# Exact solution for three particles interacting via zero-range potentials

## Abstract

Exact solutions for three identical bosons interacting via zero-range s-wave potentials are derived. The solutions are contour integrals over a product of hyperradial Bessel functions times angular functions weighted by a coefficient. The product function is a solution of the free-particle Schroedigner equation and the weight function is chosen to satisfy the zero-range boundary conditions. Scattering matrix elements for boson-dimer elastic scattering, breakup of a dimer into three particles and the time-reversed recombination process are derived. For vanishing total energy E, these quantities are given as closed-form functions of the two-body s-wave scattering length a and a three-body renormalization constant R{sub 0}. The exact results obtained by this method are compared with those obtained using other methods. Differences in the functional dependence on R{sub 0} of the order of 2% are noted. Comparison with the hidden-crossing theory finds similar agreement with the functional dependence upon a and R{sub 0}.

- Authors:

- Department of Physics and Astronomy, University of Tennessee, Knoxville, Tennessee 37996-1501, USA and Oak Ridge National Laboratory, P.O. Box 2008, Oak Ridge, Tennessee 37831 (United States)
- (8000) Bahia Blanca, Buenos Aires (Argentina)

- Publication Date:

- OSTI Identifier:
- 20786910

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Physical Review. A; Journal Volume: 73; Journal Issue: 3; Other Information: DOI: 10.1103/PhysRevA.73.032704; (c) 2006 The American Physical Society; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 74 ATOMIC AND MOLECULAR PHYSICS; BESSEL FUNCTIONS; BOSONS; BOUNDARY CONDITIONS; COMPARATIVE EVALUATIONS; DIMERS; ELASTIC SCATTERING; EXACT SOLUTIONS; INTEGRALS; MATRIX ELEMENTS; POTENTIALS; RECOMBINATION; RENORMALIZATION; S MATRIX; S WAVES; SCATTERING LENGTHS; SCHROEDINGER EQUATION; THREE-BODY PROBLEM; TWO-BODY PROBLEM; WEIGHTING FUNCTIONS

### Citation Formats

```
Macek, J. H., Yu Ovchinnikov, S., Gasaneo, G., and Departamento de Fisica, Universidad Nacional del Sur, Av. Alem1253.
```*Exact solution for three particles interacting via zero-range potentials*. United States: N. p., 2006.
Web. doi:10.1103/PHYSREVA.73.0.

```
Macek, J. H., Yu Ovchinnikov, S., Gasaneo, G., & Departamento de Fisica, Universidad Nacional del Sur, Av. Alem1253.
```*Exact solution for three particles interacting via zero-range potentials*. United States. doi:10.1103/PHYSREVA.73.0.

```
Macek, J. H., Yu Ovchinnikov, S., Gasaneo, G., and Departamento de Fisica, Universidad Nacional del Sur, Av. Alem1253. Wed .
"Exact solution for three particles interacting via zero-range potentials". United States.
doi:10.1103/PHYSREVA.73.0.
```

```
@article{osti_20786910,
```

title = {Exact solution for three particles interacting via zero-range potentials},

author = {Macek, J. H. and Yu Ovchinnikov, S. and Gasaneo, G. and Departamento de Fisica, Universidad Nacional del Sur, Av. Alem1253},

abstractNote = {Exact solutions for three identical bosons interacting via zero-range s-wave potentials are derived. The solutions are contour integrals over a product of hyperradial Bessel functions times angular functions weighted by a coefficient. The product function is a solution of the free-particle Schroedigner equation and the weight function is chosen to satisfy the zero-range boundary conditions. Scattering matrix elements for boson-dimer elastic scattering, breakup of a dimer into three particles and the time-reversed recombination process are derived. For vanishing total energy E, these quantities are given as closed-form functions of the two-body s-wave scattering length a and a three-body renormalization constant R{sub 0}. The exact results obtained by this method are compared with those obtained using other methods. Differences in the functional dependence on R{sub 0} of the order of 2% are noted. Comparison with the hidden-crossing theory finds similar agreement with the functional dependence upon a and R{sub 0}.},

doi = {10.1103/PHYSREVA.73.0},

journal = {Physical Review. A},

number = 3,

volume = 73,

place = {United States},

year = {Wed Mar 15 00:00:00 EST 2006},

month = {Wed Mar 15 00:00:00 EST 2006}

}