Exact Bohr-Sommerfeld conditions for the quantum periodic Benjamin-Ono equation (Q2297929)

The author of this article is investigating the exact Bohr-Sommerfeld conditions for the quantum Benjamin-Ono equation. In other words the author is describing the spectrum of the quantum Benjamin-Ono equation in terms of the multi-phase solutions of the classical system, known also as periodic multi-solutions. In order to obtain the desired description the author proves that the semiclassical quantization of this system given by Abanov-Wiegmann is exact and equivalent to the geometric quantization realized by Nazarov and Sklyanin. Using results previously found by Gerard-Kappeler, Dobrokhotov and Krichever, and Nazarov-Sklyanin, the author finally proves 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.