Spectrum of the operator of transverse displacements on a periodic model grid

Abstract

We introduce the operator of transverse displacements in L 2-space on a periodic singular grid F = F 0 with zero thickness and a natural measure. In the case of a model grid, we find the spectrum of this operator: a countable sequence of eigenvalues of infinite multiplicity. This result is useful for describing the asymptotic behavior of the spectrum in the problem of elasticity theory on the corresponding thin grid F h if the thickness parameter h tends to 0 and, in particular, for justifying the existence of gaps in the spectrum of the problem on the thin grid F h . Bibliography: 4 titles. Illustrations: 5 figures.