OPTIMAL LOWER ESTIMATES FOR EIGENVALUE RATIOS OF SCHRÖDINGER OPERATORS AND VIBRATING STRINGS

Abstract

We obtain optimal lower estimates for the eigenvalue ratios $(\frac{\lambda_{m}}{\lambda_{n}})$ of Dirichlet and Neumann Schrödinger operators with nonpositive potentials and Dirichlet vibrating string problems with concave and positive densities. Our results supplement those of Ashbaugh-Benguria [2] and M. J. Huang [5].