ИСТИНА

We consider the Schr ̈odinger-Poisson-Newton equations as a model of crystals. Our main results are the well posedness and dispersion decay for the linearized dynamics at the ground state. This linearization is a Hamilton system with nonselfadjoint (and even nonsymmetric) generator. We diagonalize this Hamilton generator using our theoryof spectral resolution of the Hamilton operators with positive definite energy, which is a special version of the M. Krein-H. Langer theory of selfadjoint operators in Hilbert spaces with indefinite metric. Using this spectral resolution, we establish the well posedness and the dispersion decay of the linearized dynamics with positive energy. Our key technical result is the energy positivity for the linearized dynamics under a novel Wiener-type condition on the ions positions and their charge densitities. We give examples of crystals satisfying this condition.