Optimal bounds for ratios of eigenvalues of one-dimensional Schroedinger operators with Dirichlet boundary conditions and positive potentials

Consider the Schroedinger equation -u''+V(x)u=λu on the interval Iis contained inR, where V(x) ≥ 0 for xc-I where Dirichlet boundary conditions are imposed at the endpoints of I. We prove the optimal bound λn/λ1≤n2 for n=2, 3 4,... on the ratio of the nth eigenvalue to the first eigenvalue for this problem. This leads to a complete treatment of bounds on ratios of eigenvalues for such problems. Extensions of these results to singular problems are also presented. A modified Pruefer transformation and comparison techniques are the key elements of the proof. (orig.)