On the first two eigenvalues of Sturm-Liouville operators

Among the Schrödinger operators with single-well potentials defined on ( 0 , π ) (0,π ) with transition point at π 2 π 2 , the gap between the first two eigenvalues of the Dirichlet problem is minimized when the potential is constant. This extends former results of Ashbaugh and Benguria with symmetric single-well potentials. An analogous result is given for the Dirichlet problem of vibrating strings with single-barrier densities for the ratio of the first two eigenvalues.