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ON THE RESOLVENT OF THE LAPLACE–BELTRAMI OPERATOR IN HYPERBOLIC SPACE

Part of: Spectral theory and eigenvalue problems

Published online by Cambridge University Press:  24 April 2015

GUSEIN SH. GUSEINOV
GUSEIN SH. GUSEINOV*
Affiliation:
Department of Mathematics, Atilim University, 06836 Incek, Ankara, Turkey email huseyin.huseyin@atilim.edu.tr
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Abstract

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In this paper, a detailed description of the resolvent of the Laplace–Beltrami operator in $n$-dimensional hyperbolic space is given. The resolvent is an integral operator with the kernel (Green’s function) being a solution of a hypergeometric differential equation. Asymptotic analysis of the solution of this equation is carried out.

Keywords

hyperbolic spaceLaplace–Beltrami operatorGreen’s functionresolvent

MSC classification

Secondary: 35P05: General topics in linear spectral theory
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 99 , Issue 2 , October 2015 , pp. 186 - 206
Copyright
© 2015 Australian Mathematical Publishing Association Inc. 

References

Akhiezer, N. I. and Glazman, I. M., Theory of Linear Operators in Hilbert Space, Vol. 1 (Ungar, New York, 1961), Vol. 2, 1963.Google Scholar
Beardon, A. F., The Geometry of Discrete Groups (Springer, New York, 1983).CrossRefGoogle Scholar
Faddeev, L. D., ‘Expansion in eigenfunctions of the Laplace operator in the fundamental domain of a discrete group in the Lobachevskii plane’, Trans. Moscow Math. Soc. 17 (1967), 357–386.Google Scholar
Gohberg, I. C. and Krein, M. G., Introduction to the Theory of Linear Nonselfadjoint Operators, Translations of the Mathematical Monographs, 8 (American Mathematical Society, Providence, RI, 1969).Google Scholar
Lang, S., SL(2, R) (Addison-Wesley, Reading, MA, 1975).Google Scholar
Sz.-Nagy, B. and Foias, C., Harmonic Analysis of Operators on Hilbert Space (North-Holland, Budapest, 1970).Google Scholar
Olver, F. W. J., Lozier, D. W., Boiswert, R. F. and Clark, C. W. (eds), NIST Handbook of Mathematical Functions (Cambridge University Press, Cambridge, 2010).Google Scholar
Titchmarsh, E. C., Eigenfunction Expansions Associated with Second-Order Differential Equations, Part II (Oxford University Press, Oxford, 1958).Google Scholar
Venkov, A. B., ‘Expansions in automorphic eigenfunctions of the Laplace–Beltrami operator in classical symmetric spaces of rank one and the Selberg trace formula’, Proc. Steklov Inst. Math. 125 (1973), 6–55.Google Scholar
Venkov, A. B., Spectral Theory of Automorphic Functions and its Applications (Kluwer, Dordrecht, 1990).CrossRefGoogle Scholar