We consider N atoms trapped in an isotropic harmonic potential, with s-wave interactions of infinite scattering length. In the zero-range limit, we obtain several exact analytical results: mapping between the trapped problem and the free-space zero-energy problem, separability in hyperspherical coordinates, SO(2, 1) hidden symmetry, and relations between the moments of the trapping potential energy and the moments of the total energy.

We consider either 3 spinless bosons or 3 equal mass spin-1/2 fermions, interacting via a short range potential of infinite scattering length and trapped in an isotropic harmonic potential. For a zero-range model, we obtain analytically the exact spectrum and eigenfunctions: for fermions all the states are universal; for bosons there is a coexistence of decoupled universal and efimovian states. All the universal states, even the bosonic ones, have a tiny 3-body loss rate. For a finite range model, we numerically find for bosons a coupling between zero angular momentum universal and efimovian states; the coupling is so weak that, for realistic values of the interaction range, these bosonic universal states remain long-lived and observable. With a Feshbach resonance, it is now possible to produce a stable quantum gas of fermionic atoms in the unitary limit, i.e. with an interaction of negligible range and scattering length a = ∞ [1]. The properties of this gas, including its superfluidity, are under active experimental investigation [2]. They have the remarkable feature of being universal, as was tested in particular for the zero temperature equation of state of the gas [3]. In contrast, experiments with Bose gases at a Feshbach resonance suffer from high loss rates [4,5,6], and even the existence of a unitary Bose gas phase is a very open subject [7].In this context, fully understanding the few-body unitary problem is a crucial step. In free space, the unitary 3-boson problem has an infinite number of weakly bound states, the so-called Efimov states [8]. In a trap, it has efimovian states [9, 10] but also universal states whose energy depends only on the trapping frequency [9]. Several experimental groups are currently trapping a few particles at a node of an optical lattice [11] and are controlling the interaction strength via a Feshbach resonance. Results have already been obtained for two particles per lattice node [12], a case that was solved analytically [13]. Anticipating experiments with 3 atoms per node, we derive in this Letter exact expressions for all universal and efimovian eigenstates of the 3-body problem for bosons (generalizing [9] to a non-zero angular momentum) and for equal mass fermions in a trap. We also show the long lifetime of the universal states and their observability in a real experiment, extending to universal states the numerical study of [10].If the effective range and the true range of the interaction potential are negligible as compared to the de Broglie wavelength of the 3 particles, the interaction potential can be replaced by the Bethe-Peierls contact conditions on the wavefunction ψ: it exists a function A such thatin the limit r ij ≡ |r i − r j | → 0 taken for fixed positions of the other particle k and of the center of mass R ij of i and j. In the unitary limit considered in this paper, a = ∞. When all the r ij are non zero, the wavefunction ψ obeys the non-interacting Schrödinger equationω is the oscillation frequency and m the mass of an atom.To solve this problem, we exte...

We derive exact general relations between various observables for N spin-1/2 fermions with zerorange 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 wavefunction, 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, expressions are found (in terms of the regular part of the wavefunction) for the derivative of the energy with respect to the effective range re in 3D, and to the effective range squared in 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/k 6 tail of the momentum distribution are related to ∂E/∂re (or to ∂E/∂(r 2 e ) in 2D). The second order derivative of the 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 eigenwavefunctions' 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 to 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.PACS numbers: 67.85. Lm,31.15.ac I. GENERAL INTRODUCTIONThe experimental breakthroughs of 1995 having led to the first realization of a Bose-Einstein condensate in an atomic vapor [1][2][3] have opened the era of experimental studies of ultracold gases with non-negli...