A model of scattering of three one-dimensional particles is constructed by means of the extension theory of selfadjoint operators. The dispersion equation for the three-particle bound state energy is obtained and an exact mathematical expression for the corresponding wave functions is derived
We revisit the problem of three identical bosons in free space, which exhibits a universal hierarchy of bound states (Efimov trimers). Modeling a narrow Feshbach resonance within a two-channel description, we map the integral equation for the three-body scattering amplitude to a one-dimensional Schrödinger-type single-particle equation, where an analytical solution of exponential accuracy is obtained. We give exact results for the trimer binding energies, the three-body parameter, the threshold to the three-atom continuum, and the recombination rate.
We study the wave function φ(3) of three identical bosons scattering at zero energy, zero total momentum, and zero orbital angular momentum in two dimensions, interacting via short-range potentials with a finite two-body scattering length a. We derive asymptotic expansions of φ(3) in two regimes: the 111-expansion, where all three pairwise distances are large, and the 21-expansion, where one particle is far from the other two. In the 111-expansion, the leading term grows as ln3(B/a) at large hyperradius B=(s12+s22+s32)/2. At order B−2ln−3(B/a), we identify a three-body parameter D with dimension of length squared, which we term the three-body scattering area. This quantity should be contrasted with the three-body scattering area previously studied for infinite or vanishing two-body scattering length. If the two-body interaction is attractive and supports bound states, D acquires a negative imaginary part, and we derive its relation to the probability amplitudes for the production of two-body bound states in three-body collisions. Under weak modifications of the interaction potentials, we derive the corresponding shift of D in terms of φ(3) and the changes of the two-body and three-body potentials. We also study the effects of D and φ(3) on three-body and many-body physics, including the three-body ground-state energy in a large periodic volume, the many-body energy and the three-body correlation function of the dilute two-dimensional Bose gas, and the three-body recombination rates of two-dimensional ultracold atomic Bose gases.
The spectral function of one hole in different magnetic states of the one-dimensional $t\ensuremath{-}J$ model including three-site term and frustration ${J}^{\ensuremath{'}}$ is studied. In the strong-coupling limit $\stackrel{\ensuremath{\rightarrow}}{J}0$ (corresponding to $\stackrel{\ensuremath{\rightarrow}}{U}\ensuremath{\infty}$ of the Hubbard model) a set of eigenoperators of the Liouvillian is found which allows us to derive an exact expression for the one-particle Green's function that is also applicable at finite temperature and in an arbitrary magnetic state. The spinon dispersion of the pure $t\ensuremath{-}J$ model with the ground state of the Heisenberg model can be obtained by treating the corrections due to a small exchange term by means of the projection method. The spectral function for the special frustration ${J}^{\ensuremath{'}}=J/2$ with the Majumdar-Ghosh wave function is discussed in detail. Besides the projection method, a variational ansatz with the set of eigenoperators of the t term is used. We find a symmetric spinon dispersion around the momentum $k=\ensuremath{\pi}/(2a)$ and a strong damping of the holon branch. Below the continuum a bound state is obtained with finite spectral weight and a very small separation from the continuum. Furthermore, the spectral function of the ideal paramagnetic case at a temperature ${k}_{B}T\ensuremath{\gg}J$ is discussed.