  We study theoretically the energy and spatially resolved local density of
states (LDoS) in graphene at high perpendicular magnetic field. For this
purpose, we extend from the Schr\"odinger to the Dirac case a
semicoherent-state Green's-function formalism, devised to obtain in a
quantitative way the lifting of the Landau-level degeneracy in the presence of
smooth confinement and smooth disordered potentials. Our general technique,
which rigorously describes quantum-mechanical motion in a magnetic field beyond
the semi-classical guiding center picture of vanishing magnetic length (both
for the ordinary two-dimensional electron gas and graphene), is connected to
the deformation (Weyl) quantization theory in phase space developed in
mathematical physics. For generic quadratic potentials of either scalar (i.e.,
electrostatic) or mass (i.e., associated with coupling to the substrate) types,
we exactly solve the regime of large magnetic field (yet at finite magnetic
length - formally, this amounts to considering an infinite Fermi velocity)
where Landau-level mixing becomes negligible. Hence, we obtain a closed-form
expression for the graphene Green's function in this regime, providing
analytically the discrete energy spectra for both cases of scalar and mass
parabolic confinement. Furthermore, the coherent-state representation is shown
to display a hierarchy of local energy scales ordered by powers of the magnetic
length and successive spatial derivatives of the local potential, which allows
one to devise controlled approximation schemes at finite temperature for
arbitrary and possibly disordered potential landscapes. As an application, we
derive general analytical non-perturbative expressions for the LDoS, which may
serve as a good starting point for interpreting experimental studies.
