  We discuss two physics-inspired approaches to derivation of the
eigenfunctions and eigenvalues of $A_N$ Ruijsenaars-Schneider model. First
approach which was recently proposed by the authors relies on the computations
of superconformal indices of class $\mathcal{S}$ $4d$ ${\mathcal N}=2$ theories
with the insertion of surface defects. Second approach uses computations of
Nekrasov-Shatashvili limit of $5d$ ${\mathcal N} = 1^*$ instanton partition
functions in the presence of co-dimension two defect. We compare results of
these two approaches for the low-lying levels of Ruijsenaars-Schneider model.
We also discuss different previously proposed exact quantization conditions for
the Coulomb branch parameters of the instanton partition functions and their
interpretations in terms of index calculations.
