Eigenvalue ratios for vibrating String equations with concave densities

TL;DR

This paper establishes an optimal lower bound for the ratio of eigenvalues in vibrating string equations with concave densities, extending to Sturm-Liouville problems, using a method based on eigenfunction zero restrictions.

Contribution

It provides the first proof of the optimal eigenvalue ratio bound for vibrating strings with concave densities, employing a novel approach based on eigenfunction zero restrictions.

Findings

Proved the optimal eigenvalue ratio bound for concave densities.

Extended the result to Dirichlet Sturm-Liouville problems.

Introduced a method based on restricting eigenfunctions between zeros.

Abstract

In this paper, we prove the optimal lower bound $\frac{\lambda_n}{\lambda_m}\geq(\frac{n}{m})^2$ of vibrating string $$-y''=\lambda\rho(x) y,$$ 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.