Rigorous lower bounds on the energies of the N-body problem

Jean-Louis Basdevant, André Martin, J. M. Richard +1 more

Klaus Hepp

Louis J. Kijewski, Jerome K. Percus

Variation principles for obtaining lower bounds by means of density matrices are applied to a number of many-body problems. This is done by means of restrictions derived from the study of exacly solvable models. Two classes of problems are considered: (a) obtaining a lower bound to the ground-state energy of a system of particles, and (b) obtaining a lower bound to the Helmholtz free energy. For the first case an exact lower bound is obtained for the ground-state energy of the particle-conserving Bogoliubov Hamiltonian. Of particular significance in the nonzero temperature case is a rigorous lower bound to the free energy of the Ising model with small external field at low temperatures.

Yue Zhang

Proving the existence of a lower bound on the energy of atoms is an important and noteworthy problem. Generally, people use the Heisenberg uncertainty relation to make their estimates. This article discusses the hydrogen atom and presents a new rigorous method for deriving the lower bound on the ground-state energy.

Roser Valent�, P. J. Hirschfeld, J C Anglès d'Auriac

We show that the thermodynamic potential E-\ensuremath{\mu}N of an itinerant interacting Fermi system is bounded below by the sum of the thermodynamic potentials of small subsystems into which the infinite system is partitioned. Exact lower bounds for the energy in the thermodynamic limit are therefore easily obtainable by diagonalization of small clusters. In one dimension, these bounds are remarkably close to Bethe-ansatz results for the Hubbard model. We present lower bounds for the two-dimensional Hubbard and t-J Hamiltonians. Such bounds may serve as rigorous tests for approximate treatments of this class of problems.

Achour Benslama, Abboud Metatla, A. Bachkhaznadji +5 more