EXACTLY SOLUBLE MODELS OF MANY INTERACTING FERMIONS WITH KINETIC ENERGY LINEAR IN THE MOMENTUM (ELECTRON GAS, FERMION FLUID, MODEL QUANTUM SYSTEMS)

If one assumes that the kinetic energy of a one-dimensional system is linear in the momentum, one can solve exactly the one-body Schrodinger equation in any external field as well as diagonalize the many-body hamiltonian for interacting fermions in second quantization. The validity and implications of this assumption are discussed and two slightly different additional models are introduced each of them more appropriate for a specific physical situation. The status of the Fermi surface with or without an external field is described briefly. Passing to the three-dimensional problem we find two ways to justify the exchange of a scalar (the kinetic energy) with a vector (the momentum). We present the possibilities they offer for an approach to the fermions of the Fermi sphere or at least of parts of it. We conclude by extracting useful information from this hamiltonian for experimentally measurable quantities of a many-electron gas--such as the linear response of its density to an external field--and describe how the internal potential can be varied to simulate other realistic models or even experimental data.