Functional approach to the theory of twodimensional quantum Fermi systems in an external periodic field

The behavior of an adsorbed layer of Fermi particles in a weak external field with symmetry of the centered quadratic lattice is investigated. Instability of the weakly inhomogeneous state of the system with respect to small fluctuations in the vicinity $a^\ast=2K_F$ is established; here $a^\ast$ is the magnitude of the vectors of the neighbors closest to the point in the inverse lattice; $K_F$ is the Fermi momentum.