×

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
PDFBibTeX XMLCite
Full Text: DOI

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: 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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.