# Universal low-energy behavior in three-body systems

## Abstract

We consider a pairwise interacting quantum 3-body system in 3-dimensional space with finite masses and the interaction term V{sub 12} + λ(V{sub 13} + V{sub 23}), where all pair potentials are assumed to be nonpositive. The pair interaction of the particles (1, 2) is tuned to make them have a zero energy resonance and no negative energy bound states. The coupling constant λ > 0 is allowed to take the values for which the particle pairs (1, 3) and (2, 3) have no bound states with negative energy. Let λ{sub cr} denote the critical value of the coupling constant such that E(λ) → −0 for λ → λ{sub cr}, where E(λ) is the ground state energy of the 3-body system. We prove the theorem, which states that near λ{sub cr}, one has E(λ) = C(λ − λ{sub cr})[ln(λ − λ{sub cr})]{sup −1} + h.t., where C is a constant and h.t. stands for “higher terms.” This behavior of the ground state energy is universal (up to the value of the constant C), meaning that it is independent of the form of pair interactions.

- Authors:

- FIAS, Ruth-Moufang-Straße 1, D–60438 Frankfurt am Main (Germany)

- Publication Date:

- OSTI Identifier:
- 22403098

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Journal of Mathematical Physics; Journal Volume: 56; Journal Issue: 2; Other Information: (c) 2015 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; BOUND STATE; COUPLING CONSTANTS; GROUND STATES; PAIRING INTERACTIONS; PARTICLES; THREE-BODY PROBLEM; THREE-DIMENSIONAL CALCULATIONS

### Citation Formats

```
Gridnev, Dmitry K.
```*Universal low-energy behavior in three-body systems*. United States: N. p., 2015.
Web. doi:10.1063/1.4907983.

```
Gridnev, Dmitry K.
```*Universal low-energy behavior in three-body systems*. United States. doi:10.1063/1.4907983.

```
Gridnev, Dmitry K. Sun .
"Universal low-energy behavior in three-body systems". United States.
doi:10.1063/1.4907983.
```

```
@article{osti_22403098,
```

title = {Universal low-energy behavior in three-body systems},

author = {Gridnev, Dmitry K.},

abstractNote = {We consider a pairwise interacting quantum 3-body system in 3-dimensional space with finite masses and the interaction term V{sub 12} + λ(V{sub 13} + V{sub 23}), where all pair potentials are assumed to be nonpositive. The pair interaction of the particles (1, 2) is tuned to make them have a zero energy resonance and no negative energy bound states. The coupling constant λ > 0 is allowed to take the values for which the particle pairs (1, 3) and (2, 3) have no bound states with negative energy. Let λ{sub cr} denote the critical value of the coupling constant such that E(λ) → −0 for λ → λ{sub cr}, where E(λ) is the ground state energy of the 3-body system. We prove the theorem, which states that near λ{sub cr}, one has E(λ) = C(λ − λ{sub cr})[ln(λ − λ{sub cr})]{sup −1} + h.t., where C is a constant and h.t. stands for “higher terms.” This behavior of the ground state energy is universal (up to the value of the constant C), meaning that it is independent of the form of pair interactions.},

doi = {10.1063/1.4907983},

journal = {Journal of Mathematical Physics},

number = 2,

volume = 56,

place = {United States},

year = {Sun Feb 15 00:00:00 EST 2015},

month = {Sun Feb 15 00:00:00 EST 2015}

}