Elliptic boundary value problems for Bessel operators, with applications to anti-de Sitter spacetimes

Gannot, Oran

Comptes Rendus. Mathématique, Tome 356 (2018) no. 10, pp. 988-1029.

Cette Note considère des problèmes aux limites pour une classe d'opérateurs elliptiques singuliers, qui apparaissent naturellement dans l'étude des espaces-temps asymptotiquement anti-de Sitter (aAdS). Ces problèmes impliquent un opérateur de Bessel singulier agissant dans la direction normale. Après avoir formulé une condition de Lopatinskiı̌, nous établissons des estimations elliptiques pour les fonctions dont le support est voisin du bord. La propriété de Fredholm suit d'hypothèses additionnelles à l'intérieur. Nous fournissons ici un cadre rigoureux pour l'analyse des modes sur les espaces-temps aAdS d'une large classe de conditions au bord, considérées en physique. Nous montrons que certains pinceaux d'opérateurs de Bessel ont un ensemble complet de fonctions propres.

This paper considers boundary value problems for a class of singular elliptic operators that appear naturally in the study of asymptotically anti-de Sitter (aAdS) spacetimes. These problems involve a singular Bessel operator acting in the normal direction. After formulating a Lopatinskiı̌ condition, elliptic estimates are established for functions supported near the boundary. The Fredholm property follows from additional hypotheses in the interior. This paper provides a rigorous framework for mode analysis on aAdS spacetimes for a wide range of boundary conditions considered in the physics literature. Completeness of eigenfunctions for some Bessel operator pencils is shown.

[1] Agmon, S. On the eigenfunctions and on the eigenvalues of general elliptic boundary value problems, Commun. Pure Appl. Math., Volume 15 (1962) no. 2, pp. 119-147

[2] Anderson, M.T. On the structure of asymptotically de Sitter and anti-de Sitter spaces, Adv. Theor. Math. Phys., Volume 8 (2004) no. 5, pp. 861-893

[3] Avis, S.J.; Isham, C.J.; Storey, D. Quantum field theory in anti-de Sitter space–time, Phys. Rev. D, Volume 18 (1978) no. 10, p. 3565

[4] Bachelot, A. The Klein–Gordon equation in the anti-de Sitter cosmology, J. Math. Pures Appl., Volume 96 (2011) no. 6, pp. 527-554

[5] Balasubramanian, V.; Buchel, A.; Green, S.R.; Lehner, L.; Liebling, S.L. Holographic thermalization, stability of anti-de Sitter space, and the Fermi–Pasta–Ulam paradox, Phys. Rev. Lett., Volume 113 (2014) no. 7

[6] Berkooz, M.; Sever, A.; Shomer, A. “Double-trace” deformations, boundary conditions and spacetime singularities, J. High Energy Phys., Volume 2002 (2002) no. 05

[7] Bizoń, P. Is AdS stable?, Gen. Relativ. Gravit., Volume 46 (2014) no. 5, p. 1724

[8] Bizoń, P.; Rostworowski, A. Weakly turbulent instability of anti-de Sitter spacetime, Phys. Rev. Lett., Volume 107 (2011) no. 3

[9] Bizoń, P.; Maliborski, M.; Rostworowski, A. Resonant dynamics and the instability of anti-de Sitter spacetime, Phys. Rev. Lett., Volume 115 (2015) no. 8

[10] Breitenlohner, P.; Freedman, D.Z. Positive energy in anti-de Sitter backgrounds and gauged extended supergravity, Phys. Lett. B, Volume 115 (1982) no. 3, pp. 197-201

[11] Breitenlohner, P.; Freedman, D.Z. Stability in gauged extended supergravity, Ann. Phys., Volume 144 (1982) no. 2, pp. 249-281

[12] Browder, F.E. On the eigenfunctions and eigenvalues of the general linear elliptic differential operator, Proc. Natl. Acad. Sci. USA, Volume 39 (1953) no. 5, pp. 433-439

