Singularities of the resolvent at the thresholds of a stratified operator: a general method

摘要

Our problem is about propagation of waves in stratified strips. The operators are quite general, a typical example being a coupled elasto-acoustic operator H defined in R-2 X I where I is a bounded interval of R with coefficients depending only on z epsilon I. One applies the 'conjugate operator method' to an operator obtained by a spectral decomposition of the partial Fourier transform (H) over cap of H. Around each value of the spectrum (except the eigenvalues) including the thresholds, a conjugate operator may be constructed which ensures the 'good properties' of regularity for H. A limiting absorption principle is then obtained for a large class of operators at every point of the spectrum (except eigenvalues). If the point is a threshold, the limiting absorption principle is valid in a closed subspace of the usual one (namely L-s(2), with s > 1/2) and we are interested by the behaviour of R(z), z close to a threshold, applying in the usual space L-s(2), with s > 1/2 when z tends to the threshold. Copyright (C) 2004 John Wiley Sons, Ltd.