We present the analysis of the $N$-boson spectrum computed using a soft two-body potential the strength of which has been varied in order to cover an extended range of positive and negative values of the two-body scattering length $a$ close to the un
itary limit. The spectrum shows a tree structure of two states, one shallow and one deep, attached to the ground-state of the system with one less particle. It is governed by an unique universal function, $Delta(xi)$, already known in the case of three bosons. In the three-particle system the angle $xi$, determined by the ratio of the two- and three-body binding energies $E_3/E_2=tan^2xi$, characterizes the Discrete Scale Invariance of the system. Extending the definition of the angle to the $N$-body system as $E_N/E_2=tan^2xi$, we study the $N$-boson spectrum in terms of this variable. The analysis of the results, obtained for up to $N=16$ bosons, allows us to extract a general formula for the energy levels of the system close to the unitary limit. Interestingly, a linear dependence of the universal function as a function of $N$ is observed at fixed values of $a$. We show that the finite-range nature of the calculations results in the range corrections that generate a shift of the linear relation between the scattering length $a$ and a particular form of the universal function. We also comment on the limits of applicability of the universal relations.
Universal behaviour has been found inside the window of Efimov physics for systems with $N=4,5,6$ particles. Efimov physics refers to the emergence of a number of three-body states in systems of identical bosons interacting {it via} a short-range int
eraction becoming infinite at the verge of binding two particles. These Efimov states display a discrete scale invariance symmetry, with the scaling factor independent of the microscopic interaction. Their energies in the limit of zero-range interaction can be parametrized, as a function of the scattering length, by a universal function. We have found, using a particular form of finite-range scaling, that the same universal function can be used to parametrize the energies of $Nle6$ systems inside the Efimov-physics window. Moreover, we show that the same finite-scale analysis reconciles experimental measurements of three-body binding energies with the universal theory.
We investigate universal behavior in the recombination rate of three bosons close to threshold. Using the He-He system as a reference, we solve the three-body Schrodinger equation above the dimer threshold for different potentials having large values
of the two-body scattering length $a$. To this aim we use the hyperspherical adiabatic expansion and we extract the $S$-matrix through the integral relations recently derived. The results are compared to the universal form, $alphaapprox 67.1sin^2[s_0ln(kappa_*a)+gamma]$, for different values of $a$ and selected values of the three-body parameter $kappa_*$. A good agreement with the universal formula is obtained after introducing a particular type of finite-range corrections, which have been recently proposed by two of the authors in Ref.[1]. Furthermore, we analyze the validity of the above formula in the description of a very different system: neutron-neutron-proton recombination. Our analysis confirms the universal character of the process in systems of very different scales having a large two-body scattering length.
We investigate universal behavior in elastic atom-dimer scattering below the dimer breakup threshold calculating the atom-dimer effective-range function $akcotdelta$. Using the He-He system as a reference, we solve the Schrodinger equation for a fami
ly of potentials having different values of the two-body scattering length $a$ and we compare our results to the universal zero-range form deduced by Efimov, $akcotdelta=c_1(ka)+c_2(ka)cot[s_0ln(kappa_*a)+phi(ka)]$, for selected values of the three-body parameter $kappa_*$. Using the parametrization of the universal functions $c_1,c_2,phi$ given in the literature, a good agreement with the universal formula is obtained after introducing a particular type of finite-range corrections. Furthermore, we show that the same parametrization describes a very different system: nucleon-deuteron scattering below the deuteron breakup threshold. Our analysis confirms the universal character of the process, and relates the pole energy in the effective-range function of nucleon-deuteron scattering to the three-body parameter $kappa_*$.
The Hyperspherical Harmonics basis, without a previous symmetrization step, is used to calculate binding energies of the nuclear A=6 systems using a version of the Volkov potential acting only on s-wave. The aim of this work is to illustrate the use
of the nonsymmetrized basis to deal with permutational-symmetry-breaking term in the Hamiltonian, in the present case the Coulomb interaction.
The hyperspherical harmonic basis is used to describe bound states in an $A$--body system. The approach presented here is based on the representation of the potential energy in terms of hyperspherical harmonic functions. Using this representation, th
e matrix elements between the basis elements are simple, and the potential energy is presented in a compact form, well suited for numerical implementation. The basis is neither symmetrized nor antisymmetrized, as required in the case of identical particles; however, after the diagonalization of the Hamiltonian matrix, the eigenvectors reflect the symmetries present in it, and the identification of the physical states is possible, as it will be shown in specific cases. We have in mind applications to atomic, molecular, and nuclear few-body systems in which symmetry breaking terms are present in the Hamiltonian; their inclusion is straightforward in the present method. As an example we solve the case of three and four particles interacting through a short-range central interaction and Coulomb potential.
