= Solution
There is \b[no <Landau damping>] of this branch within the completely degenerate collisionless model. Its <phase velocity> obeys $\omega/k>V_e>V_i$, while each parallel <marginal distribution> vanishes outside $|u|\leq V_\alpha$. There are no particles at the resonant <velocity>, and $F_\alpha'(\omega/k)=0$. The <Landau contour> consequently contributes no imaginary part to the <dielectric function>; a real root is allowed with $\boxed{\gamma=0}$.
If a small nonzero <temperature> supplied a tail beyond the <Fermi surface>, the conclusion would change: the resonant derivative would generally be nonzero. For the convention $p=\gamma-i\omega$ and a simple weakly damped root, expanding the continued <dielectric function> would give $\gamma=-\epsilon_I(\omega,k)/\partial_\omega\epsilon_R(\omega,k)$. This conditional formula is not needed to manufacture <damping> for the exact cutoff. Neither collisional attenuation nor nonlinear evolution is included in the given <dispersion relation>.
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