Upper bounds on the eigenvalue ratio for one-dimensional Schrödinger operators

We apply a commutation formula and the Rayleigh-Ritz inequality to study the ratio of the first two eigenvalues of one-dimensional Schrödinger operators with Dirichlet boundary conditions. The potentials we consider here are allowed to change sign. We give conditions under which the first eigenvalue is positive, and establish some new results on the upper bounds of the eigenvalue ratio.