Propagation of singularities in many-body scattering in the presence of bound states
Journées équations aux dérivées partielles (1999), article no. 16, 20 p.

In these lecture notes we describe the propagation of singularities of tempered distributional solutions u𝒮 ' of (H-λ)u=0, where H is a many-body hamiltonian H=Δ+V, Δ0, V= a V a , and λ is not a threshold of H, under the assumption that the inter-particle (e.g. two-body) interactions V a are real-valued polyhomogeneous symbols of order -1 (e.g. Coulomb-type with the singularity at the origin removed). Here the term “singularity” provides a microlocal description of the lack of decay at infinity. Our result is then that the set of singularities of u is a union of maximally extended broken bicharacteristics of H. These are curves in the characteristic variety of H, which can be quite complicated due to the existence of bound states. We use this result to describe the wave front relation of the S-matrices. Here we only present the statement of the results and sketch some of the ideas in proving them, the complete details will appear elsewhere.

@article{JEDP_1999____A16_0,
     author = {Vasy, Andr\'as},
     title = {Propagation of singularities in many-body scattering in the presence of bound states},
     journal = {Journ\'ees \'equations aux d\'eriv\'ees partielles},
     eid = {16},
     pages = {1--20},
     publisher = {Universit\'e de Nantes},
     year = {1999},
     mrnumber = {2000j:81284},
     language = {en},
     url = {http://archive.numdam.org/item/JEDP_1999____A16_0/}
}
TY  - JOUR
AU  - Vasy, András
TI  - Propagation of singularities in many-body scattering in the presence of bound states
JO  - Journées équations aux dérivées partielles
PY  - 1999
SP  - 1
EP  - 20
PB  - Université de Nantes
UR  - http://archive.numdam.org/item/JEDP_1999____A16_0/
LA  - en
ID  - JEDP_1999____A16_0
ER  - 
%0 Journal Article
%A Vasy, András
%T Propagation of singularities in many-body scattering in the presence of bound states
%J Journées équations aux dérivées partielles
%D 1999
%P 1-20
%I Université de Nantes
%U http://archive.numdam.org/item/JEDP_1999____A16_0/
%G en
%F JEDP_1999____A16_0
Vasy, András. Propagation of singularities in many-body scattering in the presence of bound states. Journées équations aux dérivées partielles (1999), article  no. 16, 20 p. http://archive.numdam.org/item/JEDP_1999____A16_0/

[1] A. Bommier. Propriétés de la matrice de diffusion, 2-amas 2-amas, pour les problèmes à N corps à longue portée. Ann. Inst. Henri Poincaré, 59:237-267, 1993. | EuDML | Numdam | MR | Zbl

[2] J. Dereziński. Asymptotic completeness of long-range N-body quantum systems. Ann. Math., 138:427-476, 1993. | MR | Zbl

[3] J. Dereziński and C. Gérard. Scattering theory of classical and quantum N-particle systems. Springer, 1997. | MR | Zbl

[4] R. G. Froese and I. Herbst. Exponential bounds and absence of positive eigen-values of N-body Schrödinger operators. Commun. Math. Phys., 87:429-447, 1982. | MR | Zbl

[5] R. G. Froese and I. Herbst. A new proof of the Mourre estimate. Duke Math. J., 49:1075-1085, 1982. | MR | Zbl

[6] C. Gérard, H. Isozaki, and E. Skibsted. Commutator algebra and resolvent estimates, volume 23 of Advanced studies in pure mathematics, pages 69-82. 1994. | MR | Zbl

[7] C. Gérard, H. Isozaki, and E. Skibsted. N-body resolvent estimates. J. Math. Soc. Japan, 48:135-160, 1996. | MR | Zbl

[8] G. M. Graf. Asymptotic completeness for N-body short range systems : a new proof. Commun. Math. Phys., 132:73-101, 1990. | MR | Zbl

[9] A. Hassell. Distorted plane waves for the 3 body Schrödinger operator. Geom. Funct. Anal., to appear. | MR | Zbl

[10] L. Hörmander. The analysis of linear partial differential operators, vol. 1-4. Springer-Verlag, 1983.

[11] M. Ikawa, editor. Spectral and scattering theory. Marcel Dekker, 1994. | MR | Zbl

[12] H. Isozaki. Structures of S-matrices for three body Schrödinger operators. Commun. Math. Phys., 146:241-258, 1992. | MR | Zbl

[13] H. Isozaki. A generalization of the radiation condition of Sommerfeld for N-body Schrödinger operators. Duke Math. J., 74:557-584, 1994. | MR | Zbl

[14] H. Isozaki. A uniqueness theorem for the N-body Schrödinger equation and its applications. In Ikawa [11], 1994. | Zbl

[15] H. Isozaki and J. Kitada. Scattering matrices for two-body schrödinger operators. Scient. Papers College Arts and Sci., Tokyo University, 35:81-107, 1985. | Zbl

[16] A. Jensen. Propagation estimates for Schrödinger-type operators. Trans. Amer. Math. Soc., 291-1:129-144, 1985. | MR | Zbl

[17] G. Lebeau. Propagation des ondes dans les variétés à coins. Ann. scient. Éc. Norm. Sup., 30:429-497, 1997. | Numdam | MR | Zbl

[18] R. B. Melrose. Differential analysis on manifolds with corners. In preparation.

[19] R. B. Melrose. Spectral and scattering theory for the Laplacian on asymptotically Euclidian spaces. In Ikawa [11], 1994. | MR | Zbl

[20] R. B. Melrose and J. Sjöstrand. Singularities of boundary value problems. I. Comm. Pure Appl. Math, 31:593-617, 1978. | MR | Zbl

[21] R. B. Melrose and M. Zworski. Scattering metrics and geodesic flow at infinity. Inventiones Mathematicae, 124:389-436, 1996. | MR | Zbl

[22] E. Mourre. Absence of singular continuous spectrum of certain self-adjoint operators. Commun. Math. Phys., 78:391-408, 1981. | MR | Zbl

[23] P. Perry, I. M. Sigal, and B. Simon. Spectral analysis of N-body Schrödinger operators. Ann. Math., 114:519-567, 1981. | MR | Zbl

[24] I. M. Sigal and A. Soffer. N-particle scattering problem : asymptotic completeness for short range systems. Ann. Math., 125:35-108, 1987. | MR | Zbl

[25] I. M. Sigal and A. Soffer. Long-range many-body scattering. Inventiones Math., 99:115-143, 1990. | MR | Zbl

[26] I. M. Sigal and A. Soffer. Asymptotic completeness of N ≤ 4-particle systems with the Coulomb-type interactions. Duke Math. J., 71:243-298, 1993. | MR | Zbl

[27] I. M. Sigal and A. Soffer. Asymptotic completeness of N-particle long-range scattering. J. Amer. Math. Soc., 7:307-334, 1994. | MR | Zbl

[28] E. Skibsted. Smoothness of N-body scattering amplitudes. Reviews in Math. Phys., 4:619-658, 1992. | MR | Zbl

[29] A. Vasy. Structure of the resolvent for three-body potentials. Duke Math. J., 90:379-434, 1997. | MR | Zbl

[30] A. Vasy. Propagation of singularities in euclidean many-body scattering in the presence of bound states. In preparation, 1999.

[31] A. Vasy. Propagation of singularities in many-body scattering. Preprint, 1999. | Zbl

[32] A. Vasy. Scattering matrices in many-body scattering. Commun. Math. Phys., 200:105-124, 1999. | MR | Zbl

[33] A. Vasy. Propagation of singularities in three-body scattering. Astérisque, To appear. | Numdam | Zbl

[34] X. P. Wang. Microlocal estimates for N-body Schrödinger operators. J. Fac. Sci. Univ. Tokyo Sect. IA, Math., 40:337-385, 1993. | Zbl

[35] D. Yafaev. Radiation conditions and scattering theory for N-particle Hamiltonians. Commun. Math. Phys., 154:523-554, 1993. | MR | Zbl