# Hyperspherical Coulomb spheroidal basis in the Coulomb three-body problem

## Abstract

A hyperspherical Coulomb spheroidal (HSCS) representation is proposed for the Coulomb three-body problem. This is a new expansion in the set of well-known Coulomb spheroidal functions. The orthogonality of Coulomb spheroidal functions on a constant-hyperradius surface {rho} = const rather than on a constant-internuclear-distance surface R = const, as in the traditional Born-Oppenheimer approach, is a distinguishing feature of the proposed approach. Owing to this, the HSCS representation proves to be consistent with the asymptotic conditions for the scattering problem at energies below the threshold for three-body breakup: only a finite number of radial functions do not vanish in the limit of {rho}{yields}{infinity}, with the result that the formulation of the scattering problem becomes substantially simpler. In the proposed approach, the HSCS basis functions are considerably simpler than those in the well-known adiabatic hyperspherical representation, which is also consistent with the asymptotic conditions. Specifically, the HSCS basis functions are completely factorized. Therefore, there arise no problems associated with avoided crossings of adiabatic hyperspherical terms.

- Authors:

- St. Petersburg State University, Fock Institute of Physics (Russian Federation)

- Publication Date:

- OSTI Identifier:
- 22156426

- Resource Type:
- Journal Article

- Journal Name:
- Physics of Atomic Nuclei

- Additional Journal Information:
- Journal Volume: 76; Journal Issue: 2; Other Information: Copyright (c) 2013 Pleiades Publishing, Ltd.; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 1063-7788

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; ASYMPTOTIC SOLUTIONS; BORN-OPPENHEIMER APPROXIMATION; EXPANSION; SCATTERING; SURFACES; THREE-BODY PROBLEM

### Citation Formats

```
Abramov, D. I., E-mail: abramov472007@yandex.ru.
```*Hyperspherical Coulomb spheroidal basis in the Coulomb three-body problem*. United States: N. p., 2013.
Web. doi:10.1134/S1063778813010018.

```
Abramov, D. I., E-mail: abramov472007@yandex.ru.
```*Hyperspherical Coulomb spheroidal basis in the Coulomb three-body problem*. United States. https://doi.org/10.1134/S1063778813010018

```
Abramov, D. I., E-mail: abramov472007@yandex.ru. Fri .
"Hyperspherical Coulomb spheroidal basis in the Coulomb three-body problem". United States. https://doi.org/10.1134/S1063778813010018.
```

```
@article{osti_22156426,
```

title = {Hyperspherical Coulomb spheroidal basis in the Coulomb three-body problem},

author = {Abramov, D. I., E-mail: abramov472007@yandex.ru},

abstractNote = {A hyperspherical Coulomb spheroidal (HSCS) representation is proposed for the Coulomb three-body problem. This is a new expansion in the set of well-known Coulomb spheroidal functions. The orthogonality of Coulomb spheroidal functions on a constant-hyperradius surface {rho} = const rather than on a constant-internuclear-distance surface R = const, as in the traditional Born-Oppenheimer approach, is a distinguishing feature of the proposed approach. Owing to this, the HSCS representation proves to be consistent with the asymptotic conditions for the scattering problem at energies below the threshold for three-body breakup: only a finite number of radial functions do not vanish in the limit of {rho}{yields}{infinity}, with the result that the formulation of the scattering problem becomes substantially simpler. In the proposed approach, the HSCS basis functions are considerably simpler than those in the well-known adiabatic hyperspherical representation, which is also consistent with the asymptotic conditions. Specifically, the HSCS basis functions are completely factorized. Therefore, there arise no problems associated with avoided crossings of adiabatic hyperspherical terms.},

doi = {10.1134/S1063778813010018},

url = {https://www.osti.gov/biblio/22156426},
journal = {Physics of Atomic Nuclei},

issn = {1063-7788},

number = 2,

volume = 76,

place = {United States},

year = {2013},

month = {2}

}