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 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.


Introduction and statement of result
In the semi-classical analysis of quantized Hamiltonian systems, a major goal is to approximate the quantum spectrum in terms of select periodic orbits of the underlying classical system. In special cases, a semi-classical approximation of the quantum spectrum may turn out to be exact. For quantizations of Liouville integrable systems, the classical energies of orbits satisfying the regular Bohr-Sommerfeld conditions give an approximation to the spectrum which is exact, e.g., for free particles on tori. Similarly, the WKB matching conditions provide an approximate spectrum which is exact, e.g., for harmonic oscillators. For quantized chaotic systems, the semi-classical approximation in the Gutzwiller trace formula is also exact in several cases, e.g., for free particles on any surface of constant negative curvature. For background on semi-classical and geometric quantization, see Kirillov [30] and Takhtajan [57].
In this paper we give an exact semi-classical description of the spectrum of the quantum periodic Benjamin-Ono equation in terms of distinguished quasi-periodic orbits of the underlying classical system known as multi-phase solutions, the periodic analogs of multi-soliton solutions. We state this result in Theorem 1.4 below. For recent mathematical surveys of classical and quantum periodic Benjamin-Ono equations, see [51,Section 5.2] and [46, Section 1.1.6], respectively.
for real v(x, t), x, t ∈ R, of spatial period 2π is Hamiltonian for the Gardner-Faddeev-Zakharov bracket as we review in Section 3. In (1.1), ε > 0 is a coefficient of dispersion whose notation we explain in Section 2.3. For T = R/2πZ, Molinet [36] proved (1.1) is globally well-posed in L 2 (T). We write v(x, t; ε) for solutions of (1.1). In their original derivation and analysis of the equation (1.1), both Benjamin [4] and Ono [47] found a 3-parameter family of periodic traveling waves that define periodic orbits of (1.1) known as 1-phase solutions: and permanent form (1.5) The parameters (1.2) define two closed intervals −∞, s ↑ 1 , s ↓ 1 , s ↑ 0 we call bands and one open interval s ↑ 1 , s ↓ 1 we call a gap. The wavespeed (1.4) is the midpoint of the band s ↓ 1 , s ↑ 0 . The wavelength of (1.5) is inversely proportional to the length of the band s ↓ 1 , s ↑ 0 and proportional to ε. As the band s ↓ 1 , s ↑ 0 shrinks or merges with the band −∞, s ↑ 1 , the permanent form (1.5) of the 1-phase solution (1.3) converges to that of a 1-soliton or constant solution of (1.1).

Multi-phase solutions of classical Benjamin-Ono
After the discovery of the family of 1-phase periodic orbits (1.3), Satsuma-Ishimori [50] found a larger family of quasi-periodic orbits of (1.1) known as multi-phase solutions. We now recall a formula for these multi-phase solutions due to Dobrokhotov-Krichever [16]. Throughout we write δ for Kronecker delta.

Bands and spatial periodicity conditions
The form of exponential terms in (1.8) implies: the length of the ith band [s ↓ i , s ↑ i−1 ] is a positive integer N i ∈ Z + multiple of ε for all i = 1, . . . , n, i.e., the ith 1-phase periodic traveling wave in the n-phase wave has N i bumps on T ∼ = R/2πZ.
The n = 1 case of Proposition 1.3 follows also by direct inspection of the cosine term in (1.5). We now show that additional conditions on lengths s ↑ i −s ↓ i of gaps s ↑ i , s ↓ i arise in quantization.

