Calculation of bounds of the eigenvalue spectrum of anharmonic phonon collision operator

Extreme bounds on the eigenvalue spectrum of anharmonic phonon collision operator in Ge are calculated using (i) an isotropic continuum model, and (ii) Born-von-Karman model. The maximum occurs for short wave transverse modes. For the isotropic model the minimum occurs for longitudinal modes of the magnitude nearly $\text{3}\text{4}$_qD down to 27 K, and q_D below it. For the dispersive model the minimum occurs for longitudinal modes with $\text{3}\text{4}$_qD for all temperatures.