Asymptotic completeness of 𝑁-particle long-range scattering

Sigal, I. M.; Soffer, A.

doi:10.1090/S0894-0347-1994-1233895-8

Asymptotic completeness of $N$-particle long-range scattering
HTML articles powered by AMS MathViewer

by I. M. Sigal and A. Soffer

J. Amer. Math. Soc. 7 (1994), 307-334

DOI: https://doi.org/10.1090/S0894-0347-1994-1233895-8

PDF | Request permission

Abstract:

We prove asymptotic completeness for $N$-particle long-range system with potentials vanishing as $O({\left | x \right |^{ - \mu }})$, where $\mu \geq 1 - {2^{ - N - 2}}$, at infinity.

References

H. L. Cycon, R. G. Froese, W. Kirsch, and B. Simon, Schrödinger operators with application to quantum mechanics and global geometry, Springer Study Edition, Texts and Monographs in Physics, Springer-Verlag, Berlin, 1987. MR 883643, DOI 10.1007/978-3-540-77522-5
Jan Dereziński, A new proof of the propagation theorem for $N$-body quantum systems, Comm. Math. Phys. 122 (1989), no. 2, 203–231. MR 994502, DOI 10.1007/BF01257413
Jan Dereziński, Algebraic approach to the $N$-body long range scattering, Rev. Math. Phys. 3 (1991), no. 1, 1–62. MR 1110555, DOI 10.1142/S0129055X91000026
Volker Enss, Asymptotic observables on scattering states, Comm. Math. Phys. 89 (1983), no. 2, 245–268. MR 709466, DOI 10.1007/BF01211831
Volker Enss, Quantum scattering theory for two- and three-body systems with potentials of short and long range, Schrödinger operators (Como, 1984) Lecture Notes in Math., vol. 1159, Springer, Berlin, 1985, pp. 39–176. MR 824987, DOI 10.1007/BFb0080332
Volker Enss, Two- and three-body quantum scattering: completeness revisited, Symposium “Partial Differential Equations” (Holzhau, 1988) Teubner-Texte Math., vol. 112, Teubner, Leipzig, 1989, pp. 108–120. MR 1105802

A remark on asymptotic clustering for

-particle quantum systems

Gian Michele Graf, Asymptotic completeness for $N$-body short-range quantum systems: a new proof, Comm. Math. Phys. 132 (1990), no. 1, 73–101. MR 1069201, DOI 10.1007/BF02278000
É. Mourre, Operateurs conjugués et propriétés de propagation, Comm. Math. Phys. 91 (1983), no. 2, 279–300 (French, with English summary). MR 723552, DOI 10.1007/BF01211163
Michael Reed and Barry Simon, Methods of modern mathematical physics. I, 2nd ed., Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1980. Functional analysis. MR 751959
I. M. Sigal, Mathematical questions of quantum many-body theory, Séminaire sur les équations aux dérivées partielles 1986–1987, École Polytech., Palaiseau, 1987, pp. Exp. No. XXIII, 24. MR 920041
I. M. Sigal, On long-range scattering, Duke Math. J. 60 (1990), no. 2, 473–496. MR 1047762, DOI 10.1215/S0012-7094-90-06019-3
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
I. M. Sigal and A. Soffer, Asymptotic completeness of multiparticle scattering, Differential equations and mathematical physics (Birmingham, Ala., 1986) Lecture Notes in Math., vol. 1285, Springer, Berlin, 1987, pp. 435–472. MR 921295, DOI 10.1007/BFb0080623
I. M. Sigal and A. Soffer, Long-range many-body scattering. Asymptotic clustering for Coulomb-type potentials, Invent. Math. 99 (1990), no. 1, 115–143. MR 1029392, DOI 10.1007/BF01234414

Local decay and velocity bounds

Asymptotic completeness for Coulomb-type

-body systems

I. M. Sigal and A. Soffer, Asymptotic completeness for $N\leq 4$ particle systems with the Coulomb-type interactions, Duke Math. J. 71 (1993), no. 1, 243–298. MR 1230292, DOI 10.1215/S0012-7094-93-07110-4

Description of the spectrum of the energy operator of quantum mechanical systems invariant under permutations of identical particles

5