where \( \begin{array} { r } { \alpha _ { \pm } \, = \, 6 \frac { V _ { b g } } { a _ { \perp } ^ { 2 } a _ { 1 D } ( B _ { \pm } ) } \, + \, \frac { V _ { b g } } { V _ { p } ^ { C I R } } } \end{array} \). In Figs. 7 and 8 we show the calculated resonance width \( \Delta _ { 1 D } \) as a func-tion of the rescaled energy \( \epsilon \) and waveguide frequency \( \omega _ { \perp } \), respectively, together with the analytical result ac-cording to Eq.(31). One observes a good agreement for \( \omega _ { \perp } < 0 . 0 1 \). However, we encounter major deviations with increasing \( \omega _ { \perp } \) except for the zero energy limit \( \epsilon \rightarrow \) 0. These deviations are due to the fact that the analytical formula (30) which we used to derive (31), was obtained in the zero energy limit and does not work for larger en-ergies. In Fig.9 we present the real part of our scattering amplitude \( R e ( f _ { p } ) \) along with the analytical results from (30) as a function of \( V _ { p } / V _ { \perp } \) for \( \epsilon \) = 0.9 (large energy) for comparison. As the trap frequency \( \omega _ { \perp } \) increases we observe deviations between the analytical and numerical results for \( f _ { p } \).

The same way as in the case of the free space reso-nance, the width \( \Delta _ { 1 D } \) of the CIR narrows with decreas-ing energy \( \epsilon \) (see Fig.7). Fig.8 shows that in a harmonic waveguide there is a region where we have a possibility for narrowing the width by decreasing the trap frequency.

# VI. CONCLUSION

We develop and analyze a theoretical model to study Feshbach resonances of identical fermions in atomic waveguides by extending the two-channel model sug-gested in [33] and adopted in [32] for confined bosons. In this model, the experimentally known parameters of Feshbach resonances in free space are used as an input. Within this approach we have calculated the shifts and widths of p-wave magnetic Feshbach resonance of \( { } ^ { 4 0 } \mathrm { K } \) atoms in the hyperfine state \|F = 9/2, mF = −7/2i and for the relative angular momentum state \|l = 1, ml = 0i emerging in harmonic waveguides as p-wave CIRs. We find a linear dependence of the resonance position on the longitudinal colliding energy below the threshold for the first transverse excitation. It is shown that in a har-monic waveguide there is the possibility to decrease the width of the p-wave Feshbach resonance by decreasing the (transversal) trap frequency which could be used in corresponding experiments. Our analysis demonstrates the importance of including the effective range terms in the computational schemes for the description of the p-wave CIRs contrary to the case of s-wave CIRs where the impact of the effective radius is negligible. In previous in-vestigations of the p-wave CIRs in harmonic waveguides[13, 17, 23] the effects due to the effective range have been neglected. The developed model can be applied for a quantitative analysis of other p-wave CIRs following a different spin structure and for confining traps of differ-ent geometry including effects due to anharmonicity and anisotropy.

# VII. ACKNOWLEDGEMENTS

Sh.S would like to thank J. Abouie and S. Abedin-pour for fruitful discussions. V.S.M.and P.S. acknowl-edge financial support by the Heisenberg-Landau Pro-gram. V.S.M. thanks the Zentrum f¨ur Optische Quan-tentechnologien of the University of Hamburg and Sh.S. thanks the Bogoliubov Laboratory of Theoretical Physics of JINR for their warm hospitality. This work was sup-ported by IASBS (Grant No.G2014IASBS12648).

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