Absolute determination of zero-energy phase shifts for multiparticle single-channel scattering: Generalized Levinson theorem

Physical Review A - Atomic, Molecular, and Optical Physics

Volumn 54, Issue 6, 1996, Pages 4978-4984

  Rosenberg, Leonard ^a   Spruch, Larry ^a  

^a POLYTECHNIC UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0038914120     PISSN: 10502947     EISSN: 10941622     Source Type: Journal    
DOI: 10.1103/PhysRevA.54.4978     Document Type: Article

References (22)

1. Z. R. Iwinski, L. Rosenberg and L. Spruch, Phys. Rev. A 31, 1229 (1985).
2. M. Reed and B. Simon, Methods of Modern Mathematical Physics (Academic, New York, 1978), Vol. IV, p. 206.
3. See, for example, R. G. Newton, Scattering Theory of Waves and Particles, 2nd ed. (Springer, New York, 1982), p. 356. A list of references relating to Levinson's theorem and various generalizations can be found on p. 397 of Newton's book. In Chap. 17 of that book a close-coupling approximation is used to derive a version of Levinson's theorem (not involving the effect of the Pauli principle, which is the focus of the present work). An extension of Levinson's theorem for three-body scattering systems with short-range interactions has been obtained;
4. see S. P. Merkuriev and A. A. Kvitsinsky, Lett. Math. Phys. 17, 307 (1989);
5. S. P. Merkuriev, V. V. Kostrykin and A. A. Kvitsinsky, Sov. J. Part. Nucl. 21, 553 (1990). The effect of the Pauli principle was not included in this work.
6. P. Swan, Proc. R. Soc. London Ser. A 228, 10 (1955).
7. Omitted end note.
8. Omitted end note.
9. L. Rosenberg and L. Spruch, following paper, Phys. Rev. A 54, 4985 (1996).
10. Z. R. Iwinski, L. Rosenberg and L. Spruch, Phys. Rev. A 33, 946 (1986).
11. L. Rosenberg, L. Spruch and T. F. O'Malley, Phys. Rev. 118, 184 (1960).
12. Z. R. Iwinski, L. Rosenberg and L. Spruch, Phys. Rev. Lett. 54, 1602 (1985).
13. Equation (2.3) represents the zero-energy limit of an identity for the tangent of the phase shift derived by T. Kato, Prog. Theor. Phys. (Kyoto) 6, 394 (1951).
14. L. Spruch, in Invited Papers of the Fifth International Conference on the Physics of Electronic and Atomic Collisions, Leningrad, 1967, edited by L. M. Branscomb (University of Colorado, Boulder, 1967), p. 89.
15. E. A. Hylleraas and B. Undheim, Z. Phys. 65, 759 (1930).
16. Use of the function (Formula presented) (r), which is not normalizable, as a trial bound-state function may be justified by the introduction of a convergence factor and the application of a continuity argument; see Ref. 9.
17. T. Ohmura, Phys. Rev. 124, 130 (1961).
18. T. F. O'Malley, L. Spruch and L. Rosenberg, J. Math. Phys. 2, 491 (1961).
19. E. P. Wigner, Phys. Rev. 73, 1002 (1948).
20. R. G. Newton, Phys. Rev. 114, 1611 (1959).
21. N. F. Mott and H. S. W. Massey, The Theory of Atomic Collisions (Clarendon, Oxford, 1965), p. 44.
22. This condition is necessary in order that φ(r) be uniquely defined in such a way that all bound states, with their associated jumps in the upper bounds on the scattering length, are to be generated by the (Formula presented) of Eq. (2.5).