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

Mathematical problems of the quantum theory of scattering for a three-particle system

L. D. Faddeev

Jan 1, 196437 citations

Applications of Functional Analysis in Quantum Scattering Theory

David Ridout

The aim of this thesis is to present a reasonable account of some of the applications that functional analysis, or more specifically operator theory, has in the theory of quantum scattering. We begin by introducing the physical scattering experiment as well as establishing the standard framework for discussing quantum theory, that of a Hilbert space. This leads to a discussion of the properties of operators acting in a Hilbert space and the role played by unbounded selfadjoint operators in quantum physics. We also spend some time discussing the position, momentum and energy operators which are of most relevance here. This in turn leads logically to the problem of formulating the scattering problem mathematically using these operators. The stationary equations of scattering theory are also rigorously derived and their suitability for two-body scattering problems is proven. We show that this analysis does not extend to three-body scattering problems, and introduce the Faddeev formulation for three-body systems to supplement the theory. This thesis concludes with a brief overview of how our scattering equations can be modified to take into account effects such as particle spin and identical particles, before discussing an example of a three-body scattering problem (incorporating these features) and deriving a mathematically satisfactory set of equations which are amenable to numerical solution.The aim of this thesis is to present a reasonable account of some of the applications that functional analysis, or more specifically operator theory, has in the theory of quantum scattering. We begin by introducing the physical scattering experiment as well as establishing the standard framework for discussing quantum theory, that of a Hilbert space. This leads to a discussion of the properties of operators acting in a Hilbert space and the role played by unbounded selfadjoint operators in quantum physics. We also spend some time discussing the position, momentum and energy operators which are of most relevance here. This in turn leads logically to the problem of formulating the scattering problem mathematically using these operators. The stationary equations of scattering theory are also rigorously derived and their suitability for two-body scattering problems is proven. We show that this analysis does not extend to three-body scattering problems, and introduce the Faddeev formulation for three-body systems to supplement the theory. This thesis concludes with a brief overview of how our scattering equations can be modified to take into account effects such as particle spin and identical particles, before discussing an example of a three-body scattering problem (incorporating these features) and deriving a mathematically satisfactory set of equations which are amenable to numerical solution.

Jan 1, 19980 citations

Mathematical Aspects of the Three-Body Problem in Quantum Scattering Theory

Richard J. Eden, L. D. Fadeev

The Mathematical GazetteDec 1, 1966391 citations

Scattering Theory in Quantum Mechanical Problems

Dimitri R. Yafaev

The aim of the lecture is to briefly describe the mathematical background of scattering theory for two- and three-particle quantum systems. We discuss basic objects of the theory: wave and scattering operators and the corresponding scattering matrix and illustrate them on the example of the Schrödinger equation. Our goal is to present time-dependent and stationary approaches and to describe the underlying mathematical methods. We also give a sketch of scattering theory for three interacting quantum particles including a difficult problem of the asymptotic completeness of scattering channels. Along with traditional results, we discuss new scattering channels arising for long-range pair interactions.The aim of the lecture is to briefly describe the mathematical background of scattering theory for two- and three-particle quantum systems. We discuss basic objects of the theory: wave and scattering operators and the corresponding scattering matrix and illustrate them on the example of the Schrödinger equation. Our goal is to present time-dependent and stationary approaches and to describe the underlying mathematical methods. We also give a sketch of scattering theory for three interacting quantum particles including a difficult problem of the asymptotic completeness of scattering channels. Along with traditional results, we discuss new scattering channels arising for long-range pair interactions.

arXiv:2205.13204May 26, 20220 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.