[13] Buchel, A.; Green, S.R.; Lehner, L.; Liebling, S.L. Conserved quantities and dual turbulent cascades in anti-de Sitter spacetime, Phys. Rev. D, Volume 91 (2015) no. 6

[14] Cardoso, V.; Dias, O.J.C.; Hartnett, G.S.; Lehner, L.; Santos, J.E. Holographic thermalization, quasinormal modes and superradiance in Kerr--AdS, J. High Energy Phys., Volume 2014 (2014) no. 4

[15] del Mar González Nogueras, M.; Qing, J. Fractional conformal laplacians and fractional Yamabe problems, Anal. PDE, Volume 6 (2013) no. 7, pp. 1535-1576

[16] Dias, O.J.C.; Santos, J.E. Boundary conditions for Kerr–AdS perturbations, J. High Energy Phys., Volume 2013 (2013) no. 10

[17] Dias, O.J.C.; Horowitz, G.T.; Marolf, D.; Santos, J.E. On the nonlinear stability of asymptotically anti-de Sitter solutions, Class. Quantum Gravity, Volume 29 (2012) no. 23

[18] Dias, O.J.C.; Horowitz, G.T.; Santos, J.E. Gravitational turbulent instability of anti-de Sitter space, Class. Quantum Gravity, Volume 29 (2012) no. 19

[19] Dunford, N.; Schwartz, J. Linear Operators. Part 2: Spectral Theory. Self Adjoint Operators in Hilbert Space, Interscience Publishers, 1963

[20] Enciso, A.; Kamran, N. A singular initial-boundary value problem for nonlinear wave equations and holography in asymptotically anti-de Sitter spaces, J. Math. Pures Appl., Volume 103 (2015) no. 4, pp. 1053-1091

[21] Evans, L.C. Partial Differential Equations, Graduate Studies in Mathematics, American Mathematical Society, 2010

[22] Gannot, O. Existence of quasinormal modes for Kerr–AdS Black Holes, Ann. Henri Poincaré, Volume 18 (2017), pp. 2757-2788

[23] Gannot, O. A global definition of quasinormal modes for Kerr–AdS Black Holes, Ann. Inst. Fourier, Volume 68 (2018) no. 3, pp. 1125-1167

[24] Graham, C.R.; Lee, J.M. Einstein metrics with prescribed conformal infinity on the ball, Adv. Math., Volume 87 (1991) no. 2, pp. 186-225

[25] Grisvard, P. Espaces intermédiaires entre espaces de Sobolev avec poids, Ann. Sc. Norm. Super. Pisa, Cl. Sci., Volume 17 (1963) no. 3, pp. 255-296

[26] Guillarmou, C. Meromorphic properties of the resolvent on asymptotically hyperbolic manifolds, Duke Math. J., Volume 129 (2005) no. 1, pp. 1-37

[27] Holzegel, G. Well-posedness for the massive wave equation on asymptotically anti-de Sitter spacetimes, J. Hyperbolic Differ. Equ., Volume 9 (2012) no. 02, pp. 239-261

[28] Holzegel, G.; Smulevici, J. Stability of Schwarzschild-AdS for the spherically symmetric Einstein–Klein--Gordon system, Commun. Math. Phys., Volume 317 (2013) no. 1, pp. 205-251

[29] Holzegel, G.H.; Warnick, C.M. Boundedness and growth for the massive wave equation on asymptotically anti-de Sitter black holes, J. Funct. Anal., Volume 266 (2014) no. 4, pp. 2436-2485

[30] Holzegel, G.; Luk, J.; Smulevici, J.; Warnick, C. Asymptotic properties of linear field equations in anti-de Sitter space, 2015 (preprint) | arXiv

[31] Horowitz, G.T.; Hubeny, V.E. Quasinormal modes of AdS black holes and the approach to thermal equilibrium, Phys. Rev. D, Volume 62 (2000) no. 2

[32] Ishibashi, A.; Wald, R.M. Dynamics in non-globally-hyperbolic static spacetimes: III. Anti-de Sitter spacetime, Class. Quantum Gravity, Volume 21 (2004) no. 12, p. 2981

