The programme of reducing N -particle problems to one-particle problems is extended to include the Pauli principle. To calculate the ground-state energy of a system of N fermions interacting by pair forces, a rigorous lower-bound shell model is derived. This shell model follows a building-up principle, and tends to improve with increasing number of particles as well as with increasing strength of interaction. For twenty particles antisymmetric in ordinary space interacting by square-well interaction of strength V 0 a 2 = 200 hslash 2 /2 m the shell model differs by less than 8% from the calculated upper bound, and hence a fortiori from the exact energy. In order to test the quality of the approximation for various interactions, it was necessary to calculate upper bounds.
A restriction on the choice of translation-invariant coordinates for a system of N identical fermions, interacting by pair forces, allows a lower bound to be established on the ground-state binding energy. This bound is the sum of the first N -1 eigenvalues of a two-particle problem. The quality of the method is found to depend on the choice of coordinates consistent with the initial restraint, the optimum set giving a bound which, in the case of Hooke's law pair forces, is [radical]3( N -1)/2( N +1) times the exact energy.
The well known and analytically solvable Lipkin–Meshkov–Glick model was devised to test several many-body approximation approaches used in nuclear physics. Here, we use this model to illustrate a method that sets lower bounds for the ground state energy of a quantum Hamiltonian. The idea of the method consists of dividing the Hamiltonian into identical groups of terms that represent subdivisions in the particle population of the system, and then to make an appropriate use of the ground energy of one of these groups. The bounds clearly improve as the size of these groups increases.
Lower bounds to the ground-state energy of a system composed of N identical particles interacting by two-body forces (and possibly also with an external potential) are derived. They are expressed in terms of the ground-state energies of (simpler) systems composed of a smaller number of particles having masses and/or interactions different from the original ones. This reduction process may be continued all the way down to systems involving one and two particles only. The generalization to systems containing some identical and some distinguishable particles is also discussed.
We establish a new iterative method for computing rigorous lower bounds for the ground-state energies of all atoms and ions. The accuracy of the resulting bounds depends on two ingredients: the quality of the initial energy estimate and the quality of the used upper bound for the maximum number of bound electrons,
Nc(Z), as a function of the nuclear charge
Z>0. In particular, this new method allows us to establish the explicit lower bound
E(N,Z)≥−0.8710N1/3Z2 for the ground-state energy
E(N,Z) of the atom/ion with
N∈N electrons and nuclear charge
Z∈(0,N], thereby considerably improving the best known constant
A=(3/2)1/3≈1.1447 in the estimate
E(N,Z)≥−AN1/3Z2 for
0<Z≤N and coming much closer to the best possible constant given by the Thomas-Fermi asymptotics
limN→∞∣E(N,N)∣N−7/3=−ETF(1)≈0.7687 derived in Adv. Math., 23(1) by Lieb and Simon. To demonstrate the potential and the limits of our method, we also compute explicit lower bounds for the ground-state energies
E(N,N) of all atoms with
2≤N≤86 and we compare them with the bounds obtained in J. Chem. Phys., 62(8) by Hertel, Lieb, Thirring and the bounds obtained by Seco in his dissertation. Finally, we also provide a uniqueness result for the expectation values of the kinetic, potential and interaction energies on the ground-state space.
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.NUCLEAR STRUCTURE Upper and lower bounds to ground state energy with soluble model; spectral properties of Lipkin model; variational methods; moment methods.