The Beliaev-Galitskii relation between the two-body t matrix and the scattering amplitude is shown to be slightly erroneous in hard-core scattering because of an incorrect interchange of limits. The correction term is computed exactly at zero momentum both in three dimensions [to relative order O(${\mathit{Ea}}^{2}$)] and in two dimensions [to relative order O(${\mathit{Ea}}^{2}$ln${\mathit{Ea}}^{2}$)].
The Beliaev-Galitskii relation between the two-body t matrix and the scattering amplitude is shown to be slightly erroneous in hard-core scattering because of an incorrect interchange of limits. The correction term is computed exactly at zero momentum both in three dimensions [to relative order O(Ea2)] and in two dimensions [to relative order O(Ea2lnEa2)].
Abstract We consider a system of three identical bosons in R3 R 3 with two-body zero-range interactions and a three-body hard-core repulsion of a given radius a>0 a > 0 . Using a quadratic form approach, we prove that the corresponding Hamiltonian is self-adjoint and bounded from below for any value of a . In particular, this means that the hard-core repulsion is sufficient to prevent the fall to the center phenomenon found by Minlos and Faddeev in their seminal work on the three-body problem in 1961. Furthermore, in the case of infinite two-body scattering length, also known as unitary limit, we prove the Efimov effect, i.e., we show that the Hamiltonian has an infinite sequence of negative eigenvalues En E n accumulating at zero and fulfilling the asymptotic geometrical law En+1/En→e−s02πforn→+∞ E n + 1 / E n → e - 2 π s 0 for n → + ∞ holds, where s0≈1.00624 s 0 ≈ 1.00624 .
For a system of fermions with a three-body contact interaction the second-order contributions to the energy per particle $\ensuremath \bar E(k_f)$ are calculated exactly. The three-particle scattering amplitude in the medium is derived in closed analytical form from the corresponding two-loop rescattering diagram. We compare the (genuine) second-order three-body contribution to $\ensuremath \bar E(k_f)\sim k_f^{10}$ with the second-order term due to the density-dependent effective two-body interaction, and find that the latter term dominates. The results of the present study are of interest for nuclear many-body calculations where chiral three-nucleon forces are treated beyond leading order via a density-dependent effective two-body interaction.
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.
We derive exact general relations between various observables for N spin-1/2 fermions with zero-range or short-range interactions, in continuous space or on a lattice, in two or three dimensions, in an arbitrary external potential. Some of our results generalize known relations between the large-momentum behavior of the momentum distribution, the short-distance behaviors of the pair distribution function and of the one-body density matrix, the norm of the regular part of the wave function, the derivative of the energy with respect to the scattering length or to time, and the interaction energy (in the case of finite-range interactions). The expression relating the energy to a functional of the momentum distribution is also generalized. Moreover, we find expressions (in terms of the regular part of the wave function) for the derivative of the energy with respect to the effective range re in three dimensions (3D), and to the effective range squared in two dimensions (2D). They express the fact that the leading corrections to the eigenenergies due to a finite-interaction range are linear in the effective range in 3D (and in its square in 2D) with model-independent coefficients. There are subtleties in the validity condition of this conclusion, for the 2D continuous space (where it is saved by factors that are only logarithmically large in the zero-range limit) and for the 3D lattice models (where it applies only for some magic dispersion relations on the lattice that sufficiently weakly break Galilean invariance and that do not have cusps at the border of the first Brillouin zone; an example of such relations is constructed). Furthermore, the subleading short-distance behavior of the pair distribution function and the subleading 1/k6 tail of the momentum distribution are related to $\ensuremath{\partial}E/\ensuremath{\partial}{r}_{e}$ [or to $\ensuremath{\partial}E/\ensuremath{\partial}({r}_{e}^{2})$ in 2D]. The second-order derivative of energy with respect to the inverse (or the logarithm in the two-dimensional case) of the scattering length is found to be expressible for any eigenstate in terms of the eigen-wave-function's regular parts; this implies that, at thermal equilibrium, this second-order derivative, taken at fixed entropy, is negative. Applications of the general relations are presented: We compute corrections to exactly solvable two-body and three-body problems and find agreement with available numerics; for the unitary gas in an isotropic harmonic trap, we determine how the finite-1/a and finite-range energy corrections vary within each energy ladder (associated with the SO(2,1) dynamical symmetry) and we deduce the frequency shift and the collapse time of the breathing mode; for the bulk unitary gas, we compare to fixed-node Monte Carlo data, and we estimate the deviation from the Bertsch parameter due to the finite interaction-range in typical experiments.