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.


35P05 General topics in linear spectral theory for PDEs
35J20 Variational methods for second-order elliptic equations
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[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: Springer-Verlag Berlin
[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) · Zbl 0687.35060
[9] Haus, H. A., Waves and Fields in Optoelectronics (1984), Prentice-Hall: Prentice-Hall Englewood Cliffs, NJ
[10] Kato, T., Perturbation Theory for Linear Operators (1966), Springer-Verlag: 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., Methods of Modern Mathematical Physics: IV—Analysis of Operators (1979), Academic Press: Academic Press New York · Zbl 0517.47006
[14] Tolstoy, I.; Clay, C. S., Oceanic Acoustics (1966), McGraw-Hill: 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: Springer-Verlag New York · Zbl 0543.76107
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