  We theoretically consider a quantum magnetic impurity coupled to a
superconductor, and obtain the local density of states at the position of the
impurity taking into account the effect of spin-fluctuations and single-ion
magnetic anisotropy. We particularly focus on the spectrum of subgap
Yu-Shiba-Rusinov (YSR or Shiba) states induced by a quantum impurity with easy-
or hard-axis uniaxial anisotropy. Although this is a relevant experimental
situation in, e.g., magnetic adatoms on the surface of clean metals, it is
customary that theoretical descriptions assume a classical-spin approximation
which is not able to account for single-ion anisotropy and other quantum
effects. Here, quantum fluctuations of the spin are taken into account in the
equations of motion of the electronic Green's function in the weak-coupling
limit, and considerably modify the energy of the Shiba states compared to the
classical-spin approximation. Our results point towards the importance of
incorporating quantum fluctuations and anisotropy effects for the correct
interpretation of scanning tunneling microscopy (STM) experiments.
