The periodic Euler-Bernoulli equation

Abstract

We continue the study of the Floquet (spectral) theory of the beam equation, namely the fourth-order eigenvalue problem \[ [ a ( x ) u ′ ′ ( x ) ] ′ ′ = λ ρ ( x ) u ( x ) , − ∞ > x > ∞ , \left [ a(x)u^{\prime \prime }(x)\right ] ^{\prime \prime }=\lambda \rho (x)u(x),\qquad -\infty >x>\infty , \] where the functions a a and ρ \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 a and ρ \rho . Here we develop a theory analogous to the theory of the Hill operator − ( d / d x ) 2 + q ( x ) -(d/dx)^2+q(x) . We first review some facts and notions from our previous works, including the concept of the pseudospectrum, or ψ \psi -spectrum. Our new analysis begins with a detailed study of the zeros of the function F ( λ ; k ) F(\lambda ;k) , for any given “quasimomentum” k ∈ C k\in \mathbb {C} , where F ( λ ; k ) = 0 F(\lambda ;k)=0 is the Floquet-Bloch variety of the beam equation (the Hill quantity corresponding to F ( λ ; k ) F(\lambda ;k) is Δ ( λ ) − 2 cos ⁡ ( k b ) \Delta (\lambda )-2\cos (kb) , where Δ ( λ ) \Delta (\lambda ) is the discriminant and b b the period of q q ). We show that the multiplicity m ( λ ∗ ) m(\lambda ^{\ast }) of any zero λ ∗ \lambda ^{\ast } of F ( λ ; k ) F(\lambda ;k) can be one or two and m ( λ ∗ ) = 2 m(\lambda ^{\ast })=2 (for some k k ) if and only if λ ∗ \lambda ^{\ast } is also a zero of another entire function D ( λ ) D(\lambda ) , independent of k k . Furthermore, we show that D ( λ ) D(\lambda ) has exactly one zero in each gap of the spectrum and two zeros (counting multiplicities) in each ψ \psi -gap. If λ ∗ \lambda ^{\ast } is a double zero of F ( λ ; k ) F(\lambda ;k) , it may happen that there is only one Floquet solution with quasimomentum k k ; thus, there are exceptional cases where the algebraic and geometric multiplicities do not agree. Next we show that if ( α , β ) (\alpha ,\beta ) is an open ψ \psi -gap of the pseudospectrum (i.e., α > β \alpha >\beta ), then the Floquet matrix T ( λ ) T(\lambda ) has a specific Jordan anomaly at λ = α \lambda =\alpha and λ = β \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 { μ n } n ∈ Z \{\mu _n\}_{n\in \mathbb {Z}} the eigenvalues of this multipoint problem and show that { μ n } n ∈ Z \{\mu _n\}_{n\in \mathbb {Z}} is also characterized as the set of values of λ \lambda for which there is a proper Floquet solution f ( x ; λ ) f(x;\lambda ) such that f ( 0 ; λ ) = 0 f(0;\lambda )=0 . We also show (Theorem 7) that each gap of the L 2 ( R ) L^{2}(\mathbb {R}) -spectrum contains exactly one μ n \mu _{n} and each ψ \psi -gap of the pseudospectrum contains exactly two μ n \mu _{n} ’s, counting multiplicities. Here when we say “gap” or “ ψ \psi -gap” we also include the endpoints (so that when two consecutive bands or ψ \psi -bands touch, the in-between collapsed gap, or ψ \psi -gap, is a point). We believe that { μ n } n ∈ Z \{\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 ν ∗ \nu ^{*} is a collapsed (“closed”) ψ \psi -gap, then the Floquet matrix T ( ν ∗ ) T(\nu ^{*}) is diagonalizable. Some of the above results were conjectured in our previous works. However, our conjecture that if all the ψ \psi -gaps are closed, then the beam operator is the square of a second-order (Hill-type) operator, is still open.