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

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...

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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
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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.

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Digital Library of Periodicals of National Academy of Sciences of Ukraine
id nasplib_isofts_kiev_ua-123456789-210290
record_format dspace
spelling Moll, A.
2025-12-05T09:22:39Z
2019
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.
1815-0659
2010 Mathematics Subject Classification: 37K40; 37K10; 53D50; 81Q20; 81Q80
arXiv: 1906.07926
https://nasplib.isofts.kiev.ua/handle/123456789/210290
https://doi.org/10.3842/SIGMA.2019.098
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.
The author would like to thank Chris Beasley, Percy Deift, Sam Johnson, Igor Krichever, Ryan Mickler, and Jonathan Weitsman for many helpful discussions. This work was supported by the Andrei Zelevinsky Research Instructorship at Northeastern University and also by the National Science Foundation RTG in Algebraic Geometry and Representation Theory under grant DMS1645877.
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Інститут математики НАН України
Symmetry, Integrability and Geometry: Methods and Applications
Exact Bohr-Sommerfeld Conditions for the Quantum Periodic Benjamin-Ono Equation
Article
published earlier
institution Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection DSpace DC
title Exact Bohr-Sommerfeld Conditions for the Quantum Periodic Benjamin-Ono Equation
spellingShingle Exact Bohr-Sommerfeld Conditions for the Quantum Periodic Benjamin-Ono Equation
Moll, A.
title_short Exact Bohr-Sommerfeld Conditions for the Quantum Periodic Benjamin-Ono Equation
title_full Exact Bohr-Sommerfeld Conditions for the Quantum Periodic Benjamin-Ono Equation
title_fullStr Exact Bohr-Sommerfeld Conditions for the Quantum Periodic Benjamin-Ono Equation
title_full_unstemmed Exact Bohr-Sommerfeld Conditions for the Quantum Periodic Benjamin-Ono Equation
title_sort exact bohr-sommerfeld conditions for the quantum periodic benjamin-ono equation
author Moll, A.
author_facet Moll, A.
publishDate 2019
language English
container_title Symmetry, Integrability and Geometry: Methods and Applications
publisher Інститут математики НАН України
format Article
description 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.
issn 1815-0659
url https://nasplib.isofts.kiev.ua/handle/123456789/210290
citation_txt 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.
work_keys_str_mv AT molla exactbohrsommerfeldconditionsforthequantumperiodicbenjaminonoequation
first_indexed 2025-12-07T21:25:02Z
last_indexed 2025-12-07T21:25:02Z
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