The behavior of the analytically continued resolvent operator near and an application to energy decay

Author:
Kazuhiro Yamamoto

Journal:
Proc. Amer. Math. Soc. **115** (1992), 985-993

MSC:
Primary 35P05; Secondary 35L05, 47F05

MathSciNet review:
1089414

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Abstract: We shall study the behavior of the analytically continued resolvent operator for perturbations of in a neighborhood of . As an application, making use of Vainberg's argument, we shall show the local energy decay of solutions to generalized wave equations whose stationary problems are not positive definite.

**[1]**L. Hörmander,*Linear partial differential operators*, Springer-Verlag, New York, 1963.**[2]**H. Iwashita and Y. Shibata,*On the analyticity of spectral functions for some exterior boundary value problems*, Glas. Mat. Ser. III**23(43)**(1988), no. 2, 291–313 (English, with Serbo-Croatian summary). MR**1012030****[3]**R. B. Melrose and J. Sjöstrand,*Singularities of boundary value problems. II*, Comm. Pure Appl. Math.**35**(1982), no. 2, 129–168. MR**644020**, 10.1002/cpa.3160350202**[4]**James Ralston,*Note on the decay of acoustic waves*, Duke Math. J.**46**(1979), no. 4, 799–804. MR**552527****[5]**B. R. Vainberg,*On exterior elliptic problems polynomial depending on a spectral parameter, and the asymptotic behavior for large time of solutions of nonstationary problems*, Math. USSR Sb.**21**(1973), 221-239.**[6]**-,*On the short wave asymptotic behavior of solutions of stationary problems*, Russian Math. Surveys**30**(1975), 1-58.**[7]**G. N. Watson,*A treatise on the theory of Bessel functions*, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1995. Reprint of the second (1944) edition. MR**1349110****[8]**Kazuhiro Yamamoto,*Scattering theory and spectral representations for general wave equations with short range perturbations*, Canad. J. Math.**43**(1991), no. 2, 435–448. MR**1113765**, 10.4153/CJM-1991-026-0**[9]**-,*Existence of outgoing solutions for perturbations for**and applications to the scattering matrix*, Math. Proc. Cambridge Philos. Soc.**111**(1992).

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DOI:
https://doi.org/10.1090/S0002-9939-1992-1089414-2

Keywords:
Exterior problem,
resolvent operator,
energy decay

Article copyright:
© Copyright 1992
American Mathematical Society