On the meromorphic continuation of the Mellin transform associated to the wave kernel on Riemannian symmetric spaces of the non-compact type

Ali Hassani; Ali Hassani

doi:10.3934/math.2024716

AIMS Mathematics

2024, Volume 9, Issue 6: 14731-14746. doi: 10.3934/math.2024716

Previous Article Next Article

Research article

On the meromorphic continuation of the Mellin transform associated to the wave kernel on Riemannian symmetric spaces of the non-compact type

Ali Hassani ^{1,2
,
,}

1.
Department of Mathematics, Al-Leith University College, Umm Al-Qura University, Mecca 24231, Saudi Arabia
2.
Institut Supérieur des Sciences Appliquées et de Technologie, Université de Sousse, 4003 Sousse, Tunisia

Received: 04 March 2024 Revised: 09 April 2024 Accepted: 15 April 2024 Published: 23 April 2024
MSC : 53C35, 53Z05, 22E30, 43A85

We considered the Mellin transform assigned to the convolution wave kernel associated to the Laplace-Beltrami operator on higher rank Riemannian symmetric spaces of the non-compact type. The occurrence of the analyticity strip of this transform can be deduced directly from the pointwise kernel estimates. Using the zeta function techniques, we established its meromorphic extension to the entire complex plane $ {{\Bbb C}} $ with simple poles on the real line.
- Riemannian symmetric spaces,
- wave kernel,
- Mellin transform,
- asymptotic expansion,
- meromorphic continuation,
- zeta function
Citation: Ali Hassani. On the meromorphic continuation of the Mellin transform associated to the wave kernel on Riemannian symmetric spaces of the non-compact type[J]. AIMS Mathematics, 2024, 9(6): 14731-14746. doi: 10.3934/math.2024716

Related Papers:

Abstract

We considered the Mellin transform assigned to the convolution wave kernel associated to the Laplace-Beltrami operator on higher rank Riemannian symmetric spaces of the non-compact type. The occurrence of the analyticity strip of this transform can be deduced directly from the pointwise kernel estimates. Using the zeta function techniques, we established its meromorphic extension to the entire complex plane $ {{\Bbb C}} $ with simple poles on the real line.

References

[1]	J. P. Anker, H. W. Zhang, Wave equation on general noncompact symmetric spaces, arXiv Preprint, https://doi.org/10.48550/arXiv.2010.08467
[2]	J. Bertrand, P. Bertrand, J. P. Ovarlez, The transforms and applications handbook, 2 Eds., CRC Press, 2000.
[3]	I. Brevik, A. Bytsenko, A. Goncalves, F. L. Williams, Zeta function regularization and the thermodynamic potential for quantum fields in symmetric spaces, J. Phys. A-Math. Gen., 31 (1988), 4437–4448. https://doi.org/10.1088/0305-4470/31/19/009 doi: 10.1088/0305-4470/31/19/009
[4]	Y. Brychkov, O. I. Marichev, N. V. Savischenko, Handbook of Mellin transforms, CRC Press, 2019. https://doi.org/10.1201/9780429434259
[5]	A. Bytsenko, E. Elizalde, S. Odintsov, A. Romeo, S. Zerbini, Zeta regularization techniques with applications, World Scientific, 1994. https://doi.org/10.1142/2065
[6]	A. Bytsenko, A. Goncalves, F. L. Williams, The conformal anomaly in general rank 1 symmetric spaces and associated operator product, Mod. Phys. Lett. A, 13 (1998), 99–108. https://doi.org/10.1142/S0217732398000140 doi: 10.1142/S0217732398000140
[7]	A. Bytsenko, F. L. Williams, Asymptotics of the heat kernel on rank-1 locally symmetric spaces, J. Phys. A-Math. Gen., 32 (1999), 5773–5779. https://doi.org/10.1088/0305-4470/32/31/303 doi: 10.1088/0305-4470/32/31/303
[8]	A. Bytsenko, A. Goncalves, S. Zerbini, One-loop effective potential for scalar and vector fields on higher-dimensional noncommutative flat manifolds, Mod. Phys. Lett., A16 (2001), 1479–1486. https://doi.org/10.1142/S0217732301004765 doi: 10.1142/S0217732301004765
[9]	R. Cahn, J. Wolf, Zeta functions and their asymptotic expansions for compact symmetric spaces of rank one, Comment. Math. Helv., 51 (1976), 1–21. https://doi.org/10.1007/BF02568140 doi: 10.1007/BF02568140
[10]	R. Cahn, P. Gilkey, J. Wolf, Heat equation, proportionality principle, and volume of fundamental domains, In: Mathematical Physics and Applied Mathematics, Dordrecht: Springer, 1976, 43–54. https://doi.org/10.1007/978-94-010-1508-0-6
[11]	R. Campores, Harmonic analysis and propagators on homogeneous spaces, Phys. Rep., 196 (1990), 1–134. https://doi.org/10.1016/0370-1573(90)90120-Q doi: 10.1016/0370-1573(90)90120-Q
[12]	R. Camporesi, $\zeta$-function regularization of one-loop effective potentials in anti-de Sitter spacetime, Phys. Rev. D, 43 (1991), 3958–3965. https://doi.org/10.1103/PhysRevD.43.3958 doi: 10.1103/PhysRevD.43.3958
[13]	R. Camporesi, A. Higuchi, Arbitrary spin effective potentials in anti-de Sitter spacetime, Phys. Rev. D, 7 (1993), 3339–3344. https://doi.org/10.1103/PhysRevD.47.3339 doi: 10.1103/PhysRevD.47.3339
[14]	R. Camporesi, On the analytic continuation of the Minakshisundaram-Pleijel zeta function for compact symmetric spaces of rank one, J. Math. Anal. Appl., 214 (1997), 524–549. https://doi.org/10.1006/jmaa.1997.5588 doi: 10.1006/jmaa.1997.5588
[15]	G. Cognola, E. Elizalde, S. Zerbini, One-loop effective potential from higher-dimensional AdS black holes, Phys. Lett. B, 585 (2004), 155–162. https://doi.org/10.1016/j.physletb.2004.02.004 doi: 10.1016/j.physletb.2004.02.004
[16]	M. Cowling, S. Guilini, S. Meda, Oscillatory multipliers related to the wave equation on noncompact symmetric spaces, J. Lond. Math. Soc., 66 (2002), 691–709. https://doi.org/10.1112/S0024610702003563 doi: 10.1112/S0024610702003563
[17]	R. Gangolli, V. S. Varadarajan, Harmonic analysis of spherical functions on real reductive groups, Springer Science & Business Media, 2012. https://doi.org/10.1007/978-3-642-72956-0
[18]	T. F. Godoy, Minakshisundaram-Pleijel coefficients for compact locally symmetric spaces of classical type with non-positive sectional curvature, Ph. D. Thesis, Argentina: National University of Córdoba, 1987.
[19]	T. F. Godoy, R. J. Miatello, F. L. Williams, The local zeta function for symmetric spaces of non-compact type, J. Geom. Phys., 61 (2011), 125–136. https://doi.org/10.1016/j.geomphys.2010.08.008 doi: 10.1016/j.geomphys.2010.08.008
[20]	I. Gradshteyn, I. Ryzhik, Table of integrals, series, and products, Corrected and Enlarged Edition, A. Jeffrey Eds., Academic Press, 1980.
[21]	A. Hassani, Wave equation on Riemannian symmetric spaces, J. Math. Phys., 52 (2011). https://doi.org/10.1063/1.3567167
[22]	S. Helgason, Geometric analysis on symmetric spaces, American Mathematical Society, 1994. https://doi.org/10.1090/surv/039/02
[23]	S. Helgason, Groups and geometric analysis: Integral geometry, invariant differential operators, and spherical functions, American Mathematical Society, 2022.
[24]	S. Lang, Complex analysis, 2 Eds., Springer Verlag, 1985. https://doi.org/10.1007/978-1-4757-1871-3
[25]	R. Miatello, On the Minakshisundaram-Pleijel coefficients for the vector-valued heat kernel on compact locally symmetric spaces of negative curvature, T. Am. Math. Soc., 260 (1980), 1–33. https://doi.org/10.2307/1999874 doi: 10.2307/1999874
[26]	S. Minakshisundaram, A. Pleijel, Some properties of the eigenfunctions of the Laplace operator on Riemannian manifolds, Can. J. Math., 1 (1949), 242–256. https://doi.org/10.4153/CJM-1949-021-5 doi: 10.4153/CJM-1949-021-5
[27]	R. B. Paris, D. Kaminsky, Asymptotics and Mellin-Barnes integrals, Cambridge University Press, 2001. https://doi.org/10.1017/CBO9780511546662
[28]	V. S. Varadarajan, The method of stationnary phase and application to geometry and analysis on Lie groups, In: Algebraic and Analytic Methods in Representation Theory, San Diego: Academic Press, 1997,167–242. https://doi.org/10.1016/B978-012625440-2/50005-7
[29]	F. L. Williams, Meromorphic continuation of Minakshisundaram-Pleijel series for semisimple Lie groups, Pac. J. Math., 182 (1998), 137–156. https://doi.org/10.2140/PJM.1998.182.137 doi: 10.2140/PJM.1998.182.137
[30]	F. L. Williams, Minakshisundaram-Pleijel coefficients for non-compact higher symmetric spaces, Anal. Math. Phys., 10 (2020). https://doi.org/10.1007/s13324-020-00396-x

Reader Comments

Your name:*

Email:*
© 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)