Statement of result: gaps and exact Bohr-Sommerfeld conditions
In Theorem 1.4 below, we give exact Bohr-Sommerfeld quantization conditions on the tori in phase space defined by the classical multi-phase solutions (1.7). As a consequence, we give a semi-classical interpretation of the results of Nazarov-Sklyanin [40] for the quantum periodic Benjamin-Ono equation and also show that the semi-classical soliton quantization of (1.1) by Abanov-Wiegmann [1] is exact. Recall for > 0 and any periodic orbit γ of a classical Hamiltonian O : M → R in a phase space (M, dα) with Liouville 1-form α, the -Bohr-Sommerfeld condition on γ is that the action γ α is a N ∈ Z + multiple of 2π : Recall also that for a self-adjoint operator O( ) in a Hilbert space (H, ·, · ) chosen to quantize O in (M, dα), the -Bohr-Sommerfeld approximation to the spectrum of O( ) is given by the classical energies O| γ of the periodic orbits γ satisfying (1.11). When O is Liouville integrable, one takes conditions (1.11) for each Hamiltonian O i in an integrable hierarchy containing O whose corresponding periodic orbits γ i are a basis of cycles on the Liouville tori. Using a recent description of (1.1) as a classical Liouville integrable Hamiltonian system in L 2 (T) by , in Section 10 we prove: Theorem 1.4. Let ε > 0 and > 0 be dimensionless coefficients of dispersion and quantization.
• [Part I: Bohr-Sommerfeld conditions] Let γ s i,n (ε) be the cycle in the phase space of (1.1) defined by varying only the ith phase χ i in the multi-phase initial data v s, χ (x, 0; ε) (1.7). The action of the Gardner-Faddeev-Zakharov Liouville 1-form α GFZ (3.4) along γ s i,n (ε) is 2πε times the length of the ith gap s ↑ i , s ↓ i . Neglecting the infinitely-many transverse directions in phase space to n-phase tori, the regular Bohr-Sommerfeld conditions (1.11) on the classical actions (1.12) are therefore that the length of the ith gap s ↑ i , s ↓ i is a positive integer N i ∈ Z + multiple of /ε > 0 for all i = 1, . . . , n with N i independent of N i in the description of band lengths in (1.10).
• [Part II: Exact Bohr-Sommerfeld conditions] The spectrum of the geometric quantization of (1.1) on T for fixed a = 2π 0 v(x) dx 2π found by Nazarov-Sklyanin [40] is the subset of classical energy levels s ↓ i , the spatial periodicity conditions (1.10), and the Bohr-Sommerfeld conditions (1.13) with ε in each replaced by the renormalized coefficient of classical dispersion determined by Abanov-Wiegmann [1]. The renormalization (1.14) in Part II of Theorem 1.4 reflects the fluctuations of the quantum system in the infinitely-many transverse directions to the Liouville tori neglected in Part I. The formula (1.14) for ε 1 can be characterized by the rectangles R(r 2 , r 1 ) of side lengths −r 2 √ 2, r 1 √ 2 for two choices of r 2 < 0 < r 1 in Fig. 1: for (1.15) (i) R(ε 2 , ε 1 ) encloses the same area 2 as R(− /ε, ε) and (ii) the intersection of R(ε 2 , ε 1 ) and the exterior of R(− /ε, ε) is a square.

Interpretation of result: Dobrokhotov-Krichever profiles and anisotropic partitions
We now interpret our Theorem 1.4 for the quantum multi-phase solutions in terms of partitions (Young diagrams) built from the rectangles in Fig. 1. For the classical multi-phase solutions, A subset of Dobrokhotov-Krichever profiles are the anisotropic partition profiles of Kerov [29]: Definition 1.6. For r 2 < 0 < r 1 and a ∈ R, a piecewise-linear real function f (c) of c ∈ R is an anisotropic partition profile of anisotropy (r 2 , r 1 ) centered at a if the region is a disjoint union of finitely-many translates of a −r 2 √ 2 × r 1 √ 2 rectangle R(r 2 , r 1 ). • [Part I] The original (approximate) regular Bohr-Sommerfeld conditions on the multi-phase v s, χ (x, t; ε) are that f (c| s) is an anisotropic partition profile of anisotropy (− /ε, ε).
Proof . f (c| s) is an anisotropic partition profile of anisotropy (r 2 , r 1 ) if and only if the band lengths s ↓ i −s ↑ i−1 = r 1 N i and gap lengths s ↑ i −s ↓ i = −r 2 N i for N i , N i ∈ Z + for i = 1, . . . , n.

Motivation
Our Theorem 1.4 establishes the presence of classical multi-phase solutions in the work of Nazarov-Sklyanin [40] and hence realizes Jack functions as quantum multi-phase states. Conversely, our Theorem 1.4 relates the results in Nazarov-Sklyanin [40] to the semi-classical studies of quantum Benjamin-Ono dynamics out of equilibrium by Abanov-Wiegmann [1], Bettelheim-Abanov-Wiegmann [5], and Wiegmann [60]. As will appear in [39], Theorem 1.4 implies that A. Moll the semi-classical and small dispersion asymptotics in the author's thesis [37] 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 as studied in Bettelheim-Abanov-Wiegmann [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, have already been discussed by the author in [38, Section 8].

Outline
In Section 2 we discuss our Theorem 1.4 and its relation to previous results. In Section 3 we review the Hamiltonian and Lax operator of (1.1). In Section 4 we recall the classical Nazarov-Sklyanin hierarchy [40] and its presentation in terms of dispersive action profiles from [38]. In Section 5 we identify the classical global action variables of Gérard-Kappeler [21] with the gaps in the dispersive action profiles from [38]. In Section 6 we define a geometric quantization of (1.1) by quantizing the Hamiltonian and Lax operator. In Section 7 we show that the renormalization of the classical coupling (1.14) in Abanov-Wiegmann [1] is implicit in the realization of Jack functions as quantum periodic Benjamin-Ono Hamiltonian eigenfunctions. In Section 8 we present the quantum Nazarov-Sklyanin hierarchy and its exact spectrum from [40]. In Section 9 we recall finite gap conditions for multi-phase solutions from [38]. In Section 10 we derive formula (1.12) for the classical actions from results of Gérard-Kappeler [21], establish the Bohr-Sommerfeld conditions (1.13), and prove Theorem 1.4.

