Scattering matrices for the quantum 𝑁 body problem

Hassell, Andrew

doi:10.1090/S0002-9947-00-02563-0

Scattering matrices for the quantum $N$ body problem
HTML articles powered by AMS MathViewer

by Andrew Hassell PDF

Trans. Amer. Math. Soc. 352 (2000), 3799-3820 Request permission

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

Christian Gérard, Hiroshi Isozaki, and Erik Skibsted, Commutator algebra and resolvent estimates, Spectral and scattering theory and applications, Adv. Stud. Pure Math., vol. 23, Math. Soc. Japan, Tokyo, 1994, pp. 69–82. MR 1275395, DOI 10.2969/aspm/02310069
Christian Gérard, Hiroshi Isozaki, and Erik Skibsted, $N$-body resolvent estimates, J. Math. Soc. Japan 48 (1996), no. 1, 135–160. MR 1361552, DOI 10.2969/jmsj/04810135
A. Hassell, Plane waves for the 3 body Schrödinger operator, Geom. and Funct. Anal., to appear.
A. Hassell and A. Vasy, Symbolic functional calculus and $N$ body resolvent estimates, J. Funct. Anal., to appear.
B. Helffer and J. Sjöstrand, Équation de Schrödinger avec champ magnétique et équation de Harper, Schrödinger operators (Sønderborg, 1988) Lecture Notes in Phys., vol. 345, Springer, Berlin, 1989, pp. 118–197 (French). MR 1037319, DOI 10.1007/3-540-51783-9_{1}9
Ira Herbst and Erik Skibsted, Free channel Fourier transform in the long-range $N$-body problem, J. Anal. Math. 65 (1995), 297–332. MR 1335378, DOI 10.1007/BF02788775
Lars Hörmander, The analysis of linear partial differential operators. I, 2nd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 256, Springer-Verlag, Berlin, 1990. Distribution theory and Fourier analysis. MR 1065993, DOI 10.1007/978-3-642-61497-2
Hiroshi Isozaki, A generalization of the radiation condition of Sommerfeld for $N$-body Schrödinger operators, Duke Math. J. 74 (1994), no. 2, 557–584. MR 1272984, DOI 10.1215/S0012-7094-94-07420-6
Arne Jensen and Hitoshi Kitada, Fundamental solutions and eigenfunction expansions for Schrödinger operators. II. Eigenfunction expansions, Math. Z. 199 (1988), no. 1, 1–13. MR 954747, DOI 10.1007/BF01160205
Richard B. Melrose, Spectral and scattering theory for the Laplacian on asymptotically Euclidian spaces, Spectral and scattering theory (Sanda, 1992) Lecture Notes in Pure and Appl. Math., vol. 161, Dekker, New York, 1994, pp. 85–130. MR 1291640
Cai Zhong Li, Chang Jiang Zhu, and Hui Jiang Zhao, Global resolvability for quasilinear hyperbolic systems, J. Partial Differential Equations 8 (1995), no. 1, 55–63. MR 1317291
Richard Melrose and Maciej Zworski, Scattering metrics and geodesic flow at infinity, Invent. Math. 124 (1996), no. 1-3, 389–436. MR 1369423, DOI 10.1007/s002220050058
P. Perry, I. M. Sigal, and B. Simon, Spectral analysis of $N$-body Schrödinger operators, Ann. of Math. (2) 114 (1981), no. 3, 519–567. MR 634428, DOI 10.2307/1971301
Michael Reed and Barry Simon, Methods of modern mathematical physics. III, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1979. Scattering theory. MR 529429
I. M. Sigal and A. Soffer, The $N$-particle scattering problem: asymptotic completeness for short-range systems, Ann. of Math. (2) 126 (1987), no. 1, 35–108. MR 898052, DOI 10.2307/1971345
András Vasy, Structure of the resolvent for three-body potentials, Duke Math. J. 90 (1997), no. 2, 379–434. MR 1484859, DOI 10.1215/S0012-7094-97-09010-4
Andras Vasy, Asymptotic behavior of generalized eigenfunctions in $N$-body scattering, J. Funct. Anal. 148 (1997), no. 1, 170–184. MR 1461498, DOI 10.1006/jfan.1996.3053
A. Vasy, Propagation of singularities in three-body scattering, M.I.T. PhD thesis, 1997.
A. Vasy, Scattering matrices in many-body scattering, Commun. Math. Phys. 200, 1999, 105-124.