On the boundedness of the eigenvalue spectrum of phonon collision operator

Anharmonicity in lattice potential leads to boundedness of the eigenvalue Spectrum of the phonon collision operator. Considering the deviation, ψ_q, in the distribution function of the phonons $\text{N}{}_{\mathrm{q}}\text{, N}{}_{\mathrm{q}}\text{=}\text{N}\text{?}{}_{\mathrm{q}}\text{+}\text{N}\text{?}{}_{\mathrm{q\psi q}}\text{(}\text{N}\text{?}\text{+ 1)}$. bar denoting equilibrium value, as an odd function of the phonon wave vector q it has been possible to obtain a lower bound, μ, on the eigenvalue spectrum of the phonon collision operator P. This satisfies the inequelity relation 0?μ?p_i?λ, where p_i are the eigenvalue of P, and λ is an upper bound on it (as given by Benin). The occurence of μ ensures for the possibility of obtaining a sequence of upper bounds on the lattice thermal transport coefficient.