= Solution
The two real branches start at $\omega=\pm\omega_{\mathrm pe}$, have zero initial slope, and approach the lines $\omega=\pm kV_e$ from outside the populated parallel-<velocity> region. The original plot solves the <degenerate-plasma dielectric response> numerically; the shaded wedge shows where particles with resonant parallel <velocities> exist.
\Image[/past-exam-of-the-kinetic-theory-course-of-the-university-of-oxford/2023-degenerate-plasma-dispersion.png]
{title=Degenerate electron-plasma branches remain outside the resonant-velocity wedge}
{description=The <frequency> is scaled by the electron <plasma frequency> and the <wavenumber> by $\omega_{\mathrm pe}/V_e$. The dashed curves show the leading long-<wavelength> approximation. At large <wavenumber> the exact branches are exponentially close to $\omega=\pm kV_e$; their visual overlap with those edges does not mean that a finite-$k$ root is inside the shaded region. Ion response, collisions and quantum recoil are omitted.}
The distinction from a <nondegenerate gas> with a <Maxwellian> <Maxwell-Boltzmann velocity distribution> is its tail. A <Maxwellian> has resonant particles at every finite <phase velocity>. Its long-<wavelength> <Langmuir waves> have exponentially weak <Landau damping>, but at shorter <wavelengths> that <damping> becomes strong. The ideal degenerate cutoff instead permits this undamped branch to persist continuously into <degenerate-plasma zero-point sound>. The coefficient $3V_e^2/5$ in <2/c> is also different from the classical thermal dispersive coefficient.
A natural additional branch is a <degenerate ion-acoustic wave>. For singly charged ions with equal electron and ion <number densities>, degenerate electron <pressure> supplies a restoring force while ion <mass> supplies inertia. The electron equation of state has $dP_e/dn_e=m_eV_e^2/3$, suggesting, at long <wavelengths>,
$$
\boxed{\omega\simeq\pm kc_s,\qquad c_s^2\simeq\frac{m_e}{3m_i}V_e^2.}
$$
Both species are degenerate here, so $V_i=(m_e/m_i)V_e$. The hierarchy $V_i\ll c_s\ll V_e$ explains the <damping>: there are no resonant ions, but there are resonant electrons inside their parallel <marginal distribution>. That electron <marginal distribution> has a small nonzero slope at $u=c_s$, even though the full three-dimensional occupation is constant inside the <Fermi sphere>. One therefore expects weak electron <Landau damping>, of relative order $\sqrt{m_e/m_i}$, rather than exact absence of <damping>. Ion degeneracy <pressure> adds a relatively small correction to $c_s^2$. This reasoning concerns an electrostatic <ion-acoustic wave>; transverse or magnetized branches require additional field dynamics or a background <magnetic field>.
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