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Singularities of the resolvent at the thresholds of a stratified operator: a general method
Author(s) -
Gado Bio Soumarou Chabi,
Durand Marc,
Goudjo Côme
Publication year - 2004
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.503
Subject(s) - mathematics , operator (biology) , mathematical analysis , spectrum (functional analysis) , eigenvalues and eigenvectors , bounded function , resolvent , complex conjugate , bounded operator , spectral theory of ordinary differential equations , pure mathematics , quasinormal operator , finite rank operator , banach space , quantum mechanics , physics , biochemistry , chemistry , repressor , transcription factor , gene
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.