Eigenvalue ratios for vibrating String equations with concave densities

Abstract

In this paper, we prove the optimal lower bound $\frac{{\lambda }_n}{{\lambda }_m}\ge (\frac{n}{m}{)}^2$ of vibrating string $$-y^{\prime \prime }=\lambda \rho (x)y,$$ with Dirichlet boundary conditions for concave densities. Our aproach is based on the method of Huang [Proc. AMS., 1999]. The main argument is to restrict the two consecutive eigenfunction $y_{n-1}$ and $y_n$ between two successive zeros of $y_{n-1}$. We also prove the same result for the Dirichlet Sturm-Liouville problems.

9. arXiv admin note: substantial text overlap with arXiv:2111.01728