  We show that the low-energy density of quasiparticle states in the mixed
state of ultra-clean d-wave superconductors is characterized by pronounced
quantum oscillations in the regime where the cyclotron frequency $\hbar\omega_c
\ll \Delta_0$, the d-wave pairing gap. Such oscillations as a function of
magnetic field B are argued to be due to the internodal scattering of the
d-wave quasiparticles near wavevectors $(\pm k_D,\pm k_D)$ by the vortex
lattice as well as their Zeeman coupling. The periodicity of the oscillations
is set by the condition $k_D \sqrt{hc/(eB)} \equiv k_D' \sqrt{hc/(eB')}\pmod
{2\pi}$. We find that there is additional structure within each period which
grows in complexity as the Dirac node anisotropy increases.