Comparison with results for the Calogero-Sutherland equation
In [1], Abanov-Wiegmann give two derivations of the renormalization (1.14). The first derivation in field theory is at 1-loop by an effective action and choice of counterterms in a semi-classical quantization of (1.1) following Jevicki [25]. Part I of our Theorem 1.4 is a Hamiltonian counterpart to the 0-loop step in this first derivation, neglecting the infinitely-many transverse directions in the phase space of classical fields. The second derivation of (1.14) in hydrodynamics in [1] uses the realization of (1.1) in Calogero-Sutherland hydrodynamics and builds upon the work of Andrić-Bardek [2], Polychronakos [48], and Awata-Matsuo-Odake-Shiraishi [3]. For the quantum Calogero-Sutherland many-body problem, the analog of Part II of our Theorem 1.4namely, that after a shift the semi-classical quantization is exact -is well-known: see reviews by Calogero [8], Etingof [18], Ruijenaars [49], and Sutherland [56]. The works of Nazarov-Sklyanin [40,41] and Sergeev-Veselov [52,53] are exact extensions of the second hydrodynamic derivation of (1.14) in [1].

Comments on Hilbert schemes of points on surfaces
Our notation ε in (1.1), ε 1 in (1.14), ε 2 in (1.15), and a in (3.2) reflect the appearance of the quantum periodic Benjamin-Ono equation in equivariant cohomology of Hilbert schemes of points in C 2 reviewed in [46, Section 1.1.6]. To interpret our Theorem 1.4 in this context, note that our coefficient of classical dispersion ε = ε 1 + ε 2 is the deformation parameter of the Maulik-Okounkov Yangian [34] while our coefficient of quantization = −ε 1 ε 2 is the handlegluing element in [34]. These ε, appear in [46, Section 1.1.2] in trading C 2 for a surface S. For other exact Bohr-Sommerfeld conditions in the related theory of Nekrasov [42], see Mironov-Morozov [35]. Note that we have related dispersive action profiles of (1.1) to profiles in Nekrasov-Shatashvili [44] and Nekrasov-Pestun-Shatashvili [43] in [38, Section 2.5].
3 Classical periodic Benjamin-Ono: Hamiltonian and Lax operator In this section we recall the formulation of the classical Benjamin-Ono equation (1.1) for v periodic in x as a classical Hamiltonian system with respect to the Gardner-Faddeev-Zakharov symplectic form ω GFZ and discuss a complex structure J on the classical phase space. We also introduce the classical Lax operator L • (v; ε) of (1.1) and express the classical Hamiltonian in terms of L • (v; ε).

Classical phase space as Sobolev space from Gardner-Faddeev-Zakharov construction
For a ∈ R, we choose as the classical phase space of (1.1) the affine subspace M (a) of the s = −1/2 real L 2 -Sobolev space of T: is symplectic on the leaf M (a) in (3.2) and defines a symplectic form ω GFZ on M (a).
In geometric quantization of (1.1), we will use a compatible complex structure J on (M (a), ω GFZ ). (3.4)

Classical Hamiltonian at criticality
The Benjamin-Ono equation (1.1) is Hamiltonian: The a priori redundant constant term a 3 in (3.5) emerges naturally in Proposition 4.7 below. The classical Hamilton equations for (3.5) in Definition 3.4 are formal: M (a) is larger than the space L 2 (T) in which (1.1) is known to be well-posed [21,36]. From the perspective of dispersive equations, it is a coincidence that the symplectic space M (a) of (1.1) corresponds to its critical regularity s c = −1/2 as in Proposition 3.2. For discussion of criticality in (1.1), see Saut [51].

Classical Lax operator as generalized Toeplitz operator
We now review the definition of the classical Lax operator for Benjamin-Ono and its restriction to L 2 periodic Hardy space H • . Throughout the paper, the subscript "•" denotes a construction defined by the Szegő projection π • . Definition 3.5. Using the realization T = {w ∈ C : |w| = 1}, the L 2 -Hardy space H • on T is the Hilbert space closure of C[w] in H = L 2 (T). Equivalently, in terms of the Szegő projection Definition 3.6. For ε > 0 and bounded v, the classical Lax operator of (1.1) is the unbounded self-adjoint operator L • (v; ε) in Hardy space H • defined to be the unique self-adjoint extension of the essentially self-adjoint operator presented in the basis |h = w h for h = 0, 1, 2, . . . of C[w]. Equivalently, the Lax operator is the generalized Toeplitz operator of order 1, where L(v) is the operator of multiplication by v, For background on Toeplitz operators, see Deift-Its-Krasovsky [15]. The classical Lax ope-

Classical Hamiltonian from classical Lax operator
By direct computation, one has: are matrix elements of powers of the classical Lax operator We use the same notation "↑" as in multi-phase parameters s ↑ i in anticipation of results in Section 9.

Classical Nazarov-Sklyanin hierarchy: dispersive action profiles
In this section we recall the classical integrable hierarchy for (1.1) from Nazarov-Sklyanin [40], a collection of Poisson commuting Hamiltonians built from the Lax operator (3.6). We also present the dispersive action profiles introduced by the author in [38] which encode this classical hierarchy through spectral shift functions.

Classical Nazarov-Sklyanin hierarchy
The following generalizes (3.9) and (3.10): Definition 4.1. The classical Nazarov-Sklyanin hierarchy is the family of Hamiltonians defined as matrix elements of the th power of the Lax operator (3.6) for |0 = w 0 = 1 ∈ H • .

A. Moll
Building upon their work [41], in [40] Nazarov-Sklyanin define a classical Baker-Akhiezer function of u ∈ C \ R, w = e ix , and prove Theorem 4.2 by showing Poisson-commutativity of all 3) is the average value of (4.2) on T. Both (4.2), (4.3) are degenerations of quantum objects in Nazarov-Sklyanin [40] as we review in Section 8.2.
In recent work, Gérard-Kappeler [21] independently discovered the generating function (4.3) and classical Baker-Akhiezer function (4.2) from [40] and gave a new proof of

Embedded principal minor of classical Lax operator
The material in this section Section 4.2 and the next section Section 4.3 is necessary in order to define in the following section Section 4.4 the dispersive action profiles for (4.3) introduced by the author in [38]. From now on, we assume v bounded.

Essential self-adjointness and perturbation determinants
The next results are from [38, Section 3] and give a generalization of Cauchy's interlacing theorem from finite-rank to essentially self-adjoint operators: for any u ∈ C \ R the resolvent matrix element is a multiple of the perturbation determinant which is well-defined by the Fredholm determinant since L • − L + is rank 2 hence trace class.
For general pairs of self-adjoint operators L • , L + whose difference L • − L + is trace class, the spectral shift function ξ(c|L • , L + ) is defined by identifying the right-hand side of (4.7) with the perturbation determinant in (4.6). For background on spectral shift functions, see Birman-Pushnitski [6]. For bounded L • , both Proposition 4.4 and Corollary 4.5 are due to Kerov [28] in his theory of profiles and interlacing measures.

Dispersive action profiles: bands and gaps
We next recall from [38] why dispersive action profiles f (c|v; ε) are piecewise-linear with slopes ±1. This motivates the next definition as in Section 1.

Definition 4.8 ([38]
). The bands of dispersive action profiles f (c|v; ε) are the closures of the connected intervals of c ∈ R in which f (c|v; ε) has slope −1. The gaps of dispersive action profiles f (c|v; ε) are the interiors of the connected intervals of c ∈ R in which f (c|v; ε) has slope +1.
bounded above with −∞ as the only point of accumulation.
and hence the dispersive action profile f (c|v; ε) is piecewise-linear with slopes ±1.

Classical integrability: Gérard-Kappeler global action variables
In this section we identify the global action variables of Gérard-Kappeler [21] in L 2 (T) ∩ M (a) with the gaps of dispersive action profiles f (c|v; ε) from Definition 4.8 introduced in [38] in the case of bounded v. We also state the characterization in Gérard-Kappeler [21] of these global action variables as integrals of the Liouville 1-form α GFZ in (3.4) along a basis of cycles Γ b h (ε) of generically infinite-dimensional tori Λ b (ε) parametrized by profiles b. This relationship between profiles and classical actions plays a key role in our proof of Theorem 1.4 in Section 10.

Principal minor and shift relation
Inspection of (4.4) immediately gives the shift relation: in the dense subspace of H + is unitarily equivalent to that of a shifted classical Lax operator As a consequence, the eigenvalues C ↓ h (v; ε) of the embedded principal minor L + (v; ε) of the classical Lax operator L • (v; ε) can be calculated from those of L • (v; ε) by the shift relation

Interlacing property and simplicity of spectrum
As a first application of the shift relation in Lemma 5.1, we give a new short proof of Proposition 2.1 in Gérard-Kappeler [21] for bounded v.
Proof . Use formula (5.1) to write the interlacing property (4.11) of Corollary 4.10 as which implies the bound and that C ↑ h (v; ε) < C ↑ h−1 (v; ε) all inequalities in (4.10) are strict.

Bands and spatial periodicity conditions II
Next, we derive for generic v a counterpart to Proposition 1.3 for multi-phase v.

Gaps as Gérard-Kappeler global action variables
We now give a description of gaps: Proof . Follows from the interlacing inequalities in Corollary 4.10 and Proposition 5.3.
After our study of gaps of dispersive action profiles in [38], the same gaps were shown to be global action variables in a comprehensive analysis by Gérard-Kappeler [21] who found global action-angle variables for (1.1) posed in the space of real L 2 functions on T. We now present Gérard-Kappeler's description of the gaps in the following theorem, which is a strict subset of Theorem 1 from [21] and stated here using relations between constructions in [21,38,40]   consisting of all v whose dispersive action profiles are equal to a fixed profile b(c).

A. Moll
• [Cycles] The map ϕ which takes v to its classical Baker-Akhiezer function (4.2) is injective and has an inverse ϕ −1 defined on the image of ϕ. A smooth global basis of cycles Γ b h (ε) ∞ h=1 on the Liouville tori Λ b (ε) is given by the pushforward along ϕ −1 of the cycle in the space of meromorphic functions in u which rotates the residue at u = C ↑ h (v; ε). • [Actions] For α GFZ the Liouville 1-form from Proposition 3.3, the classical actions

Quantum periodic Benjamin-Ono: Hamiltonian and Lax operator
In this section we quantize the classical Benjamin-Ono equation (1.1) for v periodic in x by choosing J-holomorphic quantizations of the classical phase space, Hamiltonian, and Lax operator in Section 3.

Quantum state space as Fock-Sobolev space from Segal-Bargmann construction
Recall from Proposition 3.2 that the spatial Hilbert transform J defines a complex structure on the classical phase space M (a) compatible with the metric g −1/2 associated to the L 2 -Sobolev norm of regularity s = −1/2. As a state space for the quantization of (1.1) we choose the Fock space of J-holomorphic functionals on M (a) given by the Segal-Bargmann construction.
To emphasize its dependence on the regularity s = −1/2, we may refer to this Fock space as Fock-Sobolev space. in the infinitely-many Fourier modes V k from (3.1) with inner product ·, · defined by requiring The Fourier modes V ±k are functionals on M (a) which satisfy V ±k = V ∓k and are also canonical coordinates on (M (a), ω GFZ ) as seen in (3.3). The quantum analogs of V ±k are also well-known: Definition 6.3. The creation and annihilation operators are the mutually-adjoint operators of multiplication V k = V k and differentiation V −k = k ∂ ∂V k , respectively, in F (a), ·, · satisfying the quantum canonical commutation relations of the same form as the classical relations (3.3).

Quantum Hamiltonian at criticality
We now quantize the classical Benjamin-Ono equation (1.1) in M (a) by replacing V ±k in the classical Hamiltonian (3.5) by V ±k from Definition 6.3.
Definition 6.4. For independent of ε and a, V ±k in Definition 6.3, and V 0 = a, the quantum periodic Benjamin-Ono equation is the quantum Hamiltonian system in Fock-Sobolev space F (a), ·, · determined by the quantum Hamiltonian defined without normal ordering by The procedure of directly replacing classical canonically conjugate modes by their quantum analogs usually results in an ill-defined operator in Hilbert space that must be regularized by normal ordering. An important feature of the formula (3.5) is that substituting V k → V k in (3.5) results in (6.1) which is well-defined without normal ordering.

Quantum Lax operator as generalized Fock-block Toeplitz operator
Let F be a vector space over C and H • the Hardy space of Definition 3.5. Recall that block Toeplitz operators on F ⊗ H • are Toeplitz operators in H • whose matrix elements are linear operators on F. For background on block Toeplitz operators, see Section 10 in Deift-Its-Krasovsky [15]. Using material from Section 3.3, we define the quantum Lax operator for (1.1) as a generalized block Toeplitz operator in F (a) ⊗ H • . Since F = F (a) is Fock space, we call it a generalized "Fock-block" Toeplitz operator. Definition 6.5. For ε > 0 and > 0, the quantum Benjamin-Ono Lax operator is the selfadjoint operator L • ( v(·, ); ε) in F (a) ⊗ H • realized as the unique extension of where V ±k are from Definition 6.3 and V 0 acts by the scalar a. Equivalently, is the generalized Fock-block Toeplitz operator of order 1 whose symbol is the affine defined by replacing V ±k → V ±k in the Fourier series (3.1) for the classical field v.

A. Moll
The Fock-block matrix (6.2) is essentially self-adjoint in F (a) ⊗ C[w] due to the Szegő projections. Indeed, since F (a) = C[V 1 , V 2 , . . .], the matrix (6.2) preserves the dense subspace in question and commutes with the operator ∞ k=1 V k V −k ⊗ 1 + 1 ⊗ D • with finite-dimensional eigenspaces so essential self-adjointness follows by Nussbaum's criteria. As a corollary, for any Φ out , Φ in ∈ C[w], the matrix element of th powers of the quantum Lax matrix preserves F (a). By contrast, without π • , th powers of the operator L( v(·, )) which multiplies by the current (6.3) with zero mode a and level are ill-defined on F (a), a well-known issue in the theory of vertex algebras that is discussed in Kac [26].

Quantum Hamiltonian from quantum Lax operator
As in Section 3.4, we have: Proposition 6.6. The quantum periodic Benjamin-Ono Hamiltonian can be recovered as are matrix element of powers of the quantum Lax operator (6.2) for |0 = w 0 = 1 ∈ H • .

Quantum stationary states: Jack functions and Abanov-Wiegmann renormalization
In this section we show how the renormalization ε → ε 1 in (1.14) of the classical dispersion coefficient in Abanov-Wiegmann [1] is implicit in the known realization of Jack functions as quantum periodic Benjamin-Ono stationary states.

Quantum periodic Benjamin-Ono stationary states
Recall the definition of partitions.
Proof . By the definition of the quantum Lax operator (6.2), (6.4) is independent of ε and is T 2 ( ) acts diagonally on V µ with eigenvalue j µ j . By direct calculation, it commutes with O 3 (ε, ) in (6.1), hence O 3 (ε, ) preserves the finite-dimensional eigenspaces of T 2 ( ) spanned by V µ with fixed j µ j . Since (6.1) is symmetric under the exchange V k ↔ V −k which are mutual adjoints in Fock space, O 3 (ε, ) is self-adjoint on the finite-dimensional eigenspaces of T 2 ( ). The result then follows from the spectral theorem.

Quantum Nazarov-Sklyanin hierarchy: anisotropic partition profiles
In this section we present the solution of the quantization problem for the classical integrable system (1.1) posed in M (a) by Nazarov-Sklyanin [40]. We also present the exact formula in Nazarov-Sklyanin [40] for the spectrum of their quantum integrable hierarchy in terms of anisotropic partition profiles of anisotropy (ε 2 , ε 1 ) from Definition 1.6.

The quantization problem for classical integrable systems
Given a smooth symplectic manifold (M, ω), we recall four types of quantizations of Poisson subalgebras A ⊂ C ∞ (M, R) and state the quantization problem for classical integrable systems. The definitions below are all standard. We refer the reader to the survey of Faddeev [19] and to Dubrovin [17, Definition 1.1].
Definition 8.1. For formal , a deformation quantization of A is a phase space star product For > 0, an operator quantization Q of A is a unitary representation of from Definition 8.1, i.e., a choice of a Hilbert space of quantum states (H, ·, · ) and a map to the space iu(H, ·, · ) of self-adjoint operators in (H, ·, · ) so the pullback Q of multiplication of self-adjoint operators is a deformation quantization. Below we write Q (O) = O Q ( ).

Definition 8.3.
Given an almost complex structure J on M compatible with ω and associated Riemannian metric g, a J-holomorphic quantization of A is an operator quantization Q so that the symmetric part of the first bidifferential defined by (8.1) is the inverse metric g −1 .
Definition 8.4. Given a Poisson-commutative subalgebra T ⊂ A, i.e., for all T 1 , T 2 ∈ T a T-commutative quantization of A is an operator quantization Q so that for all T 1 , T 2 ∈ T, the quantization of T is a commutative subalgebra of self-adjoint operators in iu(H, ·, · ).
The quantization problem for integrable systems is to construct an T-commutative quantization of A ⊂ C ∞ (M, R) where T is the Poisson commutative subalgebra spanned by a classical integrable hierarchy in (M, ω). For further discussion, see [17, Definition 1.1].
Remark 8.7. The proof of Theorem 8.6 in Nazarov-Sklyanin [40] relies on their earlier work [41] where they construct a different family of commuting operators A(u|ε, ) diagonalized on Jack functions (7.1). The A(u|ε, ) in [41] serve to define a quantum Baker-Akhiezer function in formula (7.1) of [40]. While we do not make use of the quantum Baker-Akhiezer function below, for completeness let us mention that in the notation of [40] the classical limit is α → 0, hence by formulas (5.5), (7.1) in [40] as α → 0 the quantum Baker-Akhiezer function degenerates to the classical Baker-Akhiezer function (4.2). Recall also that (4.2) was independently discovered by Gérard-Kappeler [21] and plays a role in their results which we reviewed in Theorem 5.5.

Partitions and anisotropic partition profiles
Lemma 8.8. For any r 2 < 0 < r 1 and a ∈ R fixed, there is a bijection between partitions and anisotropic partition profiles of anisotropy (r 2 , r 1 ) centered at a ∈ R from Definition 1.6.
Proof . The rectangles which tile the region (1.16) below an anisotropic partition profile are grouped in rows of positive slope indexed by h = 1, 2, 3, . . . starting from the right. The count λ h of the number of rectangles R(r 2 , r 1 ) in the hth row defines the necessary bijection (8.7).

Spectrum of the quantum Nazarov-Sklyanin hierarchy
Without relying on knowledge of the classical multi-phase solutions of (1.1) nor on semi-classical approximation, not only did Nazarov-Sklyanin [40] solve the quantization problem for (1.1) by constructing the hierarchy (8.2), they also found the exact quantum spectrum (Hamiltonian eigenvalues) of this hierarchy at the Jack functions P λ,a (V |ε, ) (7.1) in terms of the Jack parameter α and the parts λ h of the partition λ. One striking feature of the spectrum in [40] is that it can be presented in terms of the anisotropic partition profile f λ (c − a|ε 2 , ε 1 ) from Lemma 8.8 using the conventions from Section 7.2: Theorem 8.9 (Nazarov-Sklyanin [40]). For any partition λ, a ∈ R, and ε, > 0, the eigenvalue T ↑ (u; ε, )P λ;a (V |ε, ) = T ↑ λ;a (u|ε, )P λ;a (V |ε, ) of the generating function of the quantum periodic Benjamin-Ono hierarchy (8.2) defined by at a Jack function P λ;a (V |ε, ) ∈ F (a) in the Fock-Sobolev space associated to the classical phase space (M (a), ω GFZ ) with zero mode a = 2π 0 v(x) dx 2π is given for u ∈ C \ R by where f λ (c − a|ε 2 , ε 1 ) is the anisotropic partition profile of anisotropy (ε 2 , ε 1 ) centered at a ∈ R.

Multi-phase solutions: finite-gap conditions
In this section we recall our result from [38] that multi-phase solutions (1.7) of (1.1) are finitegap. We also discuss why this result agrees with a subsequent classification of finite-gap solutions by Gérard-Kappeler [21] in order to apply their computations of classical action integrals for arbitrary v -which we presented in Theorem 5.5 -to multi-phase v = v s, χ (x, t; ε) in Section 10.

Multi-phase solutions are finite-gap
The multi-phase solutions (1.7) have not appeared at all in Sections 3, 4, 5, 6, 7 and 8 above.
In particular, for multi-phase v = v s, χ , only finitely-many gaps s ↑ i , s ↓ i are non-empty. In the following proposition, we refine the spectral description of f (c| s) in Theorem 9.1.

Multi-phase solutions from Gérard-Kappeler classification
As a consequence of the identification of gaps in [21,38] in Section 5.4, Theorem 9.1 can also be seen to follow from a recent classification of finite gap solutions: for C 0 ∈ R and τ v a polynomial in w = e ix whose zeroes all lie outside the closed unit disk.
Proof that Theorem 9.3 implies Theorem 9.1. By the Dobrokhotov-Krichever formula (1.7), the classical multi-phase solutions of Satsuma-Ishimori [50] are of the form (9.3) for τ v (e ix ) = det M s, χ n (x, t; ε) where the entries of the n × n matrix M s, χ n in (1.8) are polynomials in w = e ix . By Lemma 1.1 in Dobrokhotov-Krichever [16], the n eigenvalues of M s, χ n in w = e ix lie outside the closed unit disk, so by Theorem 9.3, (1.7) are finite-gap.
Remark 9.4. w = e ix in Theorem 9.3 matches C[w] in Definition 3.5 since in [21] there is a relationship between Φ BA (u, w|v; ε) in (4.2) and a τ v as in (9.3) for all v ∈ M (a) ∩ L 2 (T). In [38], we verified such a relationship for multi-phase v: at v = v s, χ , Φ BA (u, w|v; ε) comes from the Baker-Akhiezer function on the singular spectral curves in Dobrokhotov-Krichever [16] defined from two solutions to two non-stationary Schrödinger equations whose time-dependent potentials determine τ v in (9.3).

Multi-phase solutions: Bohr-Sommerfeld conditions
In this section we prove Theorem 1.4 in 7 Steps. In Steps 1-5 we derive formula (1.12). In Step 6 we derive formula (1.13), completing the proof of Part I. In Steps 7-9 we prove Part II.

10.1
Step 1: 1-phase case of (1.12) For the 1-phase Benjamin-Ono periodic traveling wave (1.3) with s ↑ 1 < s ↓ 1 < s ↑ 0 , the n = 1, i = 1 case of (1.12) can be directly computed from the closed formula (1.3) and the series formula (3.4) for the Liouville 1-form to give We omit the calculation of (10.1) since it is also the n = 1 case of formula (10.6) in Step 3 below. Note that Step 3 below is logically independent of Step 2 below, so we can use (10.1) in Step 2.

10.2
Step 2: Asymptotic validity of (1.12) As in (1.10), for N = (N n , . . . , N 1 ) ∈ Z n + define As will be important below, note that both R n ≥ = [0, ∞) n and R n > = (0, ∞) n are simply-connected. If all gap lengths s ↑ j − s ↓ j → ∞ diverge, i.e., in any limit to ∞ in B reg that (1.12) holds asymptotically. The proof of (10.3) is as follows: as all gap lengths diverge, the off-diagonal entries of the matrix (1.8) vanish, hence the logarithmic derivative of the determinant in (1.7) splits into a sum indexed by j = 1, . . . , n. Since the cycle γ s i,n (ε) varies only χ i , and since χ i appears only in the term with j = i, the action integral is asymptotically given by the n = 1 case in Step 1. The asymptotic relation (10.3) appears in the proof of Lemma 1.1 in Dobrokhotov-Krichever [16] and is the regime in which the multi-phase solution becomes a linear superposition of 1-phase solutions.

12) is constant
We claim that the coefficients in (10.5) do not depend on s. This is a short but crucial step in the proof. is), so the completely integrable system associated to the n-phase solutions is monodromy-free.

10.5
Step 5: Evaluation of (1.12) By (10.6) and (10.7), to prove (1.12) it suffices to prove which is equivalent to the n relations C j,i (ε) = δ(i − j). Restating the asymptotic relation (10.3) from Step 2 using the decomposition (10.8), in any limit in which all Taking n different limits in which all gaps diverge but the jth gap grows faster than the others gives the desired n relations C j,i (ε) = δ(i − j). • ω = dα an exact symplectic form with Liouville 1-form α, • T : M → B the associated moment map to a simply-connected base B of dim R B = n, Lagrangian fibers given by the Liouville tori, • γ b i a basis of cycles of the tori Λ b indexed by i = 1, . . . , n, • b ∈ B reg ⊂ B a regular value of T , the regular Bohr-Sommerfeld conditions on b are the n conditions for i = 1, . . . , n given by 9) where N i ∈ Z + is a positive integer and > 0 is a dimensionless real parameter of quantization.

A. Moll
Bohr-Sommerfeld conditions -and associated semi-classical approximations of quantum spectra -have been long studied in mathematical physics. For background, see Takhtajan [57,Section 6.3], Vũ Ngoc [59,Section 5], and Woodhouse [61,Section 8.4]. Definition 10.1 is a special case of the definition of Bohr-Sommerfeld leaves of general real polarizations of M (whose Lagrangian leaves Λ are not necessarily tori).
In practice, the assumption [ω] = 0 that the symplectic form is exact is often weakened to [ω] ∈ H 2 (M ; Z), thus trading ω = dα for the realization ω = F ∇ of the symplectic form as the curvature 2-form of a connection ∇ on a line bundle L → M . In this setting, one reformulates the Bohr-Sommerfeld conditions as the requirement that the holonomy group of the flat connection ∇| Λ is trivial. For simply-connected B, a result of Guillemin-Sternberg [23] guarantees that this more general definition specializes to our Definition 10.1 above.
Next, we argue that the assumptions in Definition 10.1 apply to our problem. At first glance, this seems impossible: the multi-phase profiles b(c) = f (c| s) are certainly not regular values of the moment map which takes v to its dispersive action profile b(c) = f (c|v; ε) (or, equivalently, the gap lengths). As we saw in Step 3, the tori Λ f (·| s) explored by multi-phase solutions has realdimension n, but generic tori Λ b (ε) in (5.2) are infinite-dimensional (generic v are infinite-gap). However, in Part I we are to neglect the infinitely-many transverse directions in phase space to Λ f (·| s) , an assumption that will allow us to use the regular Bohr-Sommerfeld conditions. For N = (N n , . . . , N 1 ) ∈ Z n + , consider the n spectral indices h n > · · · > h 1 for h j = N j + · · · + N 1 from Proposition 9. given by taking the multi-phase profile (or, equivalently, the gap lengths). Notice that the base B N ;n (a; ε) ∼ = R n ≥0 is simply-connected, and that B reg   where f λ (c − a|ε 2 , ε 1 ) is the anisotropic partition profile of anisotropy (ε 2 , ε 1 ) centered at a in Lemma 8.8. Formula (10.14) is the coefficient of u −4 in the logarithmic derivative of (8.8).

10.9
Step 9: Exact Bohr-Sommerfeld conditions We now prove Part II of Theorem 1.4. By formula (10.14), the exact spectrum of the quantum periodic Benjamin-Ono equation in F (a) is indexed by partitions λ with quantum energy levels +∞ −∞ c 3 1 2 f λ (c − a|ε 2 , ε 1 )dc. By formula (10.12), this quantum spectrum coincides with the classical energy levels of the multi-phase solutions whose multi-phase profiles f (c| s) have s ↓ i and band and gap lengths for N i , N i ∈ Z + . These are the spatial periodicity conditions (1.10) and regular Bohr-Sommerfeld conditions (1.13) after the renormalization (1.14) of Abanov-Wiegmann [1].