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

• Published in: Symmetry, Integrability and Geometry: Methods and Applications
• Date: 2019
• Main Author: Moll, A.
• Format: Article
• Language: English
• Published: Інститут математики НАН України 2019
• Online Access: https://nasplib.isofts.kiev.ua/handle/123456789/210290
• Journal Title: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• Cite this: Exact Bohr-Sommerfeld Conditions for the Quantum Periodic Benjamin-Ono Equation / A. Moll // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 61 назв. — англ.

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.