An inversion theorem in Fermi surface theory (Q2711838)

In previous papers [see J. Stat. Phys. 84, 1209-1336 (1996; Zbl 1081.82507), Commun. Pure Appl. Math. 51, 1133-1246 (1998; Zbl 0916.35095) and ibid. 52, 273-324 (1999; Zbl 0910.35104) for reviews] the authors started a perturbative analysis of nonspherical Fermi surfaces, formed by interacting electrons in a solid. In the center of their study were certain regularity properties for the transformation from the noninteracting to the interacting surface. In the present paper, the authors complete their analysis by proving an inversion theorem for the map between the free and nonfree dispersion relations that were used in the renormalization of these models, the only condition being that the Fermi surface is strictly convex and the interaction is of short-range type. The theorem also casts some light on the physical role of the function \(K\), used as a counterterm in the renormalization procedure.