Spectrum of the free rod under tension and compression

L. Mercredi Chasman, Jooyeon Chung

Research output: Contribution to journalArticlepeer-review

1 Scopus citations


In this paper, we study the spectrum of the one-dimensional vibrating free rod equation u(4) -τu" = μu under tension (τ > 0) or compression (τ < 0). The eigenvalues μ as functions of the tension/compression parameter τ 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.

Original languageEnglish (US)
Pages (from-to)2145-2177
Number of pages33
JournalApplicable Analysis
Issue number12
StatePublished - Sep 10 2019

Bibliographical note

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

Publisher Copyright:
© 2018, © 2018 Informa UK Limited, trading as Taylor & Francis Group.


  • 74K10
  • Bi-Laplacian
  • Primary: 34L15
  • Secondary: 34L10
  • avoided crossings
  • cascading
  • fourth order


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