# Collapse to the Center and Ambiguity in the Asymptotic Behavior of the Off-Shell Scattering Amplitude in Singular Three-Body Problems

## Abstract

Some examples of equations for the there-body problem where there is an oscillating high-momentum behavior are discussed. Specifically, these are (i) the equation in the fixed-center approximation; (ii) the unitarized equation in the fixed-center approximation; (iii) the Skornyakov–Ter-Martirosyan equation; and (iv) equations involving operators that are used in effective field theory—that is, those that are expandable in positive-power series in momentum. It is shown that, in such problems, there arises a situation analogous to a collapse to the center—that is, an infinite number of bound states. The energy of these states is unbounded from below. In this sense, the situation in the models being considered is close to a collapse to the center in the two-body problem.

- Authors:

- National Research Center Kurchatov Institute, Institute for Theoretical and Experimental Physics (Russian Federation)

- Publication Date:

- OSTI Identifier:
- 22761732

- Resource Type:
- Journal Article

- Journal Name:
- Physics of Atomic Nuclei

- Additional Journal Information:
- Journal Volume: 81; Journal Issue: 3; Other Information: Copyright (c) 2018 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; APPROXIMATIONS; ASYMPTOTIC SOLUTIONS; EQUATIONS; POWER SERIES; SCATTERING AMPLITUDES; THREE-BODY PROBLEM; TWO-BODY PROBLEM

### Citation Formats

```
Kudryavtsev, A. E., E-mail: kudryavt@itep.ru, and Romanov, A. I.
```*Collapse to the Center and Ambiguity in the Asymptotic Behavior of the Off-Shell Scattering Amplitude in Singular Three-Body Problems*. United States: N. p., 2018.
Web. doi:10.1134/S1063778818030158.

```
Kudryavtsev, A. E., E-mail: kudryavt@itep.ru, & Romanov, A. I.
```*Collapse to the Center and Ambiguity in the Asymptotic Behavior of the Off-Shell Scattering Amplitude in Singular Three-Body Problems*. United States. doi:10.1134/S1063778818030158.

```
Kudryavtsev, A. E., E-mail: kudryavt@itep.ru, and Romanov, A. I. Tue .
"Collapse to the Center and Ambiguity in the Asymptotic Behavior of the Off-Shell Scattering Amplitude in Singular Three-Body Problems". United States. doi:10.1134/S1063778818030158.
```

```
@article{osti_22761732,
```

title = {Collapse to the Center and Ambiguity in the Asymptotic Behavior of the Off-Shell Scattering Amplitude in Singular Three-Body Problems},

author = {Kudryavtsev, A. E., E-mail: kudryavt@itep.ru and Romanov, A. I.},

abstractNote = {Some examples of equations for the there-body problem where there is an oscillating high-momentum behavior are discussed. Specifically, these are (i) the equation in the fixed-center approximation; (ii) the unitarized equation in the fixed-center approximation; (iii) the Skornyakov–Ter-Martirosyan equation; and (iv) equations involving operators that are used in effective field theory—that is, those that are expandable in positive-power series in momentum. It is shown that, in such problems, there arises a situation analogous to a collapse to the center—that is, an infinite number of bound states. The energy of these states is unbounded from below. In this sense, the situation in the models being considered is close to a collapse to the center in the two-body problem.},

doi = {10.1134/S1063778818030158},

journal = {Physics of Atomic Nuclei},

issn = {1063-7788},

number = 3,

volume = 81,

place = {United States},

year = {2018},

month = {5}

}