Integrable Hierarchy of the Quantum Benjamin-Ono Equation

Nazarov, Maxim

doi:10.3842/sigma.2013.078

Cited by 32 publications

(130 citation statements)

References 17 publications

(51 reference statements)

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“…In § [6] we present the Hamiltonian and Lax operator of a preliminary geometric quantization of (1.1). In § [7] we show how a modification of our preliminary quantization by the renormalization (1.9) of Abanov-Wiegmann [1] is implicit in the solution of the quantization problem for (1.1) by Nazarov-Sklyanin [40]. In § [8] we present the exact spectrum of the quantum periodic Benjamin-Ono hierarchy from [40].…”

Section: Figmentioning

confidence: 96%

“…As will appear in [37], Theorem [1.3.1] implies that the semi-classical and small dispersion asymptotics in the author's thesis [38] on Jack measures, a generalization of Okounkov's Schur measures [45], reflect the structure of quantum dispersive shock waves and quantum soliton trains emitted by coherent states from [5]. Note that a classical version of these small dispersion asymptotics, relating dispersive action profiles of the classical hierarchy in [40] to the formation of classical dispersive shock waves, has already been given by the author in [39]. 1.6.…”

Section: Figmentioning

confidence: 99%

“…is the principal minor of L • (v; ε). Note that the matrix (4.5) appears in [40] to define the generating function…”

Section: Classical Nazarov-sklyanin Hierarchy: Dispersive Action Profmentioning

confidence: 99%

“…, respectively, which are the bands and gaps of dispersive action profiles from [39] by Definition In the following theorem, we present the tori and cycles in description of the global action variables as gaps from Gérard-Kappeler [21]. The following is a strict subset of Theorem 1 from [21] and is stated here using relations between the constructions in [21,39,40] established above. • [Cycles] Λ b (ε) have a smooth global basis of cycles Γ b h (ε) indexed by h = 1, 2, .…”

Section: Classical Integrability: Gérard-kappeler Global Action Variamentioning

confidence: 99%

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

Moll

2019

SIGMA

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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 show that the renormalization of the classical dispersion coefficient in Abanov-Wiegmann is implicit in the definition of the quantum Lax operator in Nazarov-Sklyanin. Finally, we verify that the regular Bohr-Sommerfeld conditions for the multi-phase solutions in the renormalized theory give the exact quantum spectrum determined by Nazarov-Sklyanin without any Maslov index correction.

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Section: Figmentioning

confidence: 96%

Section: Figmentioning

confidence: 99%

“…is the principal minor of L • (v; ε). Note that the matrix (4.5) appears in [40] to define the generating function…”

Section: Classical Nazarov-sklyanin Hierarchy: Dispersive Action Profmentioning

confidence: 99%

Section: Classical Integrability: Gérard-kappeler Global Action Variamentioning

confidence: 99%

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

Moll

2019

SIGMA

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“…Such quantum integrable hierarchies exist e.g. for the quantum KdV system [35] and the quantum Benjamin-Ono equation [36].…”

Section: Local Spin Chainsmentioning

confidence: 99%

$$ T\overline{T} $$-deformation and long range spin chains

Pozsgay

Jiang

Takács

2020

J. High Energ. Phys.

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We point out that two classes of deformations of integrable models, developed completely independently, have deep connections and share the same algebraic origin. One class includes the TT -deformation of 1+1 dimensional integrable quantum field theory and related solvable irrelevant deformations proposed recently. The other class is a specific type of long range integrable deformation of quantum spin chains introduced a decade ago, in the context of N = 4 super-Yang-Mills theory. We show that the detailed structures of the two deformations are formally identical and therefore share many features. Both deformations preserve integrability and lead to non-local deformed theories, resulting in a change of the corresponding factorized S-matrices. We also prove a factorisation formula for the expectation value of the operators which trigger the deformation on the lattice; similar results in quantum field theory play an essential role in the solvability of such deformations. We point out that the long range deformation is a natural counterpart of the TT -deformation for integrable spin chains, and argue that this observation leads to interesting new avenues to explore.

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