Using the asymptotic Bethe Ansatz, we obtain an exact solution of the many-body problem for 1D spin-polarized fermions with resonant p-wave interactions, taking into account the effects of both scattering volume and effective range. Under typical experimental conditions, accounting for the effective range, the properties of the system are significantly modified due to the existence of "shape" resonances. The excitation spectrum of the considered model has unexpected features, such as the inverted position of the particle-and hole-like branches at small momenta, and roton-like minima. We find that the frequency of the "breathing" mode in the harmonic trap provides an unambiguous signature of the effective range. In this Letter, we obtain an exact solution for 1D spin-polarized fermions under resonant scattering conditions [10], which is relevant for 40 K and 6 Li atoms near p-wave Feshbach resonances [11,12]. Such a 1D experimental system has been realized for 40 K [13]. The related 3D problem has also received significant theoretical attention recently [14]. For spin-polarized fermions only scattering in odd partial wave channels is present, and at low energies, p-wave scattering is the strongest. If only the "scattering volume" is taken into account and the "effective range" of p-wave scattering is neglected (see Eq.(1) for definitions), then the projection to 1D [15] results in a fermionic Cheon-Shigehara (CS) model [16], which is dual to the bosonic LL model. For such a model, strongly interacting fermions with resonant interactions are mapped to weakly interacting bosons, the so-called fermionic Tonks-Girardeau (fTG) limit [17]. However, it was shown by L. Pricoupenko [10] that unlike the case of the strongly interacting bosonic TG limit [2,3], the requirements for the observation of the fTG limit are quite stringent, and the effective range of scattering needs to be taken into account. We provide an exact solution that accounts for both scattering volume and effective range, and obtain significant deviations from the CS model [16] due to "shape" resonance in the p-wave scattering. We find several new effects, such as the inversion of particleand hole-like spectra for low momenta, roton-like minima in the excitation spectrum, and we calculate density profiles and "breathing" modes in the harmonic trap.We use the asymptotic Bethe Ansatz (BA) [18] which is justified at sufficiently small densities, when only twoparticle collisions are important [19]. The underlying idea goes back to the earlier days of high energy physics and was known as S-matrix theory [20] in the 1950s. The BA method considers the scattering matrix between asymptotic states as an alternative to a Hamiltonian/Lagrangean description. The scattering of ultracold atoms in 1D gases close to resonance can naturally be described by the scattering phase shift, whereas the formulation of a microscopic quantum Hamiltonian is difficult. It can be shown that the scattering matrix close to resonance corresponds to a highly singular, although local, two...
