  Space-fractional quantum mechanics (SFQM) is a generalization of the standard
quantum mechanics when the Brownian trajectories in Feynman path integrals are
replaced by L{\'e}vy flights. We introduce L{\'e}vy quasicrystal by
discretizing the space-fractional Schr$\ddot{\text{o}}$dinger equation using
the Gr$\ddot{\text{u}}$nwald-Letnikov derivatives and adding on-site
quasiperiodic potential. The discretized version of the usual
Schr$\ddot{\text{o}}$dinger equation maps to the Aubry-Andr{\'e} Hamiltonian,
which supports localization-delocalization transition even in one dimension. We
find the similarities between L{\'e}vy quasicrystal and the Aubry-Andr{\'e}
(AA) model with power-law hopping and show that the L{\'e}vy quasicrystal
supports a delocalization-localization transition as one tunes the
quasiperiodic potential strength and shows the coexistence of localized and
delocalized states separated by mobility edge. Hence, a possible realization of
SFQM in optical experiments should be a new experimental platform to test the
predictions of AA models in the presence of power-law hopping.
