Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Scattering matrices for the quantum $N$ body problem


Author: Andrew Hassell
Journal: Trans. Amer. Math. Soc. 352 (2000), 3799-3820
MSC (2000): Primary 35P25, 81U10, 81U20, 35S05
DOI: https://doi.org/10.1090/S0002-9947-00-02563-0
Published electronically: March 27, 2000
MathSciNet review: 1695024
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract:

Let $H$ be a generalized $N$ body Schrödinger operator with very short range potentials. Using Melrose's scattering calculus, it is shown that the free channel `geometric' scattering matrix, defined via asymptotic expansions of generalized eigenfunctions of $H$, coincides (up to normalization) with the free channel `analytic' scattering matrix defined via wave operators. Along the way, it is shown that the free channel generalized eigenfunctions of Herbst-Skibsted and Jensen-Kitada coincide with the plane waves constructed by Hassell and Vasy and if the potentials are very short range.


References [Enhancements On Off] (What's this?)

  • 1. C. Gérard, H. Isozaki and E. Skibsted, Commutator algebra and resolvent estimates, Advanced Studies in Pure Mathematics, vol. 23, 1994.MR 95h:35154
  • 2. C. Gérard, H. Isozaki and E. Skibsted, $N$-body resolvent estimates, J. Math. Soc. Japan 48, 1996, 135-160. MR 96j:81131
  • 3. A. Hassell, Plane waves for the 3 body Schrödinger operator, Geom. and Funct. Anal., to appear.
  • 4. A. Hassell and A. Vasy, Symbolic functional calculus and $N$ body resolvent estimates, J. Funct. Anal., to appear.
  • 5. B. Helffer, J. Sjöstrand, Equation de Schrödinger avec champ magnétique et équation de Harper, in Lecture Notes in Physics, vol. 345, H. Holder and A. Jensen (eds.), Springer, 1989. MR 91g:35078
  • 6. I. Herbst and E. Skibsted, Free channel Fourier transform in the long range $N$-body problem, Jour. d'Anal. Math. 65, 1995, 297-332. MR 96j:81132
  • 7. L. Hörmander, The analysis of linear partial differential operators I, Springer, second edition, 1990. MR 91m:35001a
  • 8. H. Isozaki, A generalization of the radiation condition of Sommerfeld for $N$-body Schrödinger operators, Duke Math. J. 74, 1994, 557-584.MR 95d:81148
  • 9. A. Jensen and H. Kitada, Fundamental solutions and eigenfunctions expansions for Schrödinger operators II: eigenfunction expansions, Math. Z. 199, 1988, 1-13. MR 90a:35176
  • 10. R. B. Melrose, Spectral and scattering theory for the Laplacian on asymptotically Euclidean spaces, Spectral and scattering theory (M. Ikawa, editor), Marcel Dekker, 1994. MR 95k:58168
  • 11. R. B. Melrose, Geometric scattering theory, Cambridge University Press, 1995. MR 95k:35129
  • 12. R. B. Melrose and M. Zworski, Scattering metrics and geodesic flow at infinity, Invent. Math. 124, 1996, 389-436. MR 96k:58230
  • 13. P. Perry, I. M. Sigal, B. Simon, Spectral analysis of $N$-body Schrödinger equations, Annals of Math. 114, 1981, 519-567. MR 83b:81129
  • 14. M. Reed and B. Simon, Methods of modern mathematical physics III: Scattering theory, Academic Press, 1979. MR 80m:81085
  • 15. I. M. Sigal and A. Soffer, The $N$-particle scattering problem: asymptotic completeness for short-range systems, Annals of Math. 126, 1987, 35-108. MR 88m:81137
  • 16. A. Vasy, Structure of the resolvent for three-body potentials, Duke Math. J. 90, 1997, 379-434. MR 98k:81295
  • 17. A. Vasy, Asymptotic expansion of generalized eigenfunctions in $N$-body scattering, J. Funct. Anal. 148, 1997, 170-184. MR 98f:81344
  • 18. A. Vasy, Propagation of singularities in three-body scattering, M.I.T. PhD thesis, 1997.
  • 19. A. Vasy, Scattering matrices in many-body scattering, Commun. Math. Phys. 200, 1999, 105-124. CMP 99:08

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 35P25, 81U10, 81U20, 35S05

Retrieve articles in all journals with MSC (2000): 35P25, 81U10, 81U20, 35S05


Additional Information

Andrew Hassell
Affiliation: Centre for Mathematics and its Applications, Australian National University, Canberra ACT 0200, Australia
Email: hassell@maths.anu.edu.au

DOI: https://doi.org/10.1090/S0002-9947-00-02563-0
Keywords: $N$ body problem, scattering theory, scattering matrix, scattering calculus
Received by editor(s): February 11, 1998
Published electronically: March 27, 2000
Article copyright: © Copyright 2000 American Mathematical Society

American Mathematical Society