Eigenvalue ratios for vibrating String equations with concave densities

Jihed Hedhly

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arXiv · 2111.07719

Eigenvalue ratios for vibrating String equations with concave densities

Jihed Hedhly

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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.

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BibTeX RIS

Jihed Hedhly. 2021-11-15. Eigenvalue ratios for vibrating String equations with concave densities. https://arxiv.org/abs/2111.07719

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