Three bosons in one dimension with short-range interactions: Zero-range potentials

doi:https://doi.org/10.1103/PhysRevA.72.032728

Title: Three bosons in one dimension with short-range interactions: Zero-range potentials

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Abstract

We consider the three-boson problem with {delta}-function interactions in one spatial dimension. Three different approaches are used to calculate the phase shifts, which we interpret in the context of the effective range expansion, for the scattering of one free particle off a bound pair. We first follow a procedure outlined by McGuire in order to obtain an analytic expression for the desired S-matrix element. This result is then compared to a variational calculation in the adiabatic hyperspherical representation, and to a numerical solution to the momentum-space Faddeev equations. We find excellent agreement with the exact phase shifts, and comment on some of the important features in the scattering and bound-state sectors. In particular, we find that the 1+2 scattering length is divergent, marking the presence of a zero-energy resonance which appears as a feature when the pairwise interactions are short range. Finally, we consider the introduction of a three-body interaction, and comment on the cutoff dependence of the coupling.

Authors:

Mehta, Nirav P; Shepard, James R ^[1]

Department of Physics, University of Colorado, Boulder, Colorado 80309 (United States)

Publication Date:: 2005-09-15

OSTI Identifier:: 20718566

Resource Type:: Journal Article

Journal Name:: Physical Review. A

Additional Journal Information:: Journal Volume: 72; Journal Issue: 3; Other Information: DOI: 10.1103/PhysRevA.72.032728; (c) 2005 The American Physical Society; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 1050-2947

Country of Publication:: United States

Language:: English

Subject:: 74 ATOMIC AND MOLECULAR PHYSICS; BOSONS; BOUND STATE; COUPLING; DELTA FUNCTION; FADDEEV EQUATIONS; INTERACTION RANGE; NUMERICAL SOLUTION; PHASE SHIFT; POTENTIALS; RESONANCE; S MATRIX; SCATTERING; SCATTERING LENGTHS; THREE-BODY PROBLEM; VARIATIONAL METHODS

Citation Formats


                    Mehta, Nirav P, and Shepard, James R. Three bosons in one dimension with short-range interactions: Zero-range potentials.  United States: N. p., 2005. 
        Web.  doi:10.1103/PhysRevA.72.032728.

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                    Mehta, Nirav P, & Shepard, James R. Three bosons in one dimension with short-range interactions: Zero-range potentials.  United States.  https://doi.org/10.1103/PhysRevA.72.032728

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                    Mehta, Nirav P, and Shepard, James R. Thu .  
        "Three bosons in one dimension with short-range interactions: Zero-range potentials".  United States.  https://doi.org/10.1103/PhysRevA.72.032728.

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@article{osti_20718566,

  title        = {Three bosons in one dimension with short-range interactions: Zero-range potentials},

  author       = {Mehta, Nirav P and Shepard, James R},

  abstractNote = {We consider the three-boson problem with {delta}-function interactions in one spatial dimension. Three different approaches are used to calculate the phase shifts, which we interpret in the context of the effective range expansion, for the scattering of one free particle off a bound pair. We first follow a procedure outlined by McGuire in order to obtain an analytic expression for the desired S-matrix element. This result is then compared to a variational calculation in the adiabatic hyperspherical representation, and to a numerical solution to the momentum-space Faddeev equations. We find excellent agreement with the exact phase shifts, and comment on some of the important features in the scattering and bound-state sectors. In particular, we find that the 1+2 scattering length is divergent, marking the presence of a zero-energy resonance which appears as a feature when the pairwise interactions are short range. Finally, we consider the introduction of a three-body interaction, and comment on the cutoff dependence of the coupling.},

  doi          = {10.1103/PhysRevA.72.032728},

  url          = {https://www.osti.gov/biblio/20718566},
  journal      = {Physical Review. A},

  issn         = {1050-2947},

  number       = 3,

  volume       = 72,

  place        = {United States},

  year         = {2005},

  month        = {9}

}

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