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Journal of Physics A Mathematical and General

An analytical proof of saturability of an optimized lower bound forN-body Hamiltonians for some mass configurations, with arbitraryN

Kheir-Eddine Boudjemaa, Sami R. Zouzou

Published May 3, 2006• 3 citations
Publisher

Abstract

We begin by deriving explicit formulae for the energy levels of a system of N harmonic oscillators for two special mass configurations but for arbitrary N . Under the same conditions, we can perform analytically all the calculational procedure leading to an optimized lower bound for the ground state energy of an N -body system. The lower bound obtained in this way proves to be identical to the exact result. It is the first time, to our knowledge, that an explicit analytical proof of saturability has been worked out.

DOI
10.1088/0305-4470/39/20/015

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We begin by deriving explicit formulae for the energy levels of a system of N harmonic oscillators for two special mass configurations but for arbitrary N . Under the same conditions, we can perform analytically all the calculational procedure leading to an optimized lower bound for the ground state energy of an N -body system. The lower bound obtained in this way proves to be identical to the exact result. It is the first time, to our knowledge, that an explicit analytical proof of saturability has been worked out.
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A system of N identical particles with masses m1 interact with each other via a pair potential V and with an additional particle of mass m0 by the pair potential U, and the system obeys nonrelativistic quantum mechanics. A general lower bound formula is derived for the ground-state energy, which reduces to Coleman’s result in the special case m0=∞. The lower bound is compared with the exact energy for the harmonic oscillator problem V (x) =k21x2 and U (x) =k02x2, which has recently been solved exactly (Hall).
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.
Abstract The resolution of the Schrödinger equation for the translation-invariant N-body harmonic oscillator Hamiltonian in D dimensions with one-body and two-body interactions is performed by diagonalizing a matrix J of order N − 1 . It has been previously established that the diagonalization can be analytically performed in specific situations, such as for N ≤ 5 or for N identical particles. We show that the matrix J is diagonal, and thus the problem can be analytically solved, for any number of arbitrary masses provided some specific relations exist between the coupling constants and the masses. We present analytical expressions for the energies under those constraints.
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.
Studies the ground-state energy of a system of N identical bosons, each having mass m, which interact in one dimension via the pair potential V(x)= gamma f(x/a) and obey non-relativistic quantum mechanics. A recent energy lower bound based on the known exact solution for the delta potential is compared to an earlier bound provided by the equivalent two-body method. The delta bound is good whenever the potential is very narrow and deep. For any bounded potential, the earlier energy bound is always better for large N. Detailed results and graphs are given for the sech-squared potential and the square-well potential.