On-shell approximation for the -wave scattering theory
Physical Review A, 2023•APS
We investigate the scattering theory of two particles in a generic D-dimensional space. For
the s-wave problem, by adopting an on-shell approximation for the T-matrix equation, we
derive analytical formulas which connect the Fourier transform V ̃ (k) of the interaction
potential to the s-wave phase shift. In this way we obtain explicit expressions of the low-
momentum parameters g ̃ 0 and g ̃ 2 of V ̃ (k)= g ̃ 0+ g ̃ 2 k 2+⋯ in terms of the s-wave
scattering length as and the s-wave effective range rs for D= 3, D= 2, and D= 1. Our results …
the s-wave problem, by adopting an on-shell approximation for the T-matrix equation, we
derive analytical formulas which connect the Fourier transform V ̃ (k) of the interaction
potential to the s-wave phase shift. In this way we obtain explicit expressions of the low-
momentum parameters g ̃ 0 and g ̃ 2 of V ̃ (k)= g ̃ 0+ g ̃ 2 k 2+⋯ in terms of the s-wave
scattering length as and the s-wave effective range rs for D= 3, D= 2, and D= 1. Our results …
We investigate the scattering theory of two particles in a generic -dimensional space. For the -wave problem, by adopting an on-shell approximation for the -matrix equation, we derive analytical formulas which connect the Fourier transform of the interaction potential to the -wave phase shift. In this way we obtain explicit expressions of the low-momentum parameters and of in terms of the -wave scattering length and the -wave effective range for , , and . Our results, which are strongly dependent on the spatial dimension , are a useful benchmark for few-body and many-body calculations. As a specific application, we derive the zero-temperature pressure of a two-dimensional uniform interacting Bose gas with a beyond-mean-field correction which includes both scattering length and effective range.
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