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Ammann-Beenker lattice is a two-dimensional quasicrystal with eight-foldsymmetry, which can be described as a projection of a cut from afour-dimensional simple cubic lattice. We consider the vertex tight-bindingmodel on this lattice and investigate the strictly localized states at thecenter of the spectrum. We use a numerical method based on the generation offinite lattices around a given perpendicular space point and QR decompositionof the Hamiltonian to count the strictly localized states. We apply this methodto count the frequency of localized states in lattices of up to 100 000 sites.We obtain an orthogonal set of compact localized states by diagonalizing theposition operator projected onto the manifold spanned by the zero energystates. We identify twenty localized state types and calculate their exactfrequencies through their perpendicular space images. Unlike the Penroselattice, all the localized state types are eight-fold symmetric around an eightedge vertex, and all vertex types can support localized states. The totalfrequency of these twenty types gives a lower bound of$f_{LS}=30796-21776\sqrt{2} \simeq 0.08547$ for the fraction of strictlylocalized states in the spectrum. This value is in agreement with the numericalcalculation and very close to the recently conjectured exact fraction oflocalized states $f_{Ex}=3/2-\sqrt{2}\simeq 0.08579$.

Strictly localized states in the octagonal Ammann-Beenker quasicrystal