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

✍ Scribed by Bio Soumarou Chabi Gado; Marc Durand; Côme Goudjo

• Publisher: John Wiley and Sons
• Year: 2004
• Tongue: English
• Weight: 188 KB
• Volume: 27
• Category: Article
• ISSN: 0170-4214
• DOI: 10.1002/mma.503

✦ Synopsis

Abstract

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, with s>½) and we are interested by the behaviour of R(z), z close to a threshold, applying in the usual space L, with s>½ when z tends to the threshold. Copyright © 2004 John Wiley & Sons, Ltd.