The spectral theory of the vibrating periodic beam

Abstract

We study the spectral theory of the fourth-order eigenvalue problem

$$\left[ {a(x)u''(x)} \right]^{\prime \prime } = \lambda \rho (x)u(x), - \infty< x< \infty ,$$

, where the functionsa and ϱ are periodic and strictly positive. This equation models the transverse vibrations of a thin straight (periodic) beam whose physical characteristics are described bya and ϱ.

We examine the structure of the spectrum establishing the fact that the periodic and antiperiodic eigenvalues are the endpoints of the spectral bands. We also introduce an entire function, which we denote byE(λ), connected to the spectral theory, whose zeros (at least the ones of odd multiplicity) are shown to lie on the negative real axis, where they define a collection of “pseudogaps.” Next we prove some inverse results in the spirit of two old theorems of Borg for the Hill's equation. We finish with a “determinant formula” (i.e. a multiplicative trace formula) and some comments on its role in the formulation of the general inverse problem.