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 Schr\"odigner 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 ${R}_{0}$ . The exact results obtained by this method are compared with those obtained using other methods. Differences in the functional dependence on ${R}_{0}$ of the order of 2% are noted. Comparison with the hidden-crossing theory finds similar agreement with the functional dependence upon $a$ and ${R}_{0}$ .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 Schr\"odigner 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 ${R}_{0}$. The exact results obtained by this method are compared with those obtained using other methods. Differences in the functional dependence on ${R}_{0}$ of the order of 2% are noted. Comparison with the hidden-crossing theory finds similar agreement with the functional dependence upon $a$ and ${R}_{0}$.

Physical Review AMar 3, 200628 citations

An exact Hartree solution of a one-dimensional many-body problem with two- and three-body interactions

A. Sauceda, W. van Dijk

American Journal of PhysicsNov 1, 19781 citation

Many-particle systems V. Zero-range interactions

Richard L. Hall

Close upper and lowei bounds are found on the binding energy of a system of N identical bosons interacting in one dimension by an attractive square-well pair potential. In the limit in which the potential has the shape of a δ function the bounds are found to be consistent with McGuire's exact solution. In three dimensions the limit of vanishingly small range, in which the two-particle binding energy is kept constant, gives infinite binding to the N -particle system, in agreement with Thomas's theorem Consideration of other limiting Proc.esses eliminates an apparent contradiction between Thomas and Svartholm.Close upper and lowei bounds are found on the binding energy of a system of N identical bosons interacting in one dimension by an attractive square-well pair potential. In the limit in which the potential has the shape of a δ function the bounds are found to be consistent with McGuire's exact solution. In three dimensions the limit of vanishingly small range, in which the two-particle binding energy is kept constant, gives infinite binding to the N -particle system, in agreement with Thomas's theorem Consideration of other limiting Proc.esses eliminates an apparent contradiction between Thomas and Svartholm.

Proceedings of the Physical SocietyAug 1, 196715 citations

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

Nirav P. Mehta, James R. Shepard

We consider the three-boson problem with $\ensuremath{\delta}$-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.We consider the three-boson problem with $\ensuremath{\delta}$-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.

arXiv:nucl-th/0504049Physical Review ASep 28, 200516 citations

Exact solution of a many body problem with nearest and next-nearest neighbour interactions

B. Basu-Mallick, Anjan Kundu

arXiv:cond-mat/0008199Physics Letters AJan 1, 20018 citations

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 Schr\"odigner 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}_{0}

. The exact results obtained by this method are compared with those obtained using other methods. Differences in the functional dependence on

{R}_{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}_{0}

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.