A rigorous one-one correspondence is established between one-dimensional systems of bosons and of spinless fermions. This correspondence holds irrespective of the nature of the interparticle interactions, subject only to the restriction that the interaction have an impenetrable core. It is shown that the Bose and Fermi eigenfunctions are related by ψB=ψFA, where A(x1 … xn) is +1 or −1 according as the order pq … r, when the particle coordinates xj are arranged in the order xp<xq< … <xr, is an even or an odd permutation of 1 … n. The energy spectra of the two systems are identical, as are all configurational probability distributions, but the momentum distributions are quite different. The general theory is illustrated by application to the special case of impenetrable point particles; the one-one correspondence between bosons with this particular interaction and completely noninteracting fermions leads to a rigorous solution of this many-boson problem.

A harmonically trapped ultracold 1D spin-1 Bose gas with strongly repulsive or attractive 1D evenwave interactions induced by a 3D Feshbach resonance is studied. The exact ground state, a hybrid of Tonks-Girardeau (TG) and ideal Fermi gases, is constructed in the TG limit of infinite even-wave repulsion by a spinor Fermi-Bose mapping to a spinless ideal Fermi gas. It is then shown that in the limit of infinite even-wave attraction this same state remains an exact many-body eigenstate, now highly excited relative to the collapsed generalized McGuire cluster ground state, showing that the hybrid TG state is completely stable against collapse to this cluster ground state under a sudden switch from infinite repulsion to infinite attraction. It is shown to be the TG limit of a hybrid super Tonks-Girardeau (STG) state which is metastable under a sudden switch from finite but very strong repulsion to finite but very strong attraction. It should be possible to create it experimentally by a sudden switch from strongly repulsive to strongly attractive interaction, as in the recent Innsbruck experiment on a spin-polarized bosonic STG gas. In the case of strong attraction there should also exist another STG state of much lower energy, consisting of strongly bound dimers, a bosonic analog of a recently predicted STG state which is an ultracold gas of strongly bound bosonic dimers of fermionic atoms, but it is shown that this STG state cannot be created by such a switch from strong repulsion to strong attraction. If an ultracold atomic vapor is confined in a de Broglie wave guide with transverse trapping so tight and temperature so low that the transverse vibrational excitation quantum is larger than available longitudinal zero point and thermal energies, the effective dynamics becomes one-dimensional (1D) [1,2]. 3D Feshbach resonances [3] allow tuning to the neighborhood of 1D confinementinduced resonances [1,4] where the 1D interaction is very strong, leading to strong short-range correlations, breakdown of effective-field theories, and emergence of highlycorrelated N -body ground states. In the case of spinless or spin-polarized bosons with zero-range Lieb-Liniger (LL) [5] delta function repulsion g B δ(x j − x ℓ ) with coupling constant g B → +∞, the Tonks-Girardeau (TG) gas, the exact N -body ground state was determined in 1960 by a Fermi-Bose (FB) mapping to an ideal Fermi gas [6], leading to "fermionization" of many properties of this Bose system, as recently confirmed experimentally [7,8].It was predicted several years ago [9][10][11] that if g B is large and negative, there exists a highly excited gaseous state known as the super Tonks-Girardeau (STG) gas which is metastable against collapse to the McGuire cluster ground state [12], and this state was recently created by the Innsbruck group [13] by suddenly switching the interaction from strongly repulsive to strongly attractive by passing through the 1D confinement-induced resonance induced by a 3D s-wave Feshbach resonance. This initially surprising metastability w...

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The exact N -particle ground state wave function for a one-dimensional condensate of hard core bosons in a harmonic trap is employed to obtain accurate numerical results for the one-particle density matrix, occupation number distribution of the natural orbitals, and momentum distribution. Our results show that the occupation of the lowest orbital varies as N 0.59 , in contrast to N 0.5 for a spatially uniform system, and N for a true BEC. 03.75.Fi,05.30.Jp