Landau mode (${ϵ}_{\mathbf{k}}({\omega }_j)=0$)

A pole of the causally continued collective kinetic response, usually a zero of its scalar response denominator or response-matrix determinant. With time dependence $e^{-i{\omega }_{j}t}$, negative imaginary part gives damping, positive imaginary part growth, and zero imaginary part an undamped oscillation. Damped poles need not be regular discrete eigenmodes of the original collisionless operator: their contribution is obtained by continuing the initial-value response. The least damped excited pole can set the waiting time for a stationary forced response.