Rate of decay of local energy and wave operators for symmetric systems

Minoru MURATA

doi:10.2969/jmsj/03130451

Minoru MURATA

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JOURNAL FREE ACCESS

1979 Volume 31 Issue 3 Pages 451-480

DOI https://doi.org/10.2969/jmsj/03130451

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Published: 1979 Received: September 17, 1977 Available on J-STAGE: October 20, 2006 Accepted: - Advance online publication: - Revised: -
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Date of correction: October 20, 2006 Reason for correction: - Correction: CITATION Details: Wrong : 1) G.S.S. Avilla, Spectral resolution of differential operators associated with symmetric hyperbolic systems, Applicable Anal., 1 (1972), 283-299.
2) P.L. Butzer and H. Berens, Semi-groups of operators and approximations, Springer, 1967.
3) V.G. Deic, A local stationary method in the theory of scattering for a pair of spaces, Problems of Math. Phys., No. 6, 76-90, lzdat. Leningrad Univ., Leningrad, 1973 (in Russian).
4) T. Ikebe, Scattering theory for uniformly propagative systems, Proc. Internat. Confer. on Functional Anal. and Related Topics, Tokyo (1969), 225-230.
5) T. Ikebe, Remarks on non-elliptic stationary wave propagation problems, to appear.
6) T. Kato, Scattering theory with two Hilbert spaces, J. Functional Analysis, 1 (1967), 342-369.
7) T. Kato, Perturbation theory for linear operators, Springer, 1966.
8) T. Kato and S.T. Kuroda, The abstract theory of scattering, Rocky Mountain J. Math., 1 (1971), 127-171.
9) H. Komatsu, Fractional powers of operators, II, Pacific J. Math., 21 (1967), 89-111.
10) S.T. Kuroda, Scattering theory for differential operators, I and II, J. Math.Soc. Japan, 25 (1973), 75-104 and 222-234.
11) J.A. Lavita, J.R. Schulenberger and C.H. Wilcox, The scattering theory of Lax and Phillips and wave propagation problems of classical physics, Applicable Anal., 3 (1973), 57-77.
12) P.D. Lax and R.S. Phillips, Scattering theory, Academic Press, 1967.
13) J.L. Lions, Remarque sur les espaces d'interpolation, Colloques internationaux C.N.R.S., Les équations aux dérivées partielles, 1962, 75-86.
14) S. Lojasiewitz, Sur le problème de la division, Studia Math., 18 (1959), 87-136.
15) T. Muramatsu, On imbedding theorems for Besov spaces of functions defined in general regions, Publ. Res. Inst. Math. Sci., Kyoto Univ., 9 (1971), 261-285.
16) M. Murata, Asymptotic behaviors at infinity of solutions of certain linear partial differential equations, J. Fac. Sci. Univ. Tokyo, Sect. IA Math., 23 (1976), 107-148.
17) M. Murata, Rate of decay of local energy and spectral properties of elliptic operators, to appear.
18) M. Murata, Finiteness of eigenvalues of self-adjoint elliptic operators, J. Fac. Sci. Univ. Tokyo, Sect. IA Math., 25 (1978).
19) J.R. Schulenberger, A local compactness theorem for wave propagation problems of classical physics, Indiana Univ. Math. J., 22-5 (1972), 429-433.
20) T. Suzuki, The limiting absorption principle and spectral theory for a certain non-selfadjoint operator and its application, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 20 (1973), 401-412.
21) B.R. Vainberg, On the short wave asymptotic behaviour of solutions of stationary problems and the asymptotic behaviour as t→∞ of solutions of non-stationary problems, Russian Math. Surveys, 30: 2 (1975), 1-58.
22) H. Whitney, Elementary structure of real algebraic variety, Ann. of Math., 66 (1957), 545-556.
23) C.H. Wilcox, Wave operators and asymptotic solutions of wave propagation problems of classical physics, Arch. Rational Mech. Anal., 22 (1966), 37-78.
24) K. Yajima, The limiting absorption principle for uniformly propagative systems, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 21 (1974), 119-131.
25) O. Yamada, On the principle of limiting absorption for the Dirac operator, Publ. Res. Inst. Math. Sci., Kyoto Univ., 8 (1972/1973), 576-606.

