Spectrum of the free rod under tension and compression

2017/03/28 by L. Mercredi Chasman, Chasman, L. Mercredi, Jooyeon Chung +1
Mathematics · Physics and Astronomy · #34L10 #34L15 (Primary) #74K10 (Secondary) #Classical Physics (physics.class-ph) #FOS: Mathematics #FOS: Physical sciences #Spectral Theory (math.SP) #math.SP #msc:34L10 #msc:34L15 #msc:74K10 #physics.class-ph

paper · pdf · doi:10.48550/arxiv.1704.04494

38 pages with 17 figures

arxiv created 2017/03/28 · arxiv updated 2017/04/17

Abstract

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.

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