Meson states in `t Hooft model: Hamiltonian approach

We point out that the masses of the highly excited bound quark-antiquark states in QCD$_2$ in the infinite $N$ limit may be determined in the framework of a simple quantum-mechanical model with the potential $V(x) = \sigma |x|$. In the ultrarelativistic case, the masses follow the pattern $$\mu_n^2 \ =\ \frac {g^2 N}{2\pi} n,$$ which coincides with the law derived by `t Hooft by solving the Bethe--Salpeter equation. In constrast to what follows from the exact analysis, the levels of the relativistic Hamiltonian have finite widths, but these widths are exponentially suppressed, $\Gamma_n \propto \exp\{-2\pi m^2/\sigma\}$ for large quark masses. In the nonrelativistic case, the levels follow the asymptotics $\epsilon_n = \mu_n - 2m = Cn^{2/3}$, where the constant $C$ can be determined by solving the `t Hooft equation or alternatively the nonrelativistic Schr\"odinger equation.