  We derive ground state eigenfunctions and eigenvalues of various relativistic
elliptic integrable models. The models we discuss appear in computations of
superconformal indices of four-dimensional theories obtained by compactifying
six-dimensional models on Riemann surfaces. These include, among others, the
Ruijsenaars-Schneider model and the van Diejen model. The derivation of the
eigenfunctions builds on physical inputs, such as conjectured Lagrangian across
dimensions IR dualities and assumptions about the behavior of the indices in
the limit of compactifications on surfaces with large genus/number of
punctures/flux.
