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The periodic Euler-Bernoulli equation

Authors:	Vassilis G Papanicolaou

Institution:	Department of Mathematics and Statistics, Wichita State University, Wichita, Kansas 67260-0033

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 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 and $\rho$ . Here we develop a theory analogous to the theory of the Hill operator . 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 the period of ). 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 ) if and only if $\lambda ^{\ast }$ is also a zero of another entire function $D(\lambda )$ , independent of . 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 ; 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.

Keywords:	Euler-Bernoulli equation for the vibrating beam beam operator Hill operator Floquet spectrum pseudospectrum algebraic/geometric multiplicity multipoint eigenvalue problem

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