THE SELF-CONSISTENT MATHIEU PROBLEM

Abstract

The one-dimensional many-fermion system with weak attractive interaction, which is known to have self-consistent Hartree solutions corresponding to either uniform or spatially periodic mass density, is discussed as a physical model and as a mathematical problem. The Hartree problem for sinusoidal mass density leads to the self-consistent Mathieu problem, which is analyzed for the case where the first Brillouin zone boundary does not necessarily coincide with the Fermi surface (no energy gap between ground and single-particle excited states). The Mathieu equation is solved in the weak-binding approximation in the vicinity of the two lowest band gaps, the self-consistency condition is analyzed in detail, and the N-particle energy is calculated for various cases. The results suggest that the periodic state will not have lower energy than the uniform state unless the first gap in the one-particle spectrum lies near the Fermi surface. The fact that self-consistency can be obtained for a considerable range of periodicities suggests that the periodic solutions may be of importance in some many-fermion systems.