  We study the mixed state in an extreme type-II lattice d-wave superconductor
in the regime of intermediate magnetic fields H_{c1} << H << H_{c2}. We analyze
the low energy spectrum of the problem dominated by nodal Dirac-like
quasiparticles with momenta near k_F=(\pm k_D,\pm k_D) and find that the
spectrum exhibits characteristic oscillatory behavior with respect to the
product of k_D and magnetic length l. The Simon-Lee scaling, predicted in this
regime, is satisfied only on average, with the magnitude of the oscillatory
part of the spectrum displaying the same 1/l dependence as its monotonous
``envelope'' part. The oscillatory behavior of the spectrum is due to the
inter-nodal interference enhanced by the singular nature of the low energy
eigenfunctions near vortices. We also study a separate problem of a single
vortex piercing an isolated superconducting grain of size L by L. Here we find
that the periodicity of the quasiparticle energy oscillations with respect to
k_D L is doubled relative to the case where the field is zero and the vortex is
absent, both such oscillatory behaviors being present at the leading order in
1/L. Finally, we review the overall features of the tunneling conductance
experiments in YBCO and BSCCO, and suggest an interpretation of the peaks at
5-20 meV observed in the tunneling local density of states in these materials.
