Hartree-Fock Dynamics for Crystals

In the proposed project we plan to introduce dynamical Hartree-Fock equations for crystals in a novel wave-matrix representation. The wave matrix is defined as the square root of the density matrix. As a preliminary step, we plan to prove the well-posedness of the Cauchy problem for the coupled dynamical Hartree-Fock-Poisson-Newton equations in the wave-matrix representation for a finite number of particles. Further we will construct the space-periodic wave-matrix ground state of the coupled system for any fixed lattice in the wave-matrix representation minimizing the energy per cell over the wave matrix and the nuclei positions. Next we will consider the nonlinear dynamics for perturbations of the ground state and plan to prove the existence and uniqueness of global solutions. Our final goal is a detailed study of the corresponding linearized equations at the ground state proving the long-time convergence to the equilibrium distribution for the linearized dynamics and its short range perturbations under the mixing condition of Rosenblatt or Ibragimov-Linnik on the random initial state. The convergence to the equilibrium distribution for the linearized dynamics will be deduced from the dispersion decay for finite energy initial states using the Bernstein method of series and the Ibragimov-Linnik theory of weakly dependent random values. The extension to the short range perturbations will be done using the asymptotic completeness of the intertwining wave operators and the dispersion decay of finite energy solutions. The asymptotic completeness will be deduced developing methods of Gerard and Nier introduced in the context of periodic Schrödinger equations. The main difficulties in the proof of the dispersion decay are i) the diagonalization of the Bloch generators, ii) the absence of constant dispersion relations. The diagonalization will be obtained by our novel theory of spectral resolution of unbounded positive definite Hamilton operators. The positivity of the Bloch generators and the absence of the constant dispersion relations will be deduced by novel asymptotic expansions of the ground state and the dispersion relations for small electron charge. Recently we have obtained some of these results for the coupled Schrödinger-Poisson-Newton equations. The investigation is inspired by mathematical problems of Solid State Physics: absence of the quantum theory of Ohm`s Law, Fourier`s Law, etc. Our approach might be useful in various quantum dynamical problems of Solid State Physics. In particular, for the study of heat conduction, electrical conduction, photoelectric effect, thermoelectronic emission, the Hall effect, laser coherent radiation and others.

I have developed in 2015--2020 a theory of stability for finite and infinite crystals introducing i) novel general conditions on the ion charge densitites for existence of ground states and ii) general universal Jellium and Wiener conditions on the ion charge densities, which provide orbital and linear stability of the ground states, and the dispersion decay as well. For the proofs, new methods of Functional Analysis were introduced: i) theory of spectral resolution for Hamiltonian nonselfadjoint operators, ii) novel estimates for fermionic wave functions, and others. All results are summarised in the monograph prepared for publication.