Number of zero-energy eigenstates in the PXP model

The PXP model is paradigmatic in the field of quantum many-body scars. This has a number zero-energy eigenstates that exponentially large system size. Lower bounds on are obtained for both open and periodic (zero $\pi$-momentum sectors) boundary conditions. These found to be tight up sizes accessible by numerical exact diagonalization, can expected general. In addition previous results, separate lower spatial inversion-symmetric inversion-antisymmetric symmetry sectors. Furthermore, derivations improve ones as these free assumptions.