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Spectral analysis for optical fibres and stratified fluids. I: The limiting absorption principle. (English) Zbl 0731.35069
The authors prove a limiting absorption principle for the operator $H=- c^ 2(z)\rho (z)\nabla_ z\cdot \rho^{-1}(z)\nabla_ z$ on $${\mathbb{R}}^ n$$. The functions $$\rho$$ and c are supposed to satisfy many very precise assumptions concerning their regularity and their behaviour at infinity. The proof is based on Mourre theory, and the main step consists in proving a Mourre estimate on H.

##### MSC:
 35P05 General topics in linear spectral theory for PDEs 35J20 Variational methods for second-order elliptic equations
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##### References:
 [1] Agmon, S, Spectral properties of Schrödinger operators and scattering theory, Ann. scuola norm. sup. Pisa. cl. sci (2), 2, 151-218, (1975) · Zbl 0315.47007 [2] Ben-Artzi, M; Dermenjian, Y; Guillot, J.-C, Acoustic waves in perturbed stratified media, Comm. partial differential equations, 14, No. 4, 479-517, (1989) · Zbl 0675.35065 [3] Cycon, H.L; Froese, R.G; Kirsh, W; Simon, B, Schrödinger operators, (1975), Springer-Verlag Berlin [4] {\scS. DeBievre, P. D. Hislop, and I. M. Sigal}, Scattering for the wave equation on manifolds, in preparation. · Zbl 0778.58064 [5] DeBièvre, S; Pravica, D.W, Spectral analysis for optical fibres and stratified fluids II: absence of imbedded eigenvalues, (1990), University of Toronto, preprint · Zbl 0850.35067 [6] Dermenjian, Y; Guillot, J.-C, Théorie spectrale de la propagation des ondes dans un milieu stratifié perturbé, J. differential equations, 62, 357-409, (1986) · Zbl 0611.35063 [7] Froese, R.G; Herbst, I, A new proof of the Mourre estimate, Duke math. J., 49, No. 4, 1075-1085, (1982) · Zbl 0514.35025 [8] Froese, R.G; Hislop, P.D, Spectral analysis of second-order elliptic operators on non-compact manifolds, Duke math. J., 58, No. 1, 103-129, (1988) [9] Haus, H.A, Waves and fields in optoelectronics, (1984), Prentice-Hall Englewood Cliffs, NJ [10] Kato, T, Perturbation theory for linear operators, (1966), Springer-Verlag New York · Zbl 0148.12601 [11] Mourre, E, Absence of singular continuous spectrum for certain self-adjoint operators, Comm. math. phys., 78, 391-408, (1981) · Zbl 0489.47010 [12] Perry, P; Sigal, I.M; Simon, B, Spectral analysis of N-body Schrödinger operators, Ann. of math., 114, 519-567, (1981) · Zbl 0477.35069 [13] Reed, M; Simon, B; Reed, M; Simon, B; Reed, M; Simon, B; Reed, M; Simon, B, Methods of modern mathematical physics: IV—analysis of operators, (1979), Academic Press New York · Zbl 0517.47006 [14] Tolstoy, I; Clay, C.S, Oceanic acoustics, (1966), McGraw-Hill New York [15] Weder, R, Spectral and scattering theory in perturbed stratified fluids, J. math. pures appl., 64, 149-173, (1985) · Zbl 0597.76074 [16] Weder, R, Spectral and scattering theory in perturbed stratified fluids II, J. differential equations, 65, 109-131, (1986) · Zbl 0658.76078 [17] Weder, R, The limiting absorption principle at thresholds, J. math. pures appl., 67, 313-338, (1988) · Zbl 0611.76090 [18] Wilcox, C.H, Sound propagation in stratified fluids, (1984), Springer-Verlag New York · Zbl 0543.76107
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