Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 15 (2019), 098, 27 pages      arXiv:1906.07926      https://doi.org/10.3842/SIGMA.2019.098

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

Alexander Moll
Department of Mathematics, Northeastern University, Boston, MA USA

Received June 20, 2019, in final form December 12, 2019; Published online December 18, 2019

Abstract
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.

Key words: Benjamin-Ono; solitons; geometric quantization; anisotropic Young diagrams.

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