Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-24T00:04:41.593Z Has data issue: false hasContentIssue false
  • English
  • Français

The principle of limiting absorption for propagative systems in crystal optics with perturbations of long-range class

Published online by Cambridge University Press:  22 January 2016

Hideo Tamura
Hideo Tamura*
Affiliation:
Department of Engineering Mathematics Faculty of Engineering, Nagoya University, Furo-cho, Chikusa-ku, Nagoya 464, Japan
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The present paper is a continuation of [10] where we have proved the principle of limiting absorption for uniformly propagative systems with perturbations of long-range class. In this paper, we consider the Maxwell equation in crystal optics as an important example of non-uniformly propagative systems and, under the same assumptions on perturbations as in [10], we prove the principle of limiting absorption for the stationary problem associated with this equation by using a way similar to that in [10]. We here restrict our consideration to a very special class of non-uniformly propagative systems, but the method developed in this paper will be applicable to more general systems for which non-zero roots of characteristic equations of unperturbed systems are at most double. For another works on the spectral and scattering problems for non-uniformly propagative systems with perturbations of short-range class, see [1], [5], [6], [7] and [8], etc.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 84 , December 1981 , pp. 169 - 193
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1981

References

[ 1 ] Avila, G.S.S., Spectral resolution of differential operators associated with symmetric hyperbolic systems, Applicable Anal., 1 (1972), 283–299.Google Scholar
[ 2 ] Courant, R. and Hilbert, D., Methods of Mathematical Physics, 2, New York, Interscience Publishers, 1962.Google Scholar
[ 3 ] Hörmander, L., Pseudo-differential operators and hypoelliptic equations, Proc. Symposium of Singular Integrals, Amer. Math. Soc, 10 (1967), 138–183.Google Scholar
[ 4 ] Kumano-go, H., Pseudo-differential operators, Tokyo, Iwanami Publishers, 1974 (in Japanese).Google Scholar
[ 5 ] La Vita, J.A. Schulenberger, J. R. and Wilcox, C. H., The scattering theory of Lax and Phillips and wave propagation problems of classical physics, Applicable Anal., 3 (1973), 57–77.Google Scholar
[ 6 ] Murata, M., Rate of decay of local energy and wave operators for symmetric systems, J. Math. Soc. Japan, 31 (1979), 451–480.CrossRefGoogle Scholar
[ 7 ] Schechter, M., Scattering theory for elliptic systems, J. Math. Soc. Japan, 28 (1976), 71–79.Google Scholar
[ 8 ] Schulenberger, J.R., A local compactness theorem for wave propagation problems of classical physics, Indiana Univ. Math. J., 22 (1972), 429–433.Google Scholar
[ 9 ] Tamura, H., Spectral analysis for perturbed Laplace operators in cylindrical domains, Integral Equations and Operator Theory, 2 (1979), 69–115.Google Scholar
[10] Tamura, H., The principle of limiting absorption for uniformly propagative systems with perturbations of long-range class, Nagoya Math. J., 82 (1981), 141–174.Google Scholar