Right : [1] G. S. S. Avilla, Spectral resolution of differential operators associated with symmetric hyperbolic systems, Applicable Anal., 1 (1972), 283-299.
[2] P. L. Butzer and H. Berens, Semi-groups of operators and approximations, Springer, 1967.
[3] V. G. Deic, A local stationary method in the theory of scattering for a pair of spaces, Problems of Math. Phys., No. 6, 76-90, lzdat. Leningrad Univ., Leningrad, 1973 (in Russian).
[4] T. Ikebe, Scattering theory for uniformly propagative systems, Proc. Internat. Confer. on Functional Anal. and Related Topics, Tokyo (1969), 225-230.
[5] T. Ikebe, Remarks on non-elliptic stationary wave propagation problems, to appear.
[6] T. Kato, Scattering theory with two Hilbert spaces, J. Functional Analysis, 1 (1967), 342-369.
[7] T. Kato, Perturbation theory for linear operators, Springer, 1966.
[8] T. Kato and S. T. Kuroda, The abstract theory of scattering, Rocky Mountain J. Math., 1 (1971), 127-171.
[9] H. Komatsu, Fractional powers of operators, II, Pacific J. Math., 21 (1967), 89-111.
[10] S. T. Kuroda, Scattering theory for differential operators, I and II, J. Math. Soc. Japan, 25 (1973), 75-104 and 222-234.
[11] J. A. Lavita, J. R. Schulenberger and C. H. Wilcox, The scattering theory of Lax and Phillips and wave propagation problems of classical physics, Applicable Anal., 3 (1973), 57-77.
[12] P. D. Lax and R. S. Phillips, Scattering theory, Academic Press, 1967.
[13] J. L. Lions, Remarque sur les espaces d'interpolation, Colloques internationaux C. N. R. S., Les équations aux dérivées partielles, 1962, 75-86.
[14] S. Lojasiewitz, Sur le problème de la division, Studia Math., 18 (1959), 87-136.
[15] T. Muramatsu, On imbedding theorems for Besov spaces of functions defined in general regions, Publ. Res. Inst. Math. Sci., Kyoto Univ., 9 (1971), 261-285.
[16] M. Murata, Asymptotic behaviors at infinity of solutions of certain linear partial differential equations, J. Fac. Sci. Univ. Tokyo, Sect. IA Math., 23 (1976), 107-148.
[17] M. Murata, Rate of decay of local energy and spectral properties of elliptic operators, to appear.
[18] M. Murata, Finiteness of eigenvalues of self-adjoint elliptic operators, J. Fac. Sci. Univ. Tokyo, Sect. IA Math., 25 (1978).
[19] J.R. Schulenberger, A local compactness theorem for wave propagation problems of classical physics, Indiana Univ. Math. J., 22-5 (1972), 429-433.
[20] T. Suzuki, The limiting absorption principle and spectral theory for a certain non-selfadjoint operator and its application, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 20 (1973), 401-412.
[21] B. R. Vainberg, On the short wave asymptotic behaviour of solutions of stationary problems and the asymptotic behaviour as t→∞ of solutions of non-stationary problems, Russian Math. Surveys, 30: 2 (1975), 1-58.
[22] H. Whitney, Elementary structure of real algebraic variety, Ann. of Math., 66 (1957), 545-556.
[23] C.H. Wilcox, Wave operators and asymptotic solutions of wave propagation problems of classical physics, Arch. Rational Mech. Anal., 22 (1966), 37-78.
[24] K. Yajima, The limiting absorption principle for uniformly propagative systems, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 21 (1974), 119-131.
[25] O. Yamada, On the principle of limiting absorption for the Dirac operator, Publ. Res. Inst. Math. Sci., Kyoto Univ., 8 (1972/1973), 576-606.

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