On Pointwise Decay of Linear Waves on a Schwarzschild Black Hole Background

Abramowitz, M., Stegun, I.: Handbook of mathematical functions with formulas, graphs, and mathematical tables. Reprint of the 1972 edition. New York: Dover Publications, Inc., 1992

Alexandrova I., Bony J., Ramond T.: Resolvent and scattering matrix at the maximum of the potential. Serdica Math. J 34(1), 267–310 (2008)

Amrein, W., Boutet de Monvel, A., Georgescu, V.: C 0-groups, commutator methods and spectral theory of N-body Hamiltonians. Progress in Mathematics, 135. Basel: Birkhäuser Verlag, 1996

Andersson, L., Blue, P.: Hidden symmetries and decay for the wave equation on the Kerr spacetime. Preprint http://arXiv.org/abs/0908.2265v2 [math.AP], 2009

Balogh C.B.: Asymptotic expansions of the modified Bessel function of the third kind of imaginary order. SIAM J. Appl. Math 15, 1315–1323 (1967)

Bony, J.-F., Fujiié, S., Ramond, T., Zerzeri, M.: Microlocal solutions of Schrödinger equations at a maximum point of the potential, Preprint 2009

Bony J.-F., Häfner D.: Decay and non-decay of the local energy for the wave equation on the de Sitter-Schwarzschild metric. Commun. Math.Phys. 282(3), 697–719 (2008)

Briet P., Combes J.-M., Duclos P.: On the location of resonances for Schrödinger operators in the semiclassical limit. II. Barrier top resonances. Comm. Par. Diff. Eqs 12(2), 201–222 (1987)

Costin, O., Donninger, R., Schlag, W., Tanveer, S.: Semiclassical low energy scattering for one-dimensional Schrödinger operators with exponentially decaying potentials. To appear in Annales Henri Poincaré. http://arXiv.org/abs/1105.4221v1 [math.SP], 2011

Costin O., Schlag W., Staubach W., Tanveer S.: Semiclassical analysis of low and zero energy scattering for one-dimensional Schrödinger operators with inverse square potentials. J. Funct. Anal. 255(9), 2321–2362 (2008)

Dafermos, M., Rodnianski, I.: Lectures on black holes and linear waves. Preprint 2008, http://arXiv.org/abs/0811.0354v1 [gr-qc], 2008

Dafermos M., Rodnianski I.: The red-shift effect and radiation decay on black hole spacetimes. Comm. Pure Appl. Math 62(7), 859–919 (2009)

Dafermos M., Rodnianski I.: A proof of the uniform boundedness of solutions to the wave equation on slowly rotating Kerr backgrounds. Invent. Math 185, 467–559 (2011)

Dafermos, M., Rodnianski, I.: Decay for solutions of the wave equation on Kerr exterior spacetimes I-II: The cases |a| < < M or axisymmetry. Preprint http://arXiv.org/abs/1010.5132v1 [gr-qc], 2010

Davies E.B.: Spectral Theory and Differential Operators. Cambridge Univ. Press, Cambridge (1995)

Donninger R., Schlag W.: Decay estimates for the one-dimensional wave equation with an inverse power potential. Int. Math. Res. Not. 2010(22), 4276–4300 (2010)

Donninger R., Schlag W., Soffer A.: A proof of Price’s law on Schwarzschild black hole manifolds for all angular momenta. Adv. Math. 226(1), 484–540 (2011)

Finster F., Kamran N., Smoller J., Yau S.-T.: Decay of solutions of the wave equation in the Kerr geometry. Commun. Math. Phys 264(2), 465–503 (2006)

Gérard C., Grigis A.: Precise estimates of tunneling and eigenvalues near a potential barrier. J. Diff. Eqs 72(1), 149–177 (1988)

Graf G.: The Mourre estimate in the semiclassical limit. Lett. Math. Phys. 20(1), 47–54 (1990)

Gustafson S., Sigal I.M.: Mathematical concepts of quantum mechanics. Springer-Verlag, Universitext Berlin (2003)

Hawking S., Ellis G.: The large scale structure of space-time Cambridge Monographs on Mathematical Physics No 1. Cambridge University Press, London-New York (1973)

Helffer, B., Sjöstrand, J.: Semiclassical analysis of Harper’s equation III. Bull. Soc. Math. France, Memoire 39, 1990

