Completing the BLZ conjectures for quantum KdV monster potentials: degenerate-partition perturbation series and residual completeness regimes

The ODE/IM correspondence conjectures that the spectral determinants of the monster potentials - anharmonic oscillators with apparent singularities introduced by Bazhanov, Lukyanov and Zamolodchikov (BLZ) - realize all eigenstates of the quantum KdV model, with the level-N state count matching the number p(N) of integer partitions. Conti and Masoero (2021) reduced this to a perturbation problem in the large-momentum limit and proved it for non-degenerate partitions (now unconditionally, via Felder-Veselov-Nekrasov's proof of the Jacobian-eigenvalue conjecture), for completely degenerate partitions up to d=3, and for partially degenerate partitions of zero-root multiplicity at most 6 modulo an invertibility hypothesis; their follow-up (2023) established completeness for alpha>1 at large momentum conditional on the remaining cases, and Masoero-Ruzza (2026) proved completeness at the free-fermion point alpha=1. What remains open is the core of the program: existence and uniqueness of the perturbation-series solution for completely degenerate partitions with d>=4 (zero-root multiplicity at least 10), the analytic invertibility of the block matrices assumed in Proposition 8.1 for arbitrary multiplicity, the degenerate value alpha=5/2 where the leading perturbation vanishes, and unconditional completeness for 0<alpha<1 and generic momentum.

Background

The ODE/IM correspondence is a largely conjectural duality between the spectral theory of ordinary differential equations (the ODE side) and integrable quantum field theories solvable by the Bethe Ansatz (the IM side). It began with the discovery of Dorey and Tateo that the spectral determinant of the anharmonic oscillator $-d^2/dx^2+x^{2\alpha}$, $\alpha>0$, solves the nonlinear integral equations for the ground state of the quantum KdV model - the conformal field theory obtained by quantizing the second Hamiltonian structure of the classical Korteweg-de Vries equation, whose symmetry algebra is the Virasoro algebra, the infinite-dimensional algebra of generators of conformal transformations in two dimensions. Bazhanov, Lukyanov and Zamolodchikov (BLZ) extended the correspondence to all excited states: the relevant Schrodinger operators are the monster potentials $V(x)=\frac{L}{x^2}+x^{2\alpha}-2\frac{d^2}{dx^2}\sum_{k=1}^N\log\big(x^{2\alpha+2}-z_k\big),$ whose extra poles at $x^{2\alpha+2}=z_k$ must be apparent singularities (every solution of $\psi''=(V-E)\psi$ has trivial monodromy around them for every energy $E$), which is equivalent to the roots $z_k$ solving a system of $N$ coupled algebraic equations called the BLZ system. The parameters fix a Virasoro central charge $c=1-6\alpha^2/(\alpha+1)$ and highest weight, and $L=\ell(\ell+1)$ plays the role of angular momentum, so $L\to\infty$ is the large-momentum limit. BLZ conjectured a bijection between monster potentials with $N$ roots and the states at level $N$ of the quantum KdV model; since the level-$N$ subspace of a generic Virasoro module has dimension $p(N)$, the number of integer partitions of $N$, this predicts in particular that the BLZ system has exactly $p(N)$ solutions modulo permutations of the roots (the Weak BLZ conjecture), so that 'counting monster potentials' means counting solutions of this algebraic system.

Conti and Masoero attacked both conjectures through the large-momentum limit: the roots $z_k$ condensate about the complex equilibria of the ground-state potential $L/x^2+x^{2\alpha}$, and after an explicit scaling $z=\varphi(t;\varepsilon)$ with $\varepsilon=L^{-1/4}$ the BLZ system reduces to an algebraic perturbation system (their equation (5.25)) whose leading order governs the equilibria of rational extensions of the harmonic oscillator perturbed by a small cubic term $\varepsilon\kappa t^3$, with $\kappa=(5-2\alpha)/(3(2\alpha+2)^{1/4})$. Rational extensions of the harmonic oscillator with only apparent singularities are classified by Oblomkov's theorem: their pole polynomials are Wronskians of Hermite polynomials, parametrised by partitions $\nu$ of $N$. This reduces the Weak BLZ conjecture to a purely perturbative statement, Conjecture 5.8 (equivalently 5.9) of their paper: for every partition $\nu$, the perturbation system admits a unique algebraic Puiseux-series solution - an algebraic function of $\varepsilon$ with convergent expansions in fractional powers of $\varepsilon$ - with leading term $P^{[\nu]}(t)$. The case analysis rests on the Felder-Hemery-Veselov conjecture that the roots of these Wronskians are simple except possibly at the origin, which splits partitions into non-degenerate, completely degenerate ($P^{[\nu]}=t^N$ with $N=d(d+1)/2$ for $\nu=(d,d-1,\dots,1)$) and partially degenerate cases. The authors proved the non-degenerate case modulo the invertibility of a Jacobian matrix $J^{[\nu]}$ (their Conjecture 6.3), the completely degenerate cases $d\le3$, and the partially degenerate case with zero-root multiplicity at most 6 modulo an invertibility hypothesis on block matrices (their Proposition 8.1), leaving $d\ge4$ (i.e. $N\ge10$), arbitrary multiplicity, and the degenerate value $\alpha=5/2$ (where $\kappa=0$ and the perturbation ansatz changes) open, and they deferred rigorous proofs of the excited-state radial-spectrum asymptotics (their formula (5.23)) to a forthcoming study.

Subsequent work reshaped the status of these problems. Conti and Masoero (2023) classified all solutions with real positive roots of the quantum-KdV Bethe Ansatz equations by admissible hole-number sets parametrised by partitions and proved - conditional on Conjecture 5.9 - that monster potentials are complete for $\alpha>1$ and large momentum: every Bethe solution is the spectrum of a unique monster potential. Felder, Veselov and Nekrasov (2026) proved Conjecture 6.3: the eigenvalues of $J^{[\nu]}$ are twice the squares of the hook lengths of $\nu$, which settles the non-degenerate case unconditionally. Masoero and Ruