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Spectral analysis of an acoustic multistratified perturbed cylinder. (English) Zbl 0907.47046

Summary: We look into the operator of acoustic propagation \(H:=-\nabla\cdot \rho\nabla\), with Dirichlet condition, in a domain \(\Omega:= \Omega'\times \mathbb{R}\), \(\Omega'\) being bounded in \(\mathbb{R}^{n-1}\) \((n\geq 2)\). The function \(\rho\) is assumed to be real, greater than a strictly positive constant and belongs to \(L^\infty(\Omega)\). When \((-1)^j x_n\to \infty\) it is obtained by short-range and long-range perturbations of the functions \(\rho_j= \rho^{(j)}\otimes 1\) \((j= 1,2)\), where \(\rho^{(j)}\in L^\infty(\Omega')\) are real and bounded below by a strictly positive-constant. By using a variant of the conjugate operator method of E. Mourre [Commun. Math. Phys. 78, 391-408 (1981; Zbl 0489.47010)], and some ideas of the N-body problem in quantum mechanics, we develop the spectral analysis of the operator \(H\) and obtain a limiting absorption principle.

MSC:

47F05 General theory of partial differential operators
35P05 General topics in linear spectral theory for PDEs
47A10 Spectrum, resolvent
47A55 Perturbation theory of linear operators
47B25 Linear symmetric and selfadjoint operators (unbounded)

Citations:

Zbl 0489.47010
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