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Bounded multiplicity for eigenvalues of a circular vibrating clamped plate

Yuri Lvovsky, Dan Mangoubi

Abstract

We prove that no eigenvalue of the clamped disk can have multiplicity greater than six. Our method of proof is based on a new recursion formula, linear algebra arguments and a transcendency theorem due to Siegel and Shidlovskii.

Bounded multiplicity for eigenvalues of a circular vibrating clamped plate

Abstract

We prove that no eigenvalue of the clamped disk can have multiplicity greater than six. Our method of proof is based on a new recursion formula, linear algebra arguments and a transcendency theorem due to Siegel and Shidlovskii.

Paper Structure

This paper contains 11 sections, 13 theorems, 29 equations.

Table of Contents

  1. Introduction and background
  2. The vibrating membrane
  3. The vibrating clamped plate
  4. Acknowledgements
  5. Classical facts about Bessel functions
  6. A recursion formula for a cross product of Bessel functions
  7. Forbidden joint zeros
  8. A joint zero is algebraic
  9. Some elements from Siegel-Shidlovskii Theory - a zero is transcendental
  10. Proof of the main Theorem \ref{['thm:wm-zeros']}
  11. Appendix-Shortest recursion possible

Key Result

Theorem \oldthetheorem

Let $m_0, m_1, m_2, m_3$ be four distinct non-negative integers. There is no $x_0>0$ for which $W_{m_0}(x_0)=W_{m_1}(x_0)=W_{m_2}(x_0)=W_{m_3}(x_0)=0$.

Theorems & Definitions (27)

  • Theorem \oldthetheorem
  • Corollary \oldthetheorem
  • Remark \oldthetheorem
  • Proposition \oldthetheorem: wats*Ch. II.12
  • Proposition \oldthetheorem
  • Theorem \oldthetheorem
  • Lemma \oldthetheorem
  • proof : Proof of Theorem \ref{['thm:recursion']}
  • proof : Proof of Lemma \ref{['lem:Wm-formulas']}
  • Theorem \oldthetheorem
  • ...and 17 more