Eigenvalue ratios for vibrating String equations with concave densities

2021/11/15 by Jihed Hedhly, Hedhly, Jihed
Mathematics · Physics and Astronomy · #14F05 #Advanced Mathematical Physics Problems #F.2.2 #FOS: Mathematics #I.2.7 #Quantum chaos and dynamical systems #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2111.07719

openalex publication_date 2021/11/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we prove the optimal lower bound (λ_n)/(λ_m)≥((n)/(m))² of vibrating string -y''=λρ(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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