On stabilization in scattering of random and plane waves

We plan to prove the weighted energy decay i) for the Klein-Gordon equation with external potential, and ii) for the harmonic crystal coupled to the Klein-Gordon equation with external potential. For the proofs, we introduce a novel development of the Agmon-Jensen-Kato-Murata-Vainberg analytic theory of the resolvent for the considered models. We plan to apply the decay in the following directions: A. Statistical stabilization for the harmonic crystal coupled to the Klein-Gordon with external potentials. We consider random translation invariant initial data satisfying the mixing condition of the Rosenblatt or Ibragimov- Linnik. We will prove that the distribution of the solution at time t converges, in the long time limit, to the Gaussian measure. The result is a generalization of the Central Limit Theorem for the considered models. The proofs for unperturbed model, without the external potential, are based on the version of the Bernstein method of series. The extension to the perturbed model is given by a "duality method" using the "scattering theory in the mean" based on the weighted energy decay. B. Dynamical foundation of quantum differential cross section (QDCS). We introduce a novel mathematical model of the "spherical incident wave" produced by a space localized time periodic source. We will prove i) The long range asymptotics for the corresponding "spherical limiting amplitude". ii) The convergence of the "spherical" QDCS to the "plane" QDCS in the limit, when the source goes to infinity, for the Schrödinger and Klein-Gordon equations with external potentials. The convergence ii) gives the first dynamical justification of known formula for QDCS from its physical definition as the outgoing/incident flux for the case of the stationary fluxes: previously, the formula was considered as definition of the QDCS (the formulas (1.2) and (A.1.6) in V.Enss and B.Simon, Commun. Math. Phys. 76 (1980), 177-209). The advantage of spherical incident waves is that the corresponding initial state is identically zero while the one for the plane waves occupies the entire space that does not fit the spirit of scattering. The problem is discussed in M. Reed and B. Simon, "Methods of Modern Mathematical Physics", v. III, pp 355-357. C. Asymptotic stability of solitons. We plan to prove the long time convergence to a moving soliton for the solutions of relativistic nonlinear wave equations with initial data close to the solitary manifold. Similar results were obtained previously for translation invariant Schr\"odinger equations, and for the Klein-Gordon equations with external potentials. The extension to the relativistic case is based on the weighted energy decay. D. We plan to deliver the lecture course and publish lecture notes on the scattering theory and the quantum differential cross section, and to organize the conferences and Focused Semesters on the scattering and bound states in Quantum Mechanics and Quantum Electrodynamics.

I. We have proved the dispersion decay in weighted norms for the Klein-Gordon and Schroedinger eqns with scalar potentials, for discrete Klein-Gordon and Schrödinger equations and for magnetic Klein-Gordon and Schrödinger eqns. In particular, our results disprove the common prejudice on the universal slow decay of solutions to 1D wave equations. For the proof we show for the first time that the slow decay for the free 1D wave equation is caused exactly by the resonance at the edge point of the continuous spectrum. These results allow to develop further the stability theory for the nonlinear Hamilton PDEs.II. We have established the global attraction to solitary waves for the Klein-Gordon equation coupled to several nonlinear oscillators, and for the Dirac equation with mean field interaction. These results give the first mathematical model of the Bohr transitions to quantum stationary states, and disprove the common prejudice that these transitions do not allow a dynamical description. The proofs rely on a novel mathematical analysis of energy radiation with a novel application of the Titchmarsh convolution theorem. These results give new insight onto mathematical description of basic quantum phenomena (the Bohr transitions and de Broglie wave-particle duality).III. We have proved the asymptotic stability and scattering asymptotics for solitary waves for 1D relativistic Ginzburg-Landau eqns, for the 3D Dirac, Maxwell and wave eqns coupled to a particle, and for the 1D Schrödinger equn coupled to a nonlinear oscillator. These results were inspired by the problem of stability of elementary particles. The proofs rely on a novel extension of the Buslaev-Perelman-Sulem approach introduced in 1991-2003 in the context of the nonlinear translation-invariant Schrödinger equation. The problem of the extension to relativistic invariant equations remained open more than 20 years. One of the key ingredient in the proofs was our result on the dispersion decay for the linearized dynamics. Another key contribution was our theory of eigenfunction expansions which relies on a special version of the Krein-Langer theory of selfadjoint operators in the Hilbert spaces with indefinite metric.IV. We have established the convergence to equilibrium distribution for Dirac equations with potentials. Initial data are random functions satisfying the mixing condition of Rosenblatt or Ibragimov-Linnik. This result is a far reaching generalization of Central Limit Theorem to Hamiltonian PDEs. These results give an advance to the justification of equilibrium statistical physics and disprove the common prejudice that the convergence to statistical equilibrium is impossible in time-reversible Hamiltonian systems. The proofs rely on a development of our previous approach introduced in the context of the Klein-Gordon equation and recent results of N. Boussaid on the dispersion decay for the Dirac equation. The results give an advance to the justification of equilibrium statistical physics. V. We gave the dynamical justification of the scattering cross section. The problem has been stated in Reed & Simon book III. We found such justification for the first time in the context of the Schrödinger equation. Previous justifications relied on the random beams of particles. Our novel approach is completely different and relies on spherical incident waves produced by a localized source which models the heated cathode. We justify known formula for the differential cross section in double limit: first, the limiting amplitude principle holds for large times, and second, the spherical limiting amplitudes converge to plane scattering amplitudes when the source goes to infinity. Such purely wave theory should be useful for dynamical understanding of the probabilistic interpretation in quantum mechanics. The proofs rely on our results on the dispersion decay [2] and a novel application of the Ikebe uniqueness theorem for the Lippmann-Schwinger equation. An important ingredient is our novel refinement of the Povzner-Ikebe and Berezin-Schubin long-range asymptotics for the Coulomb potential.