The nonlinear scattering and stability results of Soffer and Weinstein (Comm. Math. Phys. (1990)) are extended to the case of anisotropic potentials and data. The range of nonlinearities for which the theory is shown to be valid is also extended considerably.
We prove for a class of nonlinear Schrödinger systems (NLS) having two nonlinear bound states that the (generic) large time behavior is characterized by decay of the excited state, asymptotic approach to the nonlinear ground state and dispersive radiation. Our analysis elucidates the mechanism through which initial conditions which are very near the excited state branch evolve into a (nonlinear) ground state, a phenomenon known as ground state selection. Key steps in the analysis are the introduction of a particular linearization and the derivation of a normal form which reflects the dynamics on all time scales and yields, in particular, nonlinear Master equations. Then, a novel multiple time scale dynamic stability theory is developed. Consequently, we give a detailed description of the asymptotic behavior of the two bound state NLS for all small initial data. The methods are general and can be extended to treat NLS with more than two bound states and more general nonlinearities including those of Hartree-Fock type.
Price's Law states that linear perturbations of a Schwarzschild black hole fall off as t −2ℓ−3 for t → ∞ provided the initial data decay sufficiently fast at spatial infinity. Moreover, if the perturbations are initially static (i.e., their time derivative is zero), then the decay is predicted to be t −2ℓ−4 . We give a proof of t −2ℓ−2 decay for general data in the form of weighted L 1 to L ∞ bounds for solutions of the Regge-Wheeler equation. For initially static perturbations we obtain t −2ℓ−3 . The proof is based on an integral representation of the solution which follows from self-adjoint spectral theory. We apply two different perturbative arguments in order to construct the corresponding spectral measure and the decay bounds are obtained by appropriate oscillatory integral estimates.
We prove sharp pointwise t −3 decay for scalar linear perturbations of a Schwarzschild black hole without symmetry assumptions on the data. We also consider electromagnetic and gravitational perturbations for which we obtain decay rates t −4 , and t −6 , respectively. We proceed by decomposition into angular momentum ℓ and summation of the decay estimates on the Regge-Wheeler equation for fixed ℓ. We encounter a dichotomy: the decay law in time is entirely determined by the asymptotic behavior of the Regge-Wheeler potential in the far field, whereas the growth of the constants in ℓ is dictated by the behavior of the Regge-Wheeler potential in a small neighborhood around its maximum. In other words, the tails are controlled by small energies, whereas the number of angular derivatives needed on the data is determined by energies close to the top of the Regge-Wheeler potential. This dichotomy corresponds to the well-known principle that for initial times the decay reflects the presence of complex resonances generated by the potential maximum, whereas for later times the tails are determined by the far field. However, we do not invoke complex resonances at all, but rely instead on semiclassical Sigal-Soffer type propagation estimates based on a Mourre bound near the top energy.1 1 The notation a± stands for a± ε where ε > 0 is arbitrary (the choice determines the constants involved). Also, instead of (/ ∇ 10 , / ∇ 9 ) in (1.3) one needs less, namely (/ ∇ σ+1 , / ∇ σ ) where σ > 8 is arbitrary, see the proof in Section 5 for details.2 From the point of view of the decay estimates in [17], these values need to be excluded as they are precisely the ones that give rise to a zero energy resonance.
An important class of resonance problems involves the study of perturbations of systems having embedded eigenvalues in their continuous spectrum. Problems with this mathematical structure arise in the study of many physical systems, e.g. the coupling of an atom or molecule to a photon-radiation field, and Auger states of the helium atom, as well as in spectral geometry and number theory. We present a dynamic (time-dependent) theory of such quantum resonances. The key hypotheses are (i) a resonance condition which holds generically (non-vanishing of the Fermi golden rule) and (ii) local decay estimates for the unperturbed dynamics with initial data consisting of continuum modes associated with an interval containing the embedded eigenvalue of the unperturbed Hamiltonian. No assumption of dilation analyticity of the potential is made. Our method explicitly demonstrates the flow of energy from the resonant discrete mode to continuum modes due to their coupling. The approach is also applicable to nonautonomous linear problems and to nonlinear problems. We derive the time behavior of the resonant states for intermediate and long times. Examples and applications are presented. Among them is a proof of the instability of an embedded eigenvalue at a threshold energy under suitable hypotheses.
An inequality relating averages of generalized correlations to averages of generalized susceptibilities for Gaussian field distributions is presented. This inequality is applied to random-field systems to prove under the assumption of a continuous transition the (tree level) decoupling of the quenched two-point function. By assumption of only a power-law divergence, a lower bound for T) is obtained. It rules out the possibility that some recent experimental and numerical results reflect equilibrium properties near a continuous transition.
The Nonlinear Schrödinger Equation (NLSE) with a random potential is motivated by experiments in optics and in atom optics and is a paradigm for the competition between the randomness and nonlinearity. The analysis of the NLSE with a random (Anderson like) potential has been done at various levels of control: numerical, analytical and rigorous. Yet, this model equation presents us with a highly inconclusive and often contradictory picture. We will describe the main recent results obtained in this field and propose a list of specific problems to focus on, that we hope will enable to resolve these outstanding questions.
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