  This paper deals with eigenvalue optimization problems for a family of
natural Schr\"odinger operators arising in some geometrical or physical
contexts. These operators, whose potentials are quadratic in curvature, are
considered on closed surfaces immersed in space forms and we look for
geometries that maximize the eigenvalues. We show that under suitable
assumptions on the potential, the first and the second eigenvalues are
maximized by (round) spheres.
