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 $V(x)=\sigma \mathrm{\mid }x\mathrm{\mid }$. 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 the nonrelativistic case, the energy levels follow the asymptotics $ε_n = μ_n - 2m = Cn^{2/3}$, where the constant $C$ can be determined by solving the t Hooft equation or alternatively the nonrelativistic Schrödinger equation.