Comptes Rendus
no. 10
Partial differential equations
Elliptic boundary value problems for Bessel operators, with applications to anti-de Sitter spacetimes
[Problèmes aux limites elliptiques pour les opérateurs de Bessel, avec applications aux espaces-temps anti-de Sitter]

Présenté par : the Editorial Board

Oran Gannot1
1 Department of Mathematics, Lunt Hall, Northwestern University, Evanston, IL 60208, USA
Comptes Rendus. Mathématique, Volume 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.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2018.08.003
Oran Gannot 1

1 Department of Mathematics, Lunt Hall, Northwestern University, Evanston, IL 60208, USA 
@article{CRMATH_2018__356_10_988_0,
     author = {Oran Gannot},
     title = {Elliptic boundary value problems for {Bessel} operators, with applications to anti-de {Sitter} spacetimes},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {988--1029},
     publisher = {Elsevier},
     volume = {356},
     number = {10},
     year = {2018},
     doi = {10.1016/j.crma.2018.08.003},
     language = {en},
}
TY  - JOUR
AU  - Oran Gannot
TI  - Elliptic boundary value problems for Bessel operators, with applications to anti-de Sitter spacetimes
JO  - Comptes Rendus. Mathématique
PY  - 2018
SP  - 988
EP  - 1029
VL  - 356
IS  - 10
PB  - Elsevier
DO  - 10.1016/j.crma.2018.08.003
LA  - en
ID  - CRMATH_2018__356_10_988_0
ER  - 
%0 Journal Article
%A Oran Gannot
%T Elliptic boundary value problems for Bessel operators, with applications to anti-de Sitter spacetimes
%J Comptes Rendus. Mathématique
%D 2018
%P 988-1029
%V 356
%N 10
%I Elsevier
%R 10.1016/j.crma.2018.08.003
%G en
%F CRMATH_2018__356_10_988_0
Oran Gannot. Elliptic boundary value problems for Bessel operators, with applications to anti-de Sitter spacetimes. Comptes Rendus. Mathématique, Volume 356 (2018) no. 10, pp. 988-1029. doi : 10.1016/j.crma.2018.08.003. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2018.08.003/

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[20] A. Enciso; N. Kamran 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] L.C. Evans Partial Differential Equations, Graduate Studies in Mathematics, American Mathematical Society, 2010

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Cité par Sources :

Commentaires - Politique