A gap for eigenvalues of a clamped plate problem

TL;DR

This paper provides an estimate for the eigenvalue gap in the clamped plate problem, showing it is bounded by a lower order term related to the eigenvalue index, based on asymptotic analysis.

Contribution

It introduces a new estimate for the eigenvalue gap in the clamped plate problem, extending understanding of eigenvalue distribution in higher dimensions.

Findings

Eigenvalue gap is bounded by a term with order $k^{1/n}$.

The estimate aligns with asymptotic formulas of Agmon and Pleijel.

Provides bounds for eigenvalue differences in n-dimensional domains.

Abstract

This paper studies eigenvalues of the clamped plate problem on a bounded domain in an $n$-dimensional Euclidean space. We give an estimate for the gap between $\sqrt {\Gamma_{k+1}-\Gamma_{1}}$ and $\sqrt {\Gamma_{k}-\Gamma_{1}}$, for any positive integer $k$. According to the asymptotic formula of Agmon and Pleijel, we know, the gap between $\sqrt {\Gamma_{k+1}-\Gamma_{1}}$ and $\sqrt {\Gamma_{k}-\Gamma_{1}}$ is bounded by a term with a lower order $k^{\frac1n}$ in the sense of the asymptotic formula of Agmon and Peijel, where $\Gamma_j$ denotes the $j^{^{\text{th}}}$ eigenvalue of the clamped plate problem.