A perturbative method for the spectral analysis of an acoustic multistratified strip
Elisabeth Croc, Yves Dermenjian · Mathematical Methods in the Applied Sciences · 1998
We consider the acoustic propagator A=−∇·c2∇ in the strip Ω={(x, z)∈ℝ2∣0 0. The celerity c depends for large ∣x∣ only on the variable z and describes the stratification of Ω: it is assumed to be in L∞(Ω), bounded from below by cmin>0, such that there exists M>0 with c(x, z)=c1(z) if xM. We look at the propagator A as a ‘perturbation’ of the free propagators Aj in Ω associated to the velocities cj, j=1, 2, and implement a ‘perturbative’ method, adapting ideas of Majda and Vainberg. The spectrum of A is defined in section 2, a limiting absorption principle is proved in section 3 outside of a countable set Γ(A). The points of Γ(A) can only accumulate at the left of the thresholds of the free propagators. The needed material about Aj, j=1, 2, and some technical estimates for A are given in Appendix. © 1998 B. G. Teubner Stuttgart—John Wiley & Sons, Ltd.