Eigenvalue Ratios for vibrating string equations with single-well densities

Abstract

In this paper, we prove the optimal upper bound $\frac{{\lambda }_n}{{\lambda }_m}\le (\frac{n}{m}{)}^2$ of vibrating string $$-y^{\prime \prime }=\lambda \rho (x)y,$$ with Dirichlet boundary conditions for single-well densities. The proof is based on the inequality $\frac{{\lambda }_n(\rho )}{{\lambda }_m(\rho )}\le \frac{{\lambda }_n(L)}{{\lambda }_m(L)},$ with $L$ must be a stepfunction. We also prove the same result for the Dirichlet Sturm-Liouville problems.

11 pages, 0 figures