On stability of solid state in the Schr\"odinger-Poisson-Newton model

TL;DR

This paper reviews recent advances in understanding the stability of 3D crystal structures within the Schrödinger-Poisson-Newton model, including orbital and linear stability results for finite and infinite crystals.

Contribution

It introduces new stability results for crystals, employing novel spectral theory and positivity methods, and discusses the existence and properties of ground states.

Findings

Orbital stability for finite crystals established.

Linear stability for infinite crystals proved under new conditions.

Existence of ground states and dispersive decay demonstrated.

Abstract

We survey our recent results on stability of 3D crystals in the Schr\"odinger-Poisson-Newton model. We establish orbital stability for the ground state in the case of finite crystal and linear stability for infinite crystals under novel Jellium and Wiener conditions on the charge density of ions. The corresponding examples are given. In the case of finite crystals, the proofs rely on positivity of the Hessian of Hamiltonian functional in the directions orthogonal to the manifold of ground states. The problem of spatial periodicity of the ground states is discussed. The non-periodic examples are constructed. In the case of infinite crystals the proofs rely on a novel spectral theory of Hamiltonian operators which is a special version of the Gohberg-Krein-Langer theory of selfadjoint operators in the Hilbert spaces with indefinite metric. We establish the existence of the ground states and the dispersive decay for the linearised dynamics.