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arXiv:2601.04662 · Physical Review A

Scattering of a weakly bound dimer from a hard wall in one dimension

Xican Zhang, Shina Tan

Published Mar 13, 2026• 0 citations
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Abstract

We consider a dimer formed by two particles with an attractive contact interaction in one dimension, colliding with a hard wall. We compute the scattering phase shifts and the reflection coefficients for various collision energies and various mass ratios of the two particles. For low-energy collisions (with dimer kinetic energies much smaller than the binding energy) our results are consistent with those of D. Lee and M. Pine, The European Physical Journal A 47, 41 (2011). For mass ratios much greater than 1 we use the Born-Oppenheimer approximation to show that the scattering length and the effective range of the dimer-wall collision both depend logarithmically on the mass ratio. For collision energies much greater than the binding energy, the dissociation probability is inversely proportional to the square of the incident momentum of the dimer and we find the constant of proportionality analytically, and we use a semiclassical analysis to approximately derive the ``angular distribution" of the dissociated pair, where the ``angle" $θ$ depends on the ratio of the velocities of the two outgoing unbound particles.

DOI
10.1103/df8f-nyrp

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We consider a dimer formed by two particles with an attractive contact interaction in one dimension, colliding with a hard wall. We compute the scattering phase shifts and the reflection coefficients for various collision energies and various mass ratios of the two particles. For low-energy collisions (with dimer kinetic energies much smaller than the binding energy) our results are consistent with those of D. Lee and M. Pine, The European Physical Journal A 47, 41 (2011). For mass ratios much greater than 1 we use the Born-Oppenheimer approximation to show that the scattering length and the effective range of the dimer-wall collision both depend logarithmically on the mass ratio. For collision energies much greater than the binding energy, the dissociation probability is inversely proportional to the square of the incident momentum of the dimer and we find the constant of proportionality analytically, and we use a semiclassical analysis to approximately derive the angular distribution" of the dissociated pair, where the angle" θθθ depends on the ratio of the velocities of the two outgoing unbound particles.
We consider the system of dimers formed in a one-dimensional mass-balanced Bose-Bose mixture of species $\ensuremath{\sigma}=\ensuremath{\uparrow},\ensuremath{\downarrow}$ with attractive interspecies and repulsive intraspecies contact interactions. In the plane parametrized by the ratios of the coupling constants ${g}_{\ensuremath{\uparrow}\ensuremath{\uparrow}}/|{g}_{\ensuremath{\uparrow}\ensuremath{\downarrow}}|$ and ${g}_{\ensuremath{\downarrow}\ensuremath{\downarrow}}/|{g}_{\ensuremath{\uparrow}\ensuremath{\downarrow}}|$ we trace out the curve where the dimer-dimer interaction switches from attractive to repulsive. We find this curve to be significantly (by more than a factor of 2) shifted towards larger ${g}_{\ensuremath{\sigma}\ensuremath{\sigma}}$ (or smaller $|{g}_{\ensuremath{\uparrow}\ensuremath{\downarrow}}|$) compared to the mean-field stability boundary ${g}_{\ensuremath{\uparrow}\ensuremath{\uparrow}}{g}_{\ensuremath{\downarrow}\ensuremath{\downarrow}}={g}_{\ensuremath{\uparrow}\ensuremath{\downarrow}}^{2}$. For a weak dimer-dimer attraction we predict a dilute dimerized liquid phase stabilized against collapse by a repulsive three-dimer force.
We consider systems of two-component fermions with unequal masses and interacting via a short-range attractive potential. We discuss the case where the two-component fermions form a shallow dimer with large scattering length. The three-fermion and four-fermion systems with such properties are universal and characterized by the twofermion scattering length aff and the ratio of the mass of spin-↑ fermion to the mass of spin-↓ fermion, m↑/m↓. In this study, using lattice effective field theory, we analyze fermion-dimer and dimer-dimer systems and calculate the universal fermion-dimer and dimer-dimer scattering lengths for various values of the mass ratio m↑/m↓. We find that these universal scattering lengths increase logarithmically with the mass ratio m↑/m↓.
We study scattering of three equal-mass particles in one dimension. Integrable interactions are synonymous with nondiffractive scattering, meaning that the set of incoming momenta for any scattering event coincides with the set of outgoing momenta. A system is integrable if the two-particle scattering matrix obeys the Yang-Baxter equation. Nonintegrable interactions correspond to diffractive scattering, where the set of outgoing momenta may take on all values consistent with energy and momentum conservation. Such processes play a vital role in the kinetics of one-dimensional gases, where binary collisions are unable to alter the distribution function. When integrability is broken weakly, the result is a small diffractive scattering amplitude. Our main result is a simple formula for the diffractive part of the scattering amplitude, when the violation of the Yang-Baxter equation is small. Although the derivation is given for $\ensuremath{\delta}$-function interactions, the result depends only on the two-particle scattering matrix, and should therefore also apply to finite-range interactions close to integrable.
We consider the problem of obtaining the scattering length for a fermion colliding with a dimer, formed from a fermion identical to the incident one and another different fermion. This is done in the universal regime where the range of interactions is short enough that the scattering length aaa for nonidentical fermions is the only relevant quantity. This is the generalization to fermions with different masses of the problem solved long ago by Skorniakov and Ter-Martirosian for particles with equal masses. We solve this problem analytically in the two limiting cases where the mass of the solitary fermion is very large or very small compared to the mass of the two other identical fermions. This is done for both the value of the scattering length and the function entering the Skorniakov--Ter-Martirosian integral equation, for which simple explicit expressions are obtained.
We solve the three-boson problem with contact two-and three-body interactions in one dimension and analytically calculate the ground and excited trimer-state energies.Then, by using the diffusion Monte Carlo technique we calculate the binding energy of three dimers formed in a one-dimensional Bose-Bose or Fermi-Bose mixture with attractive interspecies and repulsive intraspecies interactions.Combining these results with our three-body analytics, we extract the three-dimer scattering length close to the dimer-dimer zero crossing.In both considered cases the three-dimer interaction turns out to be repulsive.Our results constitute a concrete proposal for obtaining a one-dimensional gas with a pure three-body repulsion.
We present a diagrammatic approach for the dimer-dimer scattering problem in two or three spatial dimensions, within the resonance approximation where these dimers are in a weakly bound resonant state. This approach is first applied to the calculation of the dimer-dimer scattering length aB{a}_{B}aB​ in three spatial dimensions, for dimers made of two fermions in a spin-singlet state, with corresponding scattering length aF{a}_{F}aF​, and the already known result aB=0.60{a}_{B}=0.60aB​=0.60 aF{a}_{F}aF​ is recovered exactly. Then we make use of our approach to obtain results in two spatial dimensions for fermions as well as for bosons. Specifically, we calculate bound-state energies for three bbbbbbbbb and four bbbbbbbbbbbb resonantly interacting bosons in two dimensions. We consider also the case of a resonant interaction between fermions and bosons, and we obtain the exact bound-state energies of two bosons plus one fermion bbfbbfbbf, two bosons plus two fermions $b{f}_{\ensuremath{\uparrow}}b{f}_{\ensuremath{\downarrow}}$, and three bosons plus one fermion bbbfbbbfbbbf.