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 timeperiodic 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. Section 3: Existence theory Section 4: Isolation of the key resonant terms and formulation as a coupled finite and infinite dimensional dynamical system Section 5: Dispersive Hamiltonian normal form Section 6: Asymptotic behavior of solutions of perturbed normal form equations Section 7: Asymptotic behavior of solutions of the nonlinear Klein Gordon equation Section 8: Summary and discussionIt is interesting to contrast our results with those known for Hamiltonian partial differential equations for a function u(x, t), where x varies over a compact spatial domain, e.g. periodic or Dirichlet boundary conditions [4], [17], [37]. For nonlinear wave equations of the form, (1.1), with periodic boundary conditions in x, KAM type results have been proved; invariant tori, associated with a nonresonance condition persist under small perturbations. The nonresonance hypotheses of such results fail in the current context , a consequence of the continuous spectrum associated with unbounded spatial domains. In our situation, non-vanishing resonant coupling (condition (1.8)) provides the mechanism for the radiative decay and therefore nonpersistence of localized periodic solutions.Remarks:

We consider a class of nonlinear Schr6dinger equations (conservative and dispersive systems) with localized and dispersive solutions. We obtain a class of initial conditions, for which the asymptotic behavior (t ~ + oo) of solutions is given by a linear combination of nonlinear bound state (time periodic and spatially localized solution) of the equation and a purely dispersive part (decaying to zero with time at the free dispersion rate). We also obtain a result of asymptotic stability type: given data near a nonlinear bound state of the system, there is a nonlinear bound state of nearby energy and phase, such that the difference between the solution (adjusted by a phase) and the latter disperses to zero. It turns out that in general, the time-period (and energy) of the localized part is different for t ~ + ov from that+for t- .-or. Moreover the solution acquires an extra constant asymptotic phase e '~- .

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We prove L p estimates for charge transfer Hamiltonians, including matrix and inhomogeneous generalizations; such equations appear naturally in the study of multi-soliton systems.

We develop an approach to scattering theory for generalized N -body systems. In particular we consider a general class of three quasi-particle systems, for which we prove Asymptotic Completeness.

The Schwarzschild and Reissner-Nordstrøm solutions to Einstein's equations describe space-times which contain spherically symmetric black holes. We consider solutions to the linear wave equation in the exterior of a fixed black hole space-time of this type. We show that for solutions with initial data which decay at infinity and at the bifurcation sphere, a weighted L 6 norm in space decays like t −1/3 . This weight vanishes at the event horizon, but not at infinity. To obtain this control, we require only an loss of angular derivatives.

Recently, it has been shown that the wave equation for a scalar field on the exterior part of the Schwarzschild manifold satisfies local decay estimates useful for scattering theory and global existence [2]. The extension for the linearized Einstein equation is considered here. In 1957, Regge and Wheeler investigated spin 2 tensor fields on the Schwarzschild manifold [4]. They classified such fields into two types, which they called even and odd. For the odd fields, they were able to reduce the problem to an equation for a scalar field very similar to the wave equation for scalar fields on the Schwarzschild manifold. In 1970, Zerilli extended their results to include the even case; although, the equation for the even case is significantly more complicated and bears less resemblance to the wave equation for a scalar field [9]. Teukolsky has done a related reduction for the rotating Kerr black hole [6] which has been used to investigate the stability of the black holes [8].This paper extends the local decay estimate for the scalar wave equation of [2] to the Regge-Wheeler equation. Many of the proofs used here follow [2]. We obtain the following for r * the standard Regge-Wheeler co-ordinate and β > 3 2 , there is a constant C, depending on the initial condition through the energy norm, so that Co-ordinates and equationsThe Schwarzschild manifold describes a static black hole solution to the Einstein equation. The exterior of the black hole is most easily described by (t, r, θ, φ) ∈ R × (2M, ∞) × S 2 with the metricTo simplify the problem, Regge and Wheeler ([4]) introduced a new radial co-ordinate, r * , satisfyingThis allows the definition of a space like manifoldThe old co-ordinate r is now treated as a function of r * . In these new co-ordinates, the Regge-Wheeler equation for a scalar field u : R×M → R which determines the behavior of the odd-type tensor fields isü + Hu = 0 (1.4) 1

We prove a strict lower bound on the entropy produced when independent random variables are summed and rescaled. Using this, we develop an approach to central limit theorems from a dynamical point of view in which the entropy is a Lyapunov functional governing approach to the Gaussian limit. This dynamical approach naturally extends to cover dependent variables, and leads to new results in pure probability theory as well as in statistical mechanics. It also provides a unified framework within which many previous results are easily derived.