Abstract
The Tauberian theorem of B. M. Levitan reduces the question of the asymptotics of the spectral function of the Laplace operator on a smooth Riemannian manifold with boundary to the problem of constructing the asymptotics of a Green function possessing certain additional properties. The paper is devoted to the construction of the appropriate Green function for the case of a geodesically concave boundary.
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Literature cited
V. M. Babich and B. M. Levitan, “The problem of focusing and the asymptotics of the spectral function of the Laplace-Beltrami operator,” Dokl. Akad. Nauk SSSR,230, No. 5, 1017–1020 (1976).
V. M. Babich and B. M. Levitan, “The problem of focusing and the asymptotics of the spectral function of the Laplace-Beltrami operator,” J. Sov. Mat.,22, No. 1, (1983).
B. M. Levitan, Appendix to E. C. Titchmarsh, Eigenfunction Expansions Associated with Second-Order Differential Equations, 2nd ed., Oxford Univ. Press (1962).
V. S. Buslaev, “On the asymptotic behavior of the spectral characteristics of exterior problems for the Schrödinger operator,” Izv. Akad. Nauk SSSR, Ser. Mat.,39, No. 1, 149–235 (1975).
V. M. Babich, “On the shortwave asymptotics of the solution of the problem of a point source in an inhomogeneous medium,” Zh. Vychisl. Mat. Mat. Fiz.,5, No. 5, 949–951 (1965).
Additional information
Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 78, pp. 128–133, 1978.
The author thanks V. M. Babich for his help with the work.
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Kurylev, Y.V. A shortwave source near a smooth, convex hypersurface and the spectral function of the Laplace operator on a Riemannian manifold. J Math Sci 22, 1082–1086 (1983). https://doi.org/10.1007/BF01305290
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DOI: https://doi.org/10.1007/BF01305290