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Register a new userEffective theories of scattering with an attractive inverse-square potential and the three-body problem
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A distorted-wave version of the renormalisation group is applied to scattering by an inverse-square potential and to three-body systems. In attractive three-body systems, the short-distance wave function satisfies a Schroedinger equation with an attractive inverse-square potential, as shown by Efimov. The resulting oscillatory behaviour controls the renormalisation of the three-body interactions, with the renormalisation-group flow tending to a limit cycle as the cut-off is lowered. The approach used here leads to single-valued potentials with discontinuities as the bound states are cut off. The perturbations around the cycle start with a marginal term whose effect is simply to change the phase of the short-distance oscillations, or the self-adjoint extension of the singular Hamiltonian. The full power counting in terms of the energy and two-body scattering length is constructed for short-range three-body forces.
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We propose a three-potential formalism for the three-body Coulomb scattering problem. The corresponding integral equations are mathematically well-behaved and can succesfully be solved by the Coulomb-Sturmian separable expansion method. The results show perfect agreements with existing low-energy $n-d$ and $p-d$ scattering calculations.
A three-body scattering process in the presence of Coulomb interaction can be decomposed formally into a two-body single channel, a two-body multichannel and a genuine three-body scattering. The corresponding integral equations are coupled Lippmann-Schwinger and Faddeev-Merkuriev integral equations. We solve them by applying the Coulomb-Sturmian separable expansion method. We present elastic scattering and reaction cross sections of the $e^++H$ system both below and above the $H(n=2)$ threshold. We found excellent agreements with previous calculations in most cases.
The three-body energy-dependent effective interaction given by the Bloch-Horowitz (BH) equation is evaluated for various shell-model oscillator spaces. The results are applied to the test case of the three-body problem (triton and He3), where it is shown that the interaction reproduces the exact binding energy, regardless of the parameterization (number of oscillator quanta or value of the oscillator parameter b) of the low-energy included space. We demonstrate a non-perturbative technique for summing the excluded-space three-body ladder diagrams, but also show that accurate results can be obtained perturbatively by iterating the two-body ladders. We examine the evolution of the effective two-body and induced three-body terms as b and the size of the included space Lambda are varied, including the case of a single included shell, Lambda hw=0 hw. For typical ranges of b, the induced effective three-body interaction, essential for giving the exact three-body binding, is found to contribute ~10% to the binding energy.
For solving the $2to 2,3$ three-body Coulomb scattering problem the Faddeev-Merkuriev integral equations in discrete Hilbert-space basis representation are considered. It is shown that as far as scattering amplitudes are considered the error caused by truncating the basis can be made arbitrarily small. By this truncation also the Coulomb Greens operator is confined onto the two-body sector of the three-body configuration space and in leading order can be constructed with the help of convolution integrals of two-body Greens operators. For performing the convolution integral an integration contour is proposed that is valid for all energies, including bound-state as well as scattering energies below and above the three-body breakup threshold.
We consider quantum inverse scattering with singular potentials and calculate the Sine-Gordon model effective potential in the laboratory and centre-of-mass frames. The effective potentials are frame dependent but closely resemble the zero-momentum potential of the equivalent Ruijsenaars-Schneider model.
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