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

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 × I where I is a bounded interval of ℝ with coefficients depending only on z∈ I . One applies the ‘conjugate operator method’ to an operator obtained by a spectral decomposition of the partial Fourier transform Ĥ 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>½) 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>½ when z tends to the threshold. Copyright © 2004 John Wiley & Sons, Ltd.