A global definition of quasinormal modes for Kerr-AdS Black Holes

The quasinormal frequencies of massive scalar fields on Kerr-AdS black holes are identified with poles of a certain meromorphic family of operators, once boundary conditions are specified at the conformal boundary. Consequently, the quasinormal frequencies form a discrete subset of the complex plane and the corresponding poles are of finite rank. This result holds for a broad class of elliptic boundary conditions, with no restrictions on the rotation speed of the black hole.Comment: 37 pages; minor changes. To appear in Ann. Inst. Fourie

Feynman propagators and Hadamard states from scattering data for the Klein-Gordon equation on asymptotically Minkowski spacetimes

We consider the massive Klein-Gordon equation on a class of asymptotically static spacetimes. We prove the existence and Hadamard property of the in and out states constructed by scattering theory methods. Assuming in addition that the metric approaches that of Minkowski space at infinity in a short-range way, jointly in time and space variables, we define Feynman scattering data and prove the Fredholm property of the Klein-Gordon operator with the associated Atiyah-Patodi-Singer boundary conditions. We then construct a parametrix (with compact remainder terms) for the Fredholm problem and prove that it is also a Feynman parametrix in the sense of Duistermaat and H{\"o}rmander

Breathers in oscillator chains with Hertzian interactions

We prove nonexistence of breathers (spatially localized and time-periodic oscillations) for a class of Fermi-Pasta-Ulam lattices representing an uncompressed chain of beads interacting via Hertz's contact forces. We then consider the setting in which an additional on-site potential is present, motivated by the Newton's cradle under the effect of gravity. Using both direct numerical computations and a simplified asymptotic model of the oscillator chain, the so-called discrete p-Schr\"odinger (DpS) equation, we show the existence of discrete breathers and study their spectral properties and mobility. Due to the fully nonlinear character of Hertzian interactions, breathers are found to be much more localized than in classical nonlinear lattices and their motion occurs with less dispersion. In addition, we study numerically the excitation of a traveling breather after an impact at one end of a semi-infinite chain. This case is well described by the DpS equation when local oscillations are faster than binary collisions, a situation occuring e.g. in chains of stiff cantilevers decorated by spherical beads. When a hard anharmonic part is added to the local potential, a new type of traveling breather emerges, showing spontaneous direction-reversing in a spatially homogeneous system. Finally, the interaction of a moving breather with a point defect is also considered in the cradle system. Almost total breather reflections are observed at sufficiently high defect sizes, suggesting potential applications of such systems as shock wave reflectors

Resonances, Radiation Damping and Instability in Hamiltonian Nonlinear Wave Equations

We consider a class of nonlinear Klein-Gordon equations which are Hamiltonian and are perturbations of linear dispersive equations. The unperturbed dynamical system has a bound state, a spatially localized and time periodic solution. We show that, for generic nonlinear Hamiltonian perturbations, all small amplitude solutions decay to zero as time tends to infinity at an anomalously slow rate. In particular, spatially localized and time-periodic solutions of the linear problem are destroyed by generic nonlinear Hamiltonian perturbations via slow radiation of energy to infinity. These solutions can therefore be thought of as metastable states. The main mechanism is a nonlinear resonant interaction of bound states (eigenfunctions) and radiation (continuous spectral modes), leading to energy transfer from the discrete to continuum modes. This is in contrast to the KAM theory in which appropriate nonresonance conditions imply the persistence of invariant tori. A hypothesis ensuring that such a resonance takes place is a nonlinear analogue of the Fermi golden rule, arising in the theory of resonances in quantum mechanics. The techniques used involve: (i) a time-dependent method developed by the authors for the treatment of the quantum resonance problem and perturbations of embedded eigenvalues, (ii) a generalization of the Hamiltonian normal form appropriate for infinite dimensional dispersive systems and (iii) ideas from scattering theory. The arguments are quite general and we expect them to apply to a large class of systems which can be viewed as the interaction of finite dimensional and infinite dimensional dispersive dynamical systems, or as a system of particles coupled to a field.Comment: To appear in Inventiones Mathematica