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

SIGMA 9 (2013), 078, 14 pages      arXiv:1309.6464      https://doi.org/10.3842/SIGMA.2013.078
Contribution to the Special Issue in honor of Anatol Kirillov and Tetsuji Miwa

Integrable Hierarchy of the Quantum Benjamin-Ono Equation

Maxim Nazarov and Evgeny Sklyanin
Department of Mathematics, University of York, York YO10 5DD, United Kingdom

Received September 26, 2013, in final form December 03, 2013; Published online December 07, 2013

Abstract
A hierarchy of pairwise commuting Hamiltonians for the quantum periodic Benjamin-Ono equation is constructed by using the Lax matrix. The eigenvectors of these Hamiltonians are Jack symmetric functions of infinitely many variables $x_1,x_2,\ldots$. This construction provides explicit expressions for the Hamiltonians in terms of the power sum symmetric functions $p_n=x_1^n+x_2^n+\cdots$ and is based on our recent results from [Comm. Math. Phys. 324 (2013), 831-849].

Key words: Jack symmetric functions; quantum Benjamin-Ono equation; collective variables.

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