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Universal high-momentum asymptote and thermodynamic relations in a spinless Fermi gas with a resonant $p$-wave interaction

We investigate universal relations in a spinless Fermi gas near a $p$-wave Feshbach resonance. We show that the momentum distribution $n_{\vec{k}}$ has an asymptote proportional to $k^{-2}$ with the proportionality constant--the $p$-wave contact-- scaling with the number of closed-channel molecules. We prove the adiabatic sweep theorem for a $p$-wave resonance which reveals the physical meaning of the $p$-wave contact in thermodynamics. In contrast to the unitary Fermi gas in which Tan's contact is universal, the $p$-wave contact depends on the short-range details of the interaction.