# Spectrum of the free rod under tension and compression

## TL;DR

This paper analyzes the spectral properties of a one-dimensional free rod under tension and compression, revealing complex eigenvalue behaviors including cascading patterns and spectral crossings, with a complete characterization of eigenfunctions and eigenvalues.

## Contribution

It provides a comprehensive analysis and implicit parameterization of eigenvalues and eigenfunctions for the free rod equation under tension and compression, highlighting novel spectral behaviors.

## Key findings

- Eigenvalues exhibit three distinct behaviors depending on tension or compression.
- Eigenvalue branches show cascading patterns and barely-avoided crossings.
- Complete description and properties of eigenvalue curves are established.

## Abstract

In this paper, we study the spectrum of the one-dimensional vibrating free rod equation $u^{(4)}-\tau u"=\mu u$ under tension $(\tau>0)$ or compression $(\tau<0)$. The eigenvalues $\mu$ as functions of the tension/compression parameter $\tau$ exhibit three distinct types of behavior. In particular, eigenvalue branches in the lower half-plane exhibit a cascading pattern of barely-avoided crossings. We provide a complete description of the eigenfunctions and eigenvalues by implicitly parameterizing the eigenvalue curves. We also establish properties of the eigenvalue curves such as monotonicity, crossings, asymptotic growth, cascading and phantom spectral lines.