[33] Keldyš, M.V. On the characteristic values and characteristic functions of certain classes of non-self-adjoint equations, Dokl. Akad. Nauk SSSR, Volume 77 (1951), pp. 11-14

[34] Konoplya, R.A.; Zhidenko, A. Quasinormal modes of black holes: from astrophysics to string theory, Rev. Mod. Phys., Volume 83 (2011) no. 3, p. 793

[35] Kozlov, V.A.; Maz'ya, V.G.; Rossmann, J. Elliptic Boundary Value Problems in Domains with Point Singularities, Fields Institute Monographs, American Mathematical Society, 1997

[36] Lions, J.-L. Sur les espaces d'interpolation; dualité, Math. Scand., Volume 9 (1961) no. 1b, pp. 147-177

[37] Lions, J.-L.; Magenes, E. Non-Homogeneous Boundary Value Problems and Applications, vol. 1, Springer-Verlag, Berlin, Heidelberg, 2012

[38] Markus, A.S. Introduction to the Spectral Theory of Polynomial Operator Pencils, Translations of Mathematical Monographs, American Mathematical Society, 2012

[39] Mazzeo, R.; Melrose, R.B. Meromorphic extension of the resolvent on complete spaces with asymptotically constant negative curvature, J. Funct. Anal., Volume 75 (1987) no. 2, pp. 260-310

[40] Mazzeo, R.; Pacard, F. Constant curvature foliations in asymptotically hyperbolic spaces, Rev. Mat. Iberoam., Volume 27 (2011) no. 1, pp. 303-333

[41] Mazzeo, R.; Vertman, B. Elliptic theory of differential edge operators, ii: boundary value problems, Indiana Univ. Math. J., Volume 63 (2014), pp. 1911-1955

[42] Olver, F.W.J. Introduction to Asymptotics and Special Functions, Elsevier Science, 2014

[43] Rafe, M. Elliptic theory of differential edge operators I, Commun. Partial Differ. Equ., Volume 16 (1991) no. 10, pp. 1615-1664

[44] Roitberg, Y. Elliptic Boundary Value Problems in the Spaces of Distributions, vol. 384, Springer, Netherlands, 1996

[45] Schechter, M. Remarks on elliptic boundary value problems, Commun. Pure Appl. Math., Volume 12 (1959) no. 4, pp. 561-578

[46] Vasy, A. The wave equation on asymptotically anti de Sitter spaces, Anal. PDE, Volume 5 (2012) no. 1, pp. 81-144

[47] Vasy, A. Microlocal analysis of asymptotically hyperbolic and Kerr–de Sitter spaces (with an appendix by Semyon Dyatlov), Invent. Math., Volume 194 (2013) no. 2, pp. 381-513

[48] Vasy, A. Microlocal analysis of asymptotically hyperbolic spaces and high-energy resolvent estimates, Math. Sci. Res. Inst. Publ., Volume 60 (2013), p. 487

[49] Warnick, C.M. The massive wave equation in asymptotically AdS spacetimes, Commun. Math. Phys., Volume 321 (2013) no. 1, pp. 85-111

[50] Warnick, C.M. On quasinormal modes of asymptotically anti-de Sitter black holes, Commun. Math. Phys., Volume 333 (2015) no. 2, pp. 959-1035

[51] Winstanley, E. Classical super-radiance in Kerr–Newman–anti-de Sitter black holes, Phys. Rev. D, Volume 64 (2001) no. 10

[52] Witten, E. Multi-trace operators, boundary conditions, and AdS/CFT correspondence, 2001 (preprint) | arXiv

[53] Yakubov, S. Completeness of Root Functions of Regular Differential Operators, Monographs and Surveys in Pure and Applied Mathematics, Taylor & Francis, 1993

[54] Zettl, A. Sturm–Liouville Theory, Mathematical Surveys and Monographs, American Mathematical Society, 2010