Solution of the one-dimensional N-body problems with quadratic and/or inversely quadratic pair potentials

Journal of Mathematical Physics

Volumn 12, Issue 3, 1971, Pages 419-436

  Calogero, F ^a,b  

^a INSTITUTE FOR THEORETICAL AND EXPERIMENTAL PHYSICS   (Russian Federation)

^b SAPIENZA UNIVERSITY OF ROME   (Italy)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 29744455506     PISSN: 00222488     EISSN: None     Source Type: Journal    
DOI: 10.1063/1.1665604     Document Type: Article

References (36)

1. The 1-dimensional 3-body problem with zero-range interactions had been previously solved in some special cases.
2. A complete analysis of the S matrix describing all scattering processes between these bound states has been given by Yang (see the second of his papers, Ref. 2 above).
3. The coincidence of the energy levels (except for a constant shift) is essentially trivial, but the coincidence of the multiplicities (that probably is peculiar to the 1-dimensional case) is remarkable (see below).
4. For [formula omitted] this has been proved by C. Marchioro (unpublished).
5. See, for instance
6. Actually, the derivative of the wavefunction with respect to the variable [formula omitted] vanishes, when [formula omitted] coincide with another variable [formula omitted] only if the inversely quadratic potential is repulsive [formula omitted] but when multiplied with the wavefunction itself, it vanishes in all cases; and it is this that counts (see below and Ref. 4).
7. The solutions considered in C correspond to the subset of these solutions characterized by the condition [formula omitted]
8. Nonpolynomial solutions of Eq. (2.10) contain singularities and are therefore excluded by the requirement that the wavefunction (2.6) be physically acceptable (for an explicit illustration in the 3-body case, see Ref. 4).
9. The ansatz (2.6), when inserted in the eigenvalue equation (2.3), yields Eqs. (2.10) and (2.11), provided [formula omitted] (see Appendix A). The positive solution of this quadratic equation is the only acceptable one if g does not vanish, but both solutions [formula omitted] are acceptable (in the whole configuration space) if g vanishes.
10. While the proportionality of the partition functions implies that these systems are thermodynamically equivalent, differences would show up in observables (such as, for instance, correlation functions) depending not only on the energy spectrum of the system but also on the form of its many-body wavefunction.
11. indeed, up to a multiplicative constant, it coincides with the generating function [formula omitted] of Appendix C, with [formula omitted] The (more interesting) problem of the gas composed of N 1-dimensional particles interacting by inversely square potentials has been studied, in the [formula omitted] limit with constant density, by
12. by Bill Sutherland
13. We use for simplicity a symbolic notation, with δ functions in the denominator, whose significance should be self-evident [see Eq. (4.27)].
14. Some progress in this direction has been made by A. M. Perelomov (private communication).
15. Of course the separability of the Hamiltonian into “radial” and “angular” parts (see Appendices C and A), which obtains independently of the equality of the particles, and the simple nature of the harmonical potential, imply that the spectrum depends in all cases linearly upon the “radial” quantum number n.
16. I owe to A. M. Perelomov the suggestion to test this conjecture by perturbation theory.
17. Other, possibly more interesting, potentials, for which the basic equation of wave propagation can be solved exactly, are obtained from the above one by going over to the “Jacobi” coordinates (and eliminating the “center-of-mass coordinate” [formula omitted] from the Laplace operator, thereby reducing by one the dimensionality of the space it refers to; see Appendix A).
18. Different definitions of the Jacobi coordinates could be chosen without affecting the final result. See, e.g.
19. The exact choice of the angular coordinates [formula omitted] is immaterial, the only important property being that of Eq. (A5) below. A possible choice are the standard polar coordinates; another possible choice are the so called zonal coordinates. See, e.g.
20. Here and below z, r, a, and b are defined as in Sec. 2, Eqs. (2.7)–(2.9) and (2.13), and [formula omitted] is a translation-invariant polynomial in the N variables [formula omitted] homogeneous of degree k, and satisfying the generalized Laplace equation (2.10).
21. The first three Gegenbauer polynomials with superscript 2 are [formula omitted] [formula omitted] [formula omitted]