Comparison of upper and lower bound methods using a soluble many-Fermion model

Abstract

Upper bounds, lower bounds, and estimates based on moments are compared for the ground state energy of a soluble Hamiltonian due to Lipkin. Several bounding methods based on variational approaches and one based on the method of moments are employed. We show that lower bounds are of comparable quality with upper bounds and are no more difficult to obtain. No single bounding method is found superior over the entire range of parameters we study for the soluble Hamiltonian. However, in one region we find an upper bound from one method which coincides with the lower bound of another method, thus yielding the exact ground state energy. We also show that estimation methods based on the easily calculated first and second moments of the eigenvalue distribution are surprisingly accurate for a wide range of coupling parameters and number of active particles.