Hislop P., Nakamura S.: Semiclassical resolvent estimates. Ann.Inst. H. Poincaré Phys. Théor 51(2), 187–198 (1989)

Hunziker W., Sigal I.M.: Time-dependent scattering theory of N-body quantum systems. Rev. Math. Phys. 12(8), 1033–1084 (2000)

Hunziker W., Sigal I.M., Soffer A.: Minimal escape velocities. Comm. Par. Diff. Eqs 24(11–12), 2279–2295 (1999)

Ivrii V.Ja., Sigal I.M.: Asymptotics of the ground state energies of large Coulomb systems. Ann. of Math 138(2), 243–335 (1993)

Kay B., Wald R.: Linear stability of Schwarzschild under perturbations which are nonvanishing on the bifurcation 2-sphere. Class. Quan. Grav 4(4), 893–898 (1987)

Marzuola J., Metcalfe J., Tataru D., Tohaneanu M.: Strichartz estimates on Schwarzschild black hole backgrounds. Commun. Math. Phys 293(1), 37–83 (2010)

Metcalfe, J., Tataru, D., Tohaneanu, M.: Price’s Law on Nonstationary Spacetimes. Preprint http://arXiv.org/abs/1104.5437v2 [math.AP], 2011

Luk J.: Improved decay for solutions to the linear wave equation on a Schwarzschild black hole. Ann. Henri Poincaré 11(5), 805–880 (2010)

Luk, J.: A Vector Field Method Approach to Improved Decay for Solutions to the Wave Equation on a Slowly Rotating Kerr Black Hole. Preprint, http://arXiv.org/abs/1009.0671v2 [gr-qc], 2011

Miller, P.D.: Applied asymptotic analysis. Graduate Studies in Mathematics, 75. Providence, RI: Amer. Math. Soc., 2006

Nakamura S.: Semiclassical resolvent estimates for the barrier top energy. Commun. Par. Diff. Eq. 16(4/5), 873–883 (1991)

Olver, F.W.J.: Asymptotics and Special Functions, Wellesley, MA: A K Peters, Ltd. 1997

Mourre E.: Absence of singular continuous spectrum for certain selfadjoint operators. Commun. Math. Phys. 78(3), 391–408 (1980/81)

Price R.: Nonspherical perturbations of relativistic gravitational collapse. I. Scalar and gravitational perturbations. Phys. Rev. D 5(3), 2419–2438 (1972)

Price R.: Nonspherical perturbations of relativistic gravitational collapse. II. Integer-spin, zero-rest-mass fields. Phys. Rev. D 5(3), 2439–2454 (1972)

Ramond T.: Semiclassical study of quantum scattering on the line. Commun. Math. Phys. 177(1), 221–254 (1996)

Schlag W., Soffer A., Staubach W.: Decay for the wave and Schrödinger evolutions on manifolds with conical ends, Part I. Trans. Amer. Math. Soc. 362(1), 19–52 (2010)

Schlag W., Soffer A., Staubach W.: Decay for the wave and Schrödinger evolutions on manifolds with conical ends, Part II. Trans. Amer. Math. Soc. 362(1), 289–318 (2010)

Sigal I.M., Soffer A.: Long-range many-body scattering. Invent. Math 99, 115–143 (1990)

Sigal, I.M., Soffer, A.: Local decay and velocity bounds. Preprint, Princeton University, 1988

Sjöstrand, J.: Semiclassical Resonances Generated by Nondegenerate Critical Points. In: Pseudodifferential Operators (Oberwolfach, 1986), Lecture Notes in Math., Vol. 1256, Berlin: Springer-Verlag, 1987, pp. 402–429

Skibsted E.: Propagation estimates for N-body Schroedinger operators. Commun. Math. Phys 142(1), 67–98 (1991)

Tataru, D.: Local decay of waves on asymptotically flat stationary space-times, Preprint 2009, http://arXiv.org/abs/0910.5290v2 [math.AP], 2010

Tataru D., Tohaneanu M.: A local energy estimate on Kerr black hole backgrounds. Int. Math. Res. Not. IMRN 2011(2), 248–292 (2011)

Tohaneanu, M.: Strichartz estimates on Kerr black hole backgrounds. Preprint http://arXiv.org/abs/0910.1545v1 [math.AP], 2009

Wald R.: General relativity. University of Chicago Press, Chicago, IL (1984)