Spectrum of the free rod under tension and compression

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.

KEYWORDS:

• Bi-Laplacian
• cascading
• avoided crossings
• fourth order

AMS SUBJECT CLASSIFICATIONS:

• Primary: 34L15
• Secondary: 34L10
• 74K10

Acknowledgements

The authors would like to thank their current and past advisor Richard S. Laugesen for many helpful discussions and for his guidance, and for proposing the collaboration. Support from Simons Foundation Collaboration Grant #204296 (Laugesen) is gratefully acknowledged. The authors also wish to thank the referee for careful reading and exceptionally helpful comments.

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by the Simons Foundation Collaboration [grant number 204296].