# Scattering in multiparticle and billiard dynamical systems

## Abstract

The object of study is scattering in billiard dynamical systems and in multiparticle systems on a line (with potential interaction). There exist examples in which the scattering map for such systems is triangular, that is, the corresponding Jacobi matrix has block-triangular form. In this paper it is proved that the billiard scattering in a convex polyhedral cone is triangular if and only if the cone is the closure of some chamber of a Coxeter group. All systems of masses for which multiparticle systems with billiard interaction have triangle scattering are listed. Further, it is proved for multiparticle systems with repulsive interaction that the system of masses of the particles must belong to this list in the case of triangular scattering. Convex plane domains in which the billiards have a triangular scattering map are characterized.

- Authors:

- State Institute for Problems in Physics and Engineering, Moscow (Russian Federation)
- M.V. Lomonosov Moscow State University, Moscow (Russian Federation)

- Publication Date:

- OSTI Identifier:
- 21202877

- Resource Type:
- Journal Article

- Journal Name:
- Sbornik. Mathematics

- Additional Journal Information:
- Journal Volume: 190; Journal Issue: 7; Other Information: DOI: 10.1070/SM1999v190n07ABEH000417; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 1064-5616

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; CONES; INTERACTIONS; MAPS; MASS; MATRICES; SCATTERING

### Citation Formats

```
Plakhov, A Yu, and Stepin, A M.
```*Scattering in multiparticle and billiard dynamical systems*. United States: N. p., 1999.
Web. doi:10.1070/SM1999V190N07ABEH000417; COUNTRY OF INPUT: INTERNATIONAL ATOMIC ENERGY AGENCY (IAEA).

```
Plakhov, A Yu, & Stepin, A M.
```*Scattering in multiparticle and billiard dynamical systems*. United States. doi:10.1070/SM1999V190N07ABEH000417; COUNTRY OF INPUT: INTERNATIONAL ATOMIC ENERGY AGENCY (IAEA).

```
Plakhov, A Yu, and Stepin, A M. Tue .
"Scattering in multiparticle and billiard dynamical systems". United States. doi:10.1070/SM1999V190N07ABEH000417; COUNTRY OF INPUT: INTERNATIONAL ATOMIC ENERGY AGENCY (IAEA).
```

```
@article{osti_21202877,
```

title = {Scattering in multiparticle and billiard dynamical systems},

author = {Plakhov, A Yu and Stepin, A M},

abstractNote = {The object of study is scattering in billiard dynamical systems and in multiparticle systems on a line (with potential interaction). There exist examples in which the scattering map for such systems is triangular, that is, the corresponding Jacobi matrix has block-triangular form. In this paper it is proved that the billiard scattering in a convex polyhedral cone is triangular if and only if the cone is the closure of some chamber of a Coxeter group. All systems of masses for which multiparticle systems with billiard interaction have triangle scattering are listed. Further, it is proved for multiparticle systems with repulsive interaction that the system of masses of the particles must belong to this list in the case of triangular scattering. Convex plane domains in which the billiards have a triangular scattering map are characterized.},

doi = {10.1070/SM1999V190N07ABEH000417; COUNTRY OF INPUT: INTERNATIONAL ATOMIC ENERGY AGENCY (IAEA)},

journal = {Sbornik. Mathematics},

issn = {1064-5616},

number = 7,

volume = 190,

place = {United States},

year = {1999},

month = {8}

}