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Transactions of the American Mathematical Society

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The periodic Euler-Bernoulli equation

Author: Vassilis G. Papanicolaou
Journal: Trans. Amer. Math. Soc. 355 (2003), 3727-3759
MSC (2000): Primary 34B05, 34B10, 34B30, 34L40, 74B05
Published electronically: May 29, 2003
MathSciNet review: 1990171
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Abstract: We continue the study of the Floquet (spectral) theory of the beam equation, namely the fourth-order eigenvalue problem

\begin{displaymath}\left[ a(x)u^{\prime \prime }(x)\right] ^{\prime \prime }=\lambda \rho (x)u(x),\qquad -\infty <x<\infty , \end{displaymath}

where the functions $a$ and $\rho $ are periodic and strictly positive. This equation models the transverse vibrations of a thin straight (periodic) beam whose physical characteristics are described by $a$ and $ \rho $. Here we develop a theory analogous to the theory of the Hill operator $-(d/dx)^2+q(x)$.

We first review some facts and notions from our previous works, including the concept of the pseudospectrum, or $\psi $-spectrum.

Our new analysis begins with a detailed study of the zeros of the function $F(\lambda ;k)$, for any given ``quasimomentum'' $k\in \mathbb{C}$, where $F(\lambda ;k)=0$ is the Floquet-Bloch variety of the beam equation (the Hill quantity corresponding to $F(\lambda ;k)$ is $\Delta (\lambda )-2\cos (kb)$, where $\Delta (\lambda )$ is the discriminant and $b$ the period of $q$). We show that the multiplicity $m(\lambda ^{\ast })$ of any zero $\lambda ^{\ast }$ of $F(\lambda ;k)$ can be one or two and $m(\lambda ^{\ast })=2$ (for some $k$) if and only if $\lambda ^{\ast }$ is also a zero of another entire function $D(\lambda )$, independent of $k$. Furthermore, we show that $D(\lambda )$ has exactly one zero in each gap of the spectrum and two zeros (counting multiplicities) in each $\psi $-gap. If $\lambda ^{\ast }$ is a double zero of $F(\lambda ;k)$, it may happen that there is only one Floquet solution with quasimomentum $k$; thus, there are exceptional cases where the algebraic and geometric multiplicities do not agree.

Next we show that if $(\alpha ,\beta )$ is an open $\psi $-gap of the pseudospectrum (i.e., $\alpha <\beta $), then the Floquet matrix $T(\lambda )$has a specific Jordan anomaly at $\lambda =\alpha $ and $\lambda =\beta $.

We then introduce a multipoint (Dirichlet-type) eigenvalue problem which is the analogue of the Dirichlet problem for the Hill equation. We denote by $\{\mu _n\}_{n\in \mathbb{Z}}$ the eigenvalues of this multipoint problem and show that $\{\mu _n\}_{n\in \mathbb{Z}}$ is also characterized as the set of values of $\lambda $ for which there is a proper Floquet solution $f(x;\lambda )$ such that $f(0;\lambda )=0$.

We also show (Theorem 7) that each gap of the $L^{2}(\mathbb{R})$-spectrum contains exactly one $\mu _{n}$ and each $\psi $-gap of the pseudospectrum contains exactly two $\mu _{n}$'s, counting multiplicities. Here when we say ``gap'' or ``$\psi $-gap'' we also include the endpoints (so that when two consecutive bands or $\psi $-bands touch, the in-between collapsed gap, or $\psi $-gap, is a point). We believe that $\{\mu _{n}\}_{n\in \mathbb{Z}}$ can be used to formulate the associated inverse spectral problem.

As an application of Theorem 7, we show that if $\nu ^{*}$ is a collapsed (``closed'') $\psi $-gap, then the Floquet matrix $T(\nu ^{*})$ is diagonalizable.

Some of the above results were conjectured in our previous works. However, our conjecture that if all the $\psi $-gaps are closed, then the beam operator is the square of a second-order (Hill-type) operator, is still open.

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Additional Information

Vassilis G. Papanicolaou
Affiliation: Department of Mathematics and Statistics, Wichita State University, Wichita, Kansas 67260-0033
Address at time of publication: Department of Mathematics, National Technical University of Athens, Zografou Campus, 157 80, Athens, Greece

Keywords: Euler-Bernoulli equation for the vibrating beam, beam operator, Hill operator, Floquet spectrum, pseudospectrum, algebraic/geometric multiplicity, multipoint eigenvalue problem
Received by editor(s): November 13, 2001
Received by editor(s) in revised form: November 10, 2002
Published electronically: May 29, 2003
Article copyright: © Copyright 2003 American Mathematical Society