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We study theoretically the energy and spatially resolved local density ofstates (LDoS) in graphene at high perpendicular magnetic field. For thispurpose, we extend from the Schr\"odinger to the Dirac case asemicoherent-state Green's-function formalism, devised to obtain in aquantitative way the lifting of the Landau-level degeneracy in the presence ofsmooth confinement and smooth disordered potentials. Our general technique,which rigorously describes quantum-mechanical motion in a magnetic field beyondthe semi-classical guiding center picture of vanishing magnetic length (bothfor the ordinary two-dimensional electron gas and graphene), is connected tothe deformation (Weyl) quantization theory in phase space developed inmathematical 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 magneticlength - formally, this amounts to considering an infinite Fermi velocity)where Landau-level mixing becomes negligible. Hence, we obtain a closed-formexpression for the graphene Green's function in this regime, providinganalytically the discrete energy spectra for both cases of scalar and massparabolic confinement. Furthermore, the coherent-state representation is shownto display a hierarchy of local energy scales ordered by powers of the magneticlength and successive spatial derivatives of the local potential, which allowsone to devise controlled approximation schemes at finite temperature forarbitrary and possibly disordered potential landscapes. As an application, wederive general analytical non-perturbative expressions for the LDoS, which mayserve as a good starting point for interpreting experimental studies.

High magnetic field theory for the local density of states in graphene with smooth arbitrary potential landscapes