The variable phase approach to potential scattering with regular spherically symmetric potentials satisfying (1), and studied by Calogero in his book5, is revisited, and we show directly that it gives the absolute definition of the phase-shifts, i.e. the one which defines δℓ(k) as a continuous function of k for all k ≥ 0, up to infinity, where δℓ(∞) = 0 is automatically satisfied. This removes the usual ambiguity ±nπ, n integer, attached to the definition of the phase-shifts through the partial wave scattering amplitudes obtained from the Lippmann-Schwinger integral equation, or via the phase of the Jost functions. It is then shown rigorously, and also on several examples, that this definition of the phase-shifts is very general, and applies as well to all potentials which have a strong repulsive singularity at the origin, for instance those which behave like gr−m, g> 0, m ≥ 2, etc. We also give an example of application to the low-energy behaviour of the S-wave scattering amplitude in two dimensions, which leads to an interesting result.The variable phase approach to potential scattering with regular spherically symmetric potentials satisfying (1), and studied by Calogero in his book5, is revisited, and we show directly that it gives the absolute definition of the phase-shifts, i.e. the one which defines δℓ(k) as a continuous function of k for all k ≥ 0, up to infinity, where δℓ(∞) = 0 is automatically satisfied. This removes the usual ambiguity ±nπ, n integer, attached to the definition of the phase-shifts through the partial wave scattering amplitudes obtained from the Lippmann-Schwinger integral equation, or via the phase of the Jost functions. It is then shown rigorously, and also on several examples, that this definition of the phase-shifts is very general, and applies as well to all potentials which have a strong repulsive singularity at the origin, for instance those which behave like gr−m, g> 0, m ≥ 2, etc. We also give an example of application to the low-energy behaviour of the S-wave scattering amplitude in two dimensions, which leads to an interesting result.

Jan 1, 20017 citations

Some recent developments in n-particle scattering theory

F. S. Levin

Nuclear Physics AFeb 1, 198123 citations

Towards a Particle Theory in Two Dimensions

Laurens Vanderstraeten

Springer thesesJan 1, 20171 citation

Many particle aspects of a quantum wire

Ankit Sharma

The electronic systems in one dimension have features quite distinct from those in higher dimensions. The onedimensional Fermi surface consists of two discrete points, while in higher dimensions it is continuous. The low-energy particle-hole pairs’ spectrum, which arises from momentum and energy conservations for wave vector between zero and twice of Fermi wave vector does not exist in one dimension. In higher dimensions, this region of missing states is filled in. The Fourier transform of one-dimensional Coulomb potential diverges. This led to wide spread belief that one-dimensional interacting electronic systems should exhibit non-Fermi liquid (known as Luttinger liquid) behaviour. However, there is still no unambiguous evidence both for the real existence or non-existence of Luttinger liquid behaviour in a realistic quantum wire. This talk briefly reviews key aspects of Fermi liquid approach and Tomonaga-Luttinger liquid approach for a onedimensional system. In this paper, structure factor, pair distribution function and electron-electron scattering, which exhibit similarity between two approaches in low energy and momentum regime, is reviewed.The electronic systems in one dimension have features quite distinct from those in higher dimensions. The onedimensional Fermi surface consists of two discrete points, while in higher dimensions it is continuous. The low-energy particle-hole pairs’ spectrum, which arises from momentum and energy conservations for wave vector between zero and twice of Fermi wave vector does not exist in one dimension. In higher dimensions, this region of missing states is filled in. The Fourier transform of one-dimensional Coulomb potential diverges. This led to wide spread belief that one-dimensional interacting electronic systems should exhibit non-Fermi liquid (known as Luttinger liquid) behaviour. However, there is still no unambiguous evidence both for the real existence or non-existence of Luttinger liquid behaviour in a realistic quantum wire. This talk briefly reviews key aspects of Fermi liquid approach and Tomonaga-Luttinger liquid approach for a onedimensional system. In this paper, structure factor, pair distribution function and electron-electron scattering, which exhibit similarity between two approaches in low energy and momentum regime, is reviewed.

Indian Journal of Pure & Applied PhysicsFeb 1, 20060 citations

Study of the Asymptotic Behaviour of Many-Particle Amplitudes.

