  We provide a proposal, motivated by Separation of Variables and gauge theory
arguments, for constructing exact solutions to the quantum Baxter equation
associated to the $N$-particle relativistic Toda chain and test our proposal
against numerical results. Quantum Mechanical non-perturbative corrections,
essential in order to obtain a sensible solution, are taken into account in our
gauge theory approach by considering codimension two defects on curved
backgrounds (squashed $S^5$ and degenerate limits) rather than flat space; this
setting also naturally incorporates exact quantization conditions and energy
spectrum of the relativistic Toda chain as well as its modular dual structure.
