The periodic Euler-Bernoulli equation

We continue the study of the Floquet (spectral) theory of the beam equation, namely the fourth-order eigenvalue problem

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.