Exact Bohr–Sommerfeld Conditions for the Quantum Periodic Benjamin–Ono Equation

In this paper we describe the spectrum of the quantum periodic Benjamin–Ono equation in terms of the multi-phase solutions of the underlying classical system (the periodic multi-solitons). To do so, we show that the semi-classical quantization of this system given by Abanov–Wiegmann is exact and equivalent to the geometric quantization by Nazarov–Sklyanin. First, for the Liouville integrable subsystems defined from the multi-phase solutions, we use a result of Gérard–Kappeler to prove that if one neglects the infinitely-many transverse directions in phase space, the regular Bohr–Sommerfeld conditions on the actions are equivalent to the condition that the singularities of the Dobrokhotov–Krichever multi-phase spectral curves define an anisotropic partition (Young diagram). Next, we locate the renormalization of the classical dispersion coefficient by Abanov–Wiegmann in the realization of Jack functions as quantum periodic Benjamin–Ono stationary states. Finally, we show that the classical energies of Bohr–Sommerfeld multi-phase solutions in the renormalized theory give the exact quantum spectrum found by Nazarov–Sklyanin without any Maslov index correction.