  Motivated by recent advances in the fabrication of Josephson junctions in
which the weak link is made of a low-dimensional non-superconducting material,
we present here a systematic theoretical study of the local density of states
(LDOS) in a clean 2D normal metal (N) coupled to two s-wave superconductors
(S). To be precise, we employ the quasiclassical theory of superconductivity in
the clean limit, based on Eilenberger's equations, to investigate the
phase-dependent LDOS as function of factors such as the length or the width of
the junction, a finite reflectivity, and a weak magnetic field. We show how the
the spectrum of Andeeev bound states that appear inside the gap shape the
phase-dependent LDOS in short and long junctions. We discuss the circumstances
when a gap appears in the LDOS and when the continuum displays a significant
phase-dependence. The presence of a magnetic flux leads to a complex
interference behavior, which is also reflected in the supercurrent-phase
relation. Our results agree qualitatively with recent experiments on graphene
SNS junctions. Finally, we show how the LDOS is connected to the supercurrent
that can flow in these superconducting heterostructures and present an
analytical relation between these two basic quantities.
