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We survey our recent results on stability of 3D crystals in theSchr\"odinger-Poisson-Newton model. We establish orbital stability for theground state in the case of finite crystal and linear stability for infinitecrystals under novel Jellium and Wiener conditions on the charge density ofions. The corresponding examples are given. In the case of finite crystals, theproofs rely on positivity of the Hessian of Hamiltonian functional in thedirections orthogonal to the manifold of ground states. The problem of spatialperiodicity of the ground states is discussed. The non-periodic examples areconstructed. In the case of infinite crystals the proofs rely on a novelspectral theory of Hamiltonian operators which is a special version of theGohberg-Krein-Langer theory of selfadjoint operators in the Hilbert spaces withindefinite metric. We establish the existence of the ground states and thedispersive decay for the linearised dynamics.

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