  We study analytical properties and integrable structures of the meson
spectrum in large $N_c$ QCD$_2$. We show that the integral equation that
determines the masses of the mesons, often called the 't Hooft equation, is
equivalent to finding solutions to a TQ-Baxter equation. Using the Baxter
equation, we extract systematic expansions of the energy levels as well as
analytic asymptotic expressions for wavefunctions. Our analysis extends
previous results for a special quark mass by Fateev et al. to arbitrary quark
masses. This reformulation, together with its relation to an inhomogeneous
Fredholm equation, is particularly suited for analytical treatments and makes
accessible the analytic structure of the spectrum in the complex plane of the
quark masses. We also comment on applications of our techniques to
non-perturbative topological string partition functions.
