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Background: The inner crust of neutron stars consists of a Coulomb lattice ofneutron-rich nuclei, immersed in a sea of superfluid neutrons with backgroundrelativistic electron gas. A proper quantum mechanical treatment for such asystem under a periodic potential is the band theory of solids. The effect ofband structure on the effective mass of dripped neutrons, the so-called\textit{entrainment effect}, is currently in a debatable situation, and it hasbeen highly desired to develop a nuclear band theory taking into accountneutron superfluidity in a fully self-consistent manner.  Purpose: The main purpose of the present work is twofold: 1) to develop aformalism of the time-dependent self-consistent band theory, taking fullaccount of nuclear superfluidity, based on time-dependent density functionaltheory (TDDFT) extended for superfluid systems, and 2) to quantify the effectsof band structure and superfluidity on crustal properties, applying theformalism to the slab phase of nuclear matter in the $\beta$ equilibrium.  Results: Static calculations have been performed for a range of baryon(nucleon) number density ($n_b=0.04$--0.07 fm$^{-3}$) under the$\beta$-equilibrium condition with and without superfluidity, for variousinter-slab spacings. From a dynamic response to an external potential, weextract the collective mass of a slab and that of protons immersed in neutronsuperfluid. From the results, we find that the collective mass of a slab issubstantially reduced by 57.5--82.5\% for $n_b=0.04$--0.07 fm$^{-3}$, whichcorresponds to an enhancement of conduction neutron number density and, thus,to a reduction of the neutron effective mass, which we call theanti-entrainment effect. We discuss novel phenomena associated withsuperfluidity, quasiparticle resonances in the inner crust, which are absent innormal systems.  *shortened due to the arXiv word limit.

Superfluid extension of the self-consistent time-dependent band theory  for neutron star matter: Anti-entrainment vs. superfluid effects in the slab  phase