Eigenvalue Ratios for vibrating string equations with single-well densities

TL;DR

This paper establishes the optimal upper bounds for eigenvalue ratios in vibrating string equations with single-well densities, extending results to Sturm-Liouville problems, and introduces a new inequality involving stepfunction densities.

Contribution

The paper proves the optimal eigenvalue ratio bounds for vibrating string equations with single-well densities and extends these results to Sturm-Liouville problems, using a novel inequality involving stepfunctions.

Findings

Proved the optimal eigenvalue ratio bound $rac{ abla_n}{ abla_m}\, ext{for vibrating strings.

Extended the eigenvalue ratio bounds to Sturm-Liouville problems.

Introduced a key inequality relating eigenvalues of general densities to stepfunctions.

Abstract

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