On linear stability and dispersion for crystals in the Schroedinger-Poisson model

Alexander Komech, Elena Kopylova

Published 2015-05-26Version 1

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 theory of 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 the 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. The key result of present paper is the energy positivity for the linearized dynamics with small elementary charge $e>0$ under a novel Wiener-type condition on the ions positions and their charge densitities. We give examples of the crystals satisfying this condition. The main difficulty in the proof ofr the positivity is due to the fact that for $e=0$ the minimal spectral point $E_0=0$ is an eigenvalue of infinite multiplicity for the energy operator. To prove the positivity we study the asymptotics of the ground state as $e\to 0$ and show that the zero eigenvalue $E_0=0$ bifurcates into $E_e\sim e^2$.