Long Time Behavior and Scattering for Dirac Equation

The main goal of the project is the long time behavior of solutions to linear and nonlinear Dirac equation. We plan the following directions of research: Long time decay We plan to prove the dispersive long time decay in weighed energy norms for solutions to discrete and continuous Schrödinger, Klein-Gordon and Dirac equations with electromagnetic potential. For the proofs, we plan to extend the Agmon-Jensen-Kato-Murata analytic theory of the resolvent of the corresponding operators. Namely, we plan to prove a smoothness and high energy decay of the resolvent and the (generalized) Puiseux expansions at the thresholds. We plan to obtain the high energy decay by factorization of the magnetic Schrödinger operator in its spectral representation. The factorization relies on absence of the singular spectrum. Soliton scattering asymptotics We consider Dirac equation coupled to nonlinear oscillator, and relativistic nonlinear Dirac equation in one space dimension. We suppose that the initial data are close to the solitary manifold. We plan to prove that the solution, asymptotically in time, is the sum of some solitary wave and dispersive wave which is a solution to the free Dirac equation. The remainder decays in a global norm. We also plan to obtain similar result for discrete versions of the equations. The proofs will rely on the dispersive long time decay for the linearised Dirac equation which will be obtained in the framework of this project. Convergence to equilibrium distribution We consider discrete and continuous electromagnetic Dirac equations with 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 a Gaussian measure. This result is a generalization of the Central Limit Theorem for hyperbolic PDEs. The proofs for unperturbed model, without the potentials, will rely on the corresponding development of the Bernstein method of series. The extension to the perturbed model will be done with the "scattering theory in the mean" based on the dispersive long time decay for electromagnetic Dirac equation which will be obtained in the framework of this project. Lecture Notes We plan to publish lecture notes on the Agmon-Jensen-Kato-Murata spectral and scattering theory for the Schrödinger equations, and its extension to the Klein-Gordon equations.

The aim of this project was an analysis of the long-time asymptotic behavior of solutions to linear and nonlinear hyperbolic partial differential equations. These investigations are motivated by mathematical problems of quantum mechanics.The results of the project are published in 1 monograph, 12 papers in refereed journals, 2 preprints, 9 abstracts in conference proceedings and were presented at 9 conferences and in 4 seminars.The results focus on i) long time dispersive decay in weighted norms; ii) asymptotic stability of solitary waves and soliton scattering asymptotics; iii) convergence to equilibrium distribution.For the first time we have proved dispersive decay in weighted norms a) for magnetic 3D Schrödinger and Klein-Gordon equations; b) for 1D and 2D Dirac equations with hermitian matrix potentials; For the proof we extended the Agmon-Jensen-Kato theory of dispersive decay for the Schrödinger equation with a scalar potential to the magnetic Schrödinger and Klein-Gordon equations and to the Dirac equation. The extension was not straightforward because the corresponding resolvents do not decay at large energies.For the first time we established soliton scattering asymptotics a) for the Dirac equation coupled to a particle; b) for the Schrödinger equation coupled to nonlinear oscillator. We apply the general strategy of Buslaev and Perelman developed in the case of the Schrödinger equation: symplectic orthogonal decomposition of the dynamics near the solitary manifold, modulation equations for the symplectic projection onto the manifold, time decay in transversal directions, etc.We introduced a new class of piece-wise quadratic potentials for nonlinear wave equations, which allow an exact description of the spectral properties for the linearized equation at the solitons (kinks) which is necessary for the study of the stability properties of the solitons. We also obtained an eigenfunction expansion of solutions to the Hamilton equations and used the expansion for the calculation of the Fermi Golden Rule in the context of the nonlinear equations. This condition means a strong coupling of discrete and continuous spectral components of solutions providing the radiation of energy to infinity which results in the asymptotic stability of solitary waves. For the first time we have proved the convergence to an equilibrium distribution for the 3D Dirac equation with a hermitian potential. To reduce the case of the perturbed equation to the case of the free equation we construct a scattering theory for the solutions of infinite global charges using duality arguments and scattering theory for finite charge solutions: this version of scattering theory is based on the dispersive decay.