The Floquet theory of the periodic Euler-Bernoulli equation

We continue the study (initiated in [17]) of the spectral theory of the fou rth-order eigenvalue problem [a(x) u "(x)]" = lambda rho(x) u(x), - infinity <x <infinity, where the functions a and p are periodic and strictly positive. This equati on models the transverse vibrations of a thin straight (periodic) beam whos e physical characteristics are described by a and p. The equality of the algebraic and geometric multiplicities of the periodic and antiperiodic eigenvalues is established. Also a spectrum-like set, that we called "pseudospectrum" or "psi-spectrum," is introduced (or, rather, d iscovered). This psi-spectrum is shown to lie on the negative real axis and have a band-gap structure. Various open questions and conjectures are mentioned at the end of the pape r. (C) 1998 Academic Press.