Adrian Patrascioiu

Jan 1, 19720 citations

The scattering of a few particles with short-range interactions at low energies

Shangguo Zhu

The low-energy scattering of three bosons or distinguishable particles with short-range interactions is characterized by a fundamental parameter, the three-body scattering hypervolume, which is responsible for the nonuniversal effects in dilute Bose-Einstein condensates. We derive an analytical formula of the three-body scattering hypervolumes for weak interactions. When the interaction supports two-body bound states, the three-body scattering hypervolume gains a negative imaginary part, which is directly related to the rate constant for three-body recombinations. We develop a numerical method to calculate the three-body scattering hypervolumes for some model potentials with variable strengths. For real atoms with van der Waals potential, the three-body scattering hypervolumes are much harder to compute, because the three-body wave function is highly oscillatory at smaller inter-atomic distances. However, they may be extractable from precision data such as the collective-mode frequencies of trapped Bose-Einstein condensates. In many numerical simulations, the system being simulated is put into a large but finite volume, such as a large periodic box of size $L$ . To extract the low-energy scattering properties of two particles in infinite space from such simulations, L\\"uscher's formula must be used. In the second part of this thesis, we generalize L\\"uscher's formula to $d$ spatial dimensions. We obtain its $s$ -wave and $p$ -wave approximations and the systematic expansions of the energies of the low-lying states. At a $s$ -wave resonance in $d\\ge4$ dimensions, we identify two low-lying states close to the threshold with nearly opposite energies, $E\\sim \\pm 1/L^{d/2}$ when $d\\ge 5$ , or $E \\sim \\pm 1/L^2 \\sqrt{\\ln L}$ when $d= 4$ . These calculations provide important insights to the physics of three particles at a three-body resonance in finite volumes in two or three dimensions. Three-body resonances are important phenomena not yet completely explored in the experiments on cold atoms.The low-energy scattering of three bosons or distinguishable particles with short-range interactions is characterized by a fundamental parameter, the three-body scattering hypervolume, which is responsible for the nonuniversal effects in dilute Bose-Einstein condensates. We derive an analytical formula of the three-body scattering hypervolumes for weak interactions. When the interaction supports two-body bound states, the three-body scattering hypervolume gains a negative imaginary part, which is directly related to the rate constant for three-body recombinations. We develop a numerical method to calculate the three-body scattering hypervolumes for some model potentials with variable strengths. For real atoms with van der Waals potential, the three-body scattering hypervolumes are much harder to compute, because the three-body wave function is highly oscillatory at smaller inter-atomic distances. However, they may be extractable from precision data such as the collective-mode frequencies of trapped Bose-Einstein condensates. In many numerical simulations, the system being simulated is put into a large but finite volume, such as a large periodic box of size $L$. To extract the low-energy scattering properties of two particles in infinite space from such simulations, L\\"uscher's formula must be used. In the second part of this thesis, we generalize L\\"uscher's formula to $d$ spatial dimensions. We obtain its $s$-wave and $p$-wave approximations and the systematic expansions of the energies of the low-lying states. At a $s$-wave resonance in $d\\ge4$ dimensions, we identify two low-lying states close to the threshold with nearly opposite energies, $E\\sim \\pm 1/L^{d/2}$ when $d\\ge 5$, or $E \\sim \\pm 1/L^2 \\sqrt{\\ln L}$ when $d= 4$. These calculations provide important insights to the physics of three particles at a three-body resonance in finite volumes in two or three dimensions. Three-body resonances are important phenomena not yet completely explored in the experiments on cold atoms.

Nov 8, 20180 citations

L

. To extract the low-energy scattering properties of two particles in infinite space from such simulations, L\\"uscher's formula must be used. In the second part of this thesis, we generalize L\\"uscher's formula to

d

spatial dimensions. We obtain its

s

-wave and

p

-wave approximations and the systematic expansions of the energies of the low-lying states. At a

s

-wave resonance in

d\\ge4

dimensions, we identify two low-lying states close to the threshold with nearly opposite energies,

E\\sim \\pm 1/L^{d/2}

when

d\\ge 5

, or

E \\sim \\pm 1/L^2 \\sqrt{\\ln L}

when

d= 4

. These calculations provide important insights to the physics of three particles at a three-body resonance in finite volumes in two or three dimensions. Three-body resonances are important phenomena not yet completely explored in the experiments on cold atoms.