On the dispersion decay for crystals in the linearized Schr\"odinger-Poisson model

TL;DR

This paper analyzes the linearized Schr"odinger-Poisson model for crystals, proving dispersion decay and spectral properties of the Hamiltonian using advanced spectral and asymptotic techniques.

Contribution

It establishes dispersion decay in weighted Sobolev norms and proves absence of singular spectrum for the linearized model, using novel bounds and spectral analysis methods.

Findings

Dispersion decay in weighted Sobolev norms for solutions.

Absence of singular spectrum and limiting absorption principle.

Infinite multiplicity of eigenvalues.

Abstract

The Schr\"odinger-Poisson-Newton equations for crystals with a cubic lattice and one ion per cell are considered. The ion charge density is assumed i) to satisfy the Wiener and Jellium conditions introduced in our previous paper [28], and ii) to be exponentially decaying at infinity. The corresponding examples are given. We study the linearized dynamics at the ground state. The dispersion relations are introduced via spectral resolution for the non-selfadjoint Hamilton generator using the positivity of the energy established in [28]. Our main result is the dispersion decay in the weighted Sobolev norms for solutions with initial states from the space of continuous spectrum of the Hamilton generator. We also prove the absence of singular spectrum and limiting absorption principle. The multiplicity of every eigenvalue is shown to be infinite. The proofs rely on novel exact bounds and compactness for the inversion of the Bloch generators and on uniform asymptotics for the dispersion relations. We derive the bounds by the energy positivity from [28]. We also use the theory of analytic sets.