ORIGINAL ARTICLE
Resonances of the d'Alembertian on the anti-de Sitter space
Simon Roby,
Corresponding Author
Simon Roby
Institut Elie Cartan de Lorraine, Université de Lorraine, Metz, France
Correspondence
Simon Roby, Institut Elie Cartan de Lorraine, Université de Lorraine, 57070 Metz, France.
Email: [email protected]
Search for more papers by this authorSimon Roby,
Corresponding Author
Simon Roby
Institut Elie Cartan de Lorraine, Université de Lorraine, Metz, France
Correspondence
Simon Roby, Institut Elie Cartan de Lorraine, Université de Lorraine, 57070 Metz, France.
Email: [email protected]
Search for more papers by this authorFirst published: 30 June 2024
Abstract
We consider the action of the d'Alembertian on functions on the pseudo-Riemannian three-dimensional anti-de Sitter space. We determine the resonances of this operator. With each resonance one can associate a residue representation. We give an explicit description of these representations via Langlands parameters.
REFERENCES
- 1N. B. Andersen, Paley–Wiener theorems for hyperbolic spaces, J. Funct. Anal. 179 (2001), no. 1, 66–119.
- 2S. Dyatlov and M. Zworski, Mathematical theory of scattering resonances, Graduate Studies in Mathematics, vol. 200, American Mathematical Society, Providence, RI, 2019.
10.1090/gsm/200 Google Scholar
- 3J. Frahm and S. Polyxeni, Resonances and residue operators for pseudo-Riemannian hyperbolic spaces, J. Mat. Pures Appl. 177 (2023), 178–197.
- 4L. Guillopé and M. Zworski, Upper bounds on the number of resonances for non-compact Riemann surfaces, J. Funct. Anal. 129 (1995), no. 2, 364–389.
- 5E. M. II Harrell, Perturbation theory and atomic resonances since Schrödinger's time, Spectral theory and mathematical physics. A Festschrift in honor of Barry Simon's 60th birthday. Quantum field theory, statistical mechanics, and nonrelativistic quantum systems. Based on the SimonFest Conference, Pasadena, CA, USA, March 27–31, 2006, American Mathematical Society, Providence, RI, 2007, pp. 227–248.
- 6H. Chandra, Harmonic analysis on real reductive groups. III. The Maass–Selberg relations and the Plancherel formula, Ann. Math. (2) 104 (1976), no. 1, 117–201.
- 7P. D. Hislop, Fundamentals of scattering theory and resonances in quantum mechanics, Cubo 14 (2012), no. 3, 1–39.
10.4067/S0719-06462012000300001 Google Scholar
- 8J. Hilgert, A. Pasquale, and T. Przebinda, Resonances for the Laplacian: the cases and (except with odd), Geometric methods in physics, Trends Math, Birkhäuser, Cham, 2016, pp. 159–182.
- 9J. Hilgert, A. Pasquale, and T. Przebinda, Resonances for the Laplacian on products of two rank one Riemannian symmetric spaces, J. Funct. Anal. 272 (2017), no. 4, 1477–1523.
- 10J. Hilgert, A. Pasquale, and T. Przebinda, Resonances for the Laplacian on Riemannian symmetric spaces: the case of , Represent. Theory 21 (2017), 416–457.
- 11J. Hilgert and A. Pasquale, Resonances and residue operators for symmetric spaces of rank one, J. Math. Pures Appl. 91 (2009), no. 5, 495–507.
- 12A. W. Knapp, Representation theory of semisimple groups: an overview based on examples, Princeton Mathematical Series, Princeton University Press, 2001.
- 13S. Lang, , Graduate texts in mathematics, vol. 105, Springer-Verlag, New York, 1985. Reprint of the 1975 edition.
- 14R. Mazzeo and A. Vasy, Analytic continuation of the resolvent of the Laplacian on symmetric spaces of noncompact type, J. Funct. Anal. 228 (2005), no. 2, 311–368.
- 15R. J. Miatello and C. E. Will, The residues of the resolvent on Damek–Ricci spaces, Proc. Am. Math. Soc. 128 (2000), no. 4, 1221–1229.
- 16T. Przebinda, The oscillator duality correspondence for the pair O(2,2), Sp(2,R). ProQuest LLC, Ann Arbor, MI, Ph.D. thesis, Yale University, 1987.
- 17S. Roby, Resonances of the Laplace operator on homogeneous vector bundles on symmetric spaces of real rank one, thesis, Université de Lorraine, Metz, 2021.
- 18S. Roby, Resonances of the Laplace operator on homogeneous vector bundles on symmetric spaces of real rank-one, Adv. Math. 408 (2022), 108555.
- 19A. Strohmaier, Analytic continuation of resolvent kernels on noncompact symmetric spaces, Math. Z. 250 (2005), no. 2, 411–425.
- 20V. S. Varadarajan, An introduction to harmonic analysis on semisimple Lie groups, Camb. Stud. Adv. Math., vol. 16, Cambridge University Press, Cambridge, 1989.
- 21D. A. Vogan, Jr. Representations of real reductive Lie groups, Progress in Mathematics, vol. 15, Birkhäuser, Boston, MA, 1981.
- 22N. R. Wallach, Harmonic analysis on homogeneous spaces, Pure and Applied Mathematics, vol. 19, Marcel Dekker, Inc., New York, 1973.
- 23G. Warner, Harmonic analysis on semi-simple Lie groups. II, Springer-Verlag, New York, 1972. Die Grundlehren der mathematischen Wissenschaften, Band 189.
10.1007/978-3-642-51640-5 Google Scholar
- 24C. E. Will, The meromorphic continuation of the resolvent of the Laplacian on line bundles over , Pacific J. Math. 209 (2003), no. 1, 157–173.
