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Journal of Mathematical Physics

Lower Bounds of the Energy Eigenvalues of Systems Containing Identical Particles

Robert H. T. Yeh

Published Feb 1, 1972• 0 citations
Publisher

Abstract

Lower bounds of the energy eigenvalues of systems containing identical particles were obtained by a generalization of the method of Calogero and Marchioro [J. Math. Phys. 10, 562 (1969)].
DOI
10.1063/1.1665959

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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.
For a system of NNN identical particles in a pure bound state, the density matrices of positive orders ppp and $N\ensuremath{-}p$ have the same nonzero eigenvalues with the same multiplicities. If the number of nonzero eigenvalues is finite, these density matrices are unitarily equivalent.
A method based on the envelope theory is presented to compute approximate solutions for N-body Hamiltonians with identical particles in D dimensions (D ⩾ 2). In some favorable cases, the approximate eigenvalues can be analytically determined and can be lower or upper bounds. The accuracy of the method is tested with several examples, and an application to a N-body system with a minimal length is studied. Finally, a semiclassical interpretation is given for the generic formula of the eigenvalues.