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The Wigner-Kirkwoodℏ-expansion of the Wigner transform of the Bloch density can be resummed in the case of non local potentials if we keep only up to second order derivatives of the Wigner transform of the non-local potential with respect to the phase space variables. We also investigate a second approximation to the Bloch density where care has been taken with respect to a consistentℏ expansion. For a one dimensional example we calculate the smooth part of the density and the corresponding energy demonstrating that both approximations to the Bloch-density yield well defined average densitiesand energies.

Partial resummation of?-expansion of the Bloch density for non local potentials