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The principle of limiting absorption for uniformly propagative systems 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
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Extract

The aim of this paper is to establish the principle of limiting absorption for uniformly propagative systems Λ(x, Dx) = E(x)-1 AjDj, Dj = — i∂/∂xj , with perturbations of long-range class, where the perturbation of long-range class, roughly speaking, means that E(x) approaches to E0, E0 being the N × N identity matrix, as | x | → ∞ with order O(| x |), 0 < δ ≦ 1. (The more precise assumptions will be stated below and we require some additional assumptions on the derivatives of E(x).) The spectral and scattering problem for uniformly propagative systems was first formulated by Wilcox [10]. Since then, the principle of limiting absorption has been proved by many authors ([5], [7], [8], [11] etc.). The perturbations discussed in their works belong to the short-range class with δ > 1.

Type
Research Article
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1981

References

[1] Hörmander, L., Pseudo-differential operators and hypoelliptic equations, Proc. Symposium of Singular Integrals, Amer. Math. Soc., 10 (1967), 138183.CrossRefGoogle Scholar
[2] Ikebe, T. and Saitō, Y., Limiting absorption method and absolute continuity for the Schrödinger operators, J. Math. Kyoto Univ., 7 (1972), 513542.CrossRefGoogle Scholar
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[4] Lions, J. L. and Magenes, E., Problèmes aux limites non homogènes et applications, Vol. I, Dunod, Paris, 1968.Google Scholar
[5] Mochizuki, K., Spectral and scattering theory for symmetric hyperbolic system in an exterior domain, Publ. RIMS, Kyoto Univ., 5 (1969), 219258.CrossRefGoogle Scholar
[6] Mochizuki, K. and Uchiyama, J., Radiation conditions and spectral theory for 2-body Schrödinger operators with oscillating long-range potentials I, —the principle of limiting absorption—, J. Math. Kyoto Univ., 18 (1978), 377408.CrossRefGoogle Scholar
[7] Schulenberger, J. R. and Wilcox, C. H., The limiting absorption principle and spectral theory for steady state wave propagation in inhomogeneous anisotropic media, Arch. Rat. Mech. Anal., 41 (1972), 4665.CrossRefGoogle Scholar
[8] Suzuki, T., The limiting absorption principle for a certain non-self ad joint operator and its application, J. Fac. Sci. Univ. Tokyo, Sect. I A, 20 (1973), 401412.Google Scholar
[9] Tamura, H., Spectral analysis for perturbed Laplace operator in cylindrical domains, Integral Equation and Operator Theory, 2 (1979), 69115.CrossRefGoogle Scholar
[10] Wilcox, C. H., Wave operators and asymptotic solutions of wave propagation problem of classical physics, Arch. Rat. Mech. Anal., 22 (1966), 3778.CrossRefGoogle Scholar
[11] Yajima, K., The limiting absorption principle for uniformly propagative systems, J. Fac. Sci. Univ. Tokyo, Sect. IA, 21 (1974), 119131.Google Scholar

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