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A comment on the paper ``Lower Bounds on the Energy Eigenvalues of Systems Containing Identical Particles''

F. Calogero

^{1}and C. Marchioro^{3}
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### Abstract

It is pointed out that the extension by Yeh [J. Math. Phys. **13**, 227 (1972)] of the result by Calogero and Marchioro [J. Math. Phys. **10**, 562 (1969)] is invalid. Counterexamples are indicated; the fault hidden in the proof of the extended result is spotted.

© 1974 The American Institute of Physics

Received 17 January 1973
Published online 04 November 2003

/content/aip/journal/jmp/15/1/10.1063/1.1666492

1.

1.F. Calogero and C. Marchioro, J. Math. Phys. 10, 562 (1969). For previous, but less general, versions of this result see the other papers quoted in this reference.

2.

2.R. H. T. Yeh, J. Math. Phys. 13, 227 (1972).

3.

3.If the lack of normalizability of the wavefunctions describing the free particles is considered objectionable, the systems can be enclosed into a large, but finite, box; then the number of negative‐energy states in the many‐body case is not infinite, but it is still arbitrarily large (since the size of the box can be chosen arbitrarily large), and this is clearly inconsistent with Eq. (2).

4.

4.F. Calogero, J. Math. Phys. 12, 419 (1971).

5.

5.The lower bound result can be extended to excited states, using comparison systems involving N particles not all of which interact among themselves. For instance the JMP referee has pointed out that, “for an even number N of particles, where the notation is that used in (this paper), and the are to be chosen so that the sum on the right‐hand side achieves the i’ th‐lowest value possible; e.g., for the first excited state, the inequality becomes This result follows at once from the observation that the exact eigenfunctions for the full N‐particle system are legitimate trial eigenfunctions for a system of independent particle pairs, each having the Hamiltonian ” It should be noted that this result, while certainly correct, is nontrivial [namely, it yields a lower bound for the first excited state that is higher than the bound for the ground state given by Eq. (1)] only if the two‐body problem characterized by the Hamiltonian has a nondegenerate ground state. A necessary condition for that is the presence of the external potential W, for otherwise translation invariance implies the well known (and, in this context, highly relevant) degeneracy associated with the localization of the center of mass. Thus this lower bound can be stringent only for systems where the external potential plays a rôle, in binding the system, at least comparable to that of the interparticle potential. (This f

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