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

## TL;DR

This paper establishes exact Bohr-Sommerfeld quantization conditions for the quantum periodic Benjamin-Ono equation, linking classical multi-phase solutions with quantum spectra through geometric and semi-classical methods.

## Contribution

It proves the semi-classical quantization is exact and connects classical spectral curves with quantum spectra, extending previous semi-classical approaches to an exact quantum description.

## Key findings

- Semi-classical quantization is exact for the quantum Benjamin-Ono equation.
- Classical spectral curves determine the quantum spectrum via Young diagrams.
- Classical energies match the quantum spectrum without Maslov correction.

## 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 G\'{e}rard-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.