Inverse spectral problem for Hamiltonians of interacting fermionic systems in almost-periodic media

Abstract

We consider the inverse spectral problem for two classes of Hamiltonians of interacting Fermi particles on an integer lattice of arbitrary dimension subject to a common almost-periodic potential. We show first that the inverse problem can be solved with the help of the KAM (Kolmogorov–Arnold–Moser) techniques from Craig (1983) and Pöschel (1983), initially developed for single-particle models, for a particular class of limit-periodic potentials, and then comment on the adaptations to make in the case of quasi-periodic potentials. The solution is obtained in the class of Hamiltonians with a small amplitude of the kinetic energy operator (i.e., with a low mobility of the particles) featuring a uniform exponential decay of all eigenfunctions which prove to be unimodal.