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

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 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.

Gaussian Asymptotics of Jack Measures on Partitions from Weighted Enumeration of Ribbon Paths

“…In §8 of [43], we verified that the formulation of Theorem [4.3.5] given here agrees with the original Theorem 2 of Nazarov-Sklyanin in [45]. We refer to the collection of commuting operators (4.24) as the Nazarov-Sklyanin hierarchy.…”

“…Proposition [3.3.1] from [44] was independently proven by Gérard-Kappeler [21] in the more general case ∞ k=1 |V k | 2 < ∞ as discussed in [43,44]. We now define dispersive action profiles.…”

“…Dispersive action profiles were introduced in Definition 1.2.2 in [44] and were implicitly studied in Gérard-Kappeler [21] as shown in [43,44]. In [44], many properties of dispersive action profiles f (c|v; ε) are established.…”

A nonlinear Fourier transform for the Benjamin–Ono equation on the torus and applications

“…Consequently, if f n+1 |Sf n = 0, then • Either 1|uSf n = 0, so that, coming back to (21), Sf n is an eigenfunction of L u with the eigenvalue λ n + 1, which must be λ n+1 because of Lemma 1. • Or f n+1 |1 = 0, in which case γ n+1 = 0 from (20), and f n+1 = Sg n where, from (21),…”

“…The quantum Benjamin-Ono equation. Our nonlinear Fourier transform was recently used in [21] for establishing Bohr-Sommerfeld conditions for the quantum Benjamin-Ono equation. We refer to [21] for more references on this topic.…”