r+1) -th mean curvature of Σ n \Sigma ^n , which we will ask to be positive. If this eigenvalue is null, we will have a characterization of the sphere. This theorem generalizes the result of Harrell and Loss proved to the Laplace-Beltrame operator penalized by the square of the mean curvature.