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In this work, we use McGuire's model to describe scattering of three spinless identical particles in one spatial dimension; we first present analytic solutions of Faddeev's equation for scattering of three spinless particles in free space. The three particles interaction in finite volume is derived subsequently, and the quantization conditions by matching wave functions in free space and finite volume are presented in terms of two-body scattering phase shifts. The quantization conditions obtained in this work for the short-range interaction are L\"uscher's formula-like and consistent with Yang's results [Phys. Rev. Lett. 19, 1312 (1967)].
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Peng Guo (2017) studied this question.
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