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

Abstract

Consider the Schrodinger equation −u″+V(x)u=λu on the intervalI⊂ℝ, whereV(x)≧0 forx∈I and where Dirichlet boundary conditions are imposed at the endpoints ofI. We prove the optimal bound $$\frac{{\lambda _n }}{{\lambda _1 }} \leqq n^2 for n = 2,3,4,...$$ on the ratio of thenth 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 Prufer transformation and comparison techniques are the key elements of the proof.