ASYMPTOTIC ANALYSIS OF THE WAVE-EQUATION SPECTRUM - COMPLETENESS OF THE BLOCH SPECTRUM

In a previous Note [1] we began to study the asymptotic behaviour of t he spectrum of the wave equation in a bounded periodic heterogeneous m edium Omega when its period goes to zero. By means of a Bloch wave hom ogenization method, part of the limit spectrum was characterized as th e Bloch spectrum. In this second Note, we investigate the ''completnes s'' of our previous analysis, i.e. the discrepancy between the complet e limit spectrum and the Bloch spectrum. We prove that they differ onl y by a so-called boundary layer spectrum which is made of limit eigenv alues corresponding to sequences of eigenvectors which concentrate on the boundary of the domain Omega. In the case of a domain without boun dary (for example, a cube with periodic boundary conditions), the limi t spectrum does indeed coincide with the Bloch spectrum. The proof rel ies on a notion of Bloch measures which can be seen as ad hoc Wigner m easures in the context of semi-classical analysis.