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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
Affiliation:
Department of Engineering Mathematics Faculty of Engineering, Nagoya University, Furo-cho, Chikusa-ku, Nagoya 464, Japan
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Extract

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
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), 283299.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), 138183.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), 5777.CrossRefGoogle Scholar
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[ 7 ] Schechter, M., Scattering theory for elliptic systems, J. Math. Soc. Japan, 28 (1976), 7179.CrossRefGoogle Scholar
[ 8 ] Schulenberger, J.R., A local compactness theorem for wave propagation problems of classical physics, Indiana Univ. Math. J., 22 (1972), 429433.CrossRefGoogle Scholar
[ 9 ] Tamura, H., Spectral analysis for perturbed Laplace operators in cylindrical domains, Integral Equations and Operator Theory, 2 (1979), 69115.CrossRefGoogle Scholar
[10] Tamura, H., The principle of limiting absorption for uniformly propagative systems with perturbations of long-range class, Nagoya Math. J., 82 (1981), 141174.CrossRefGoogle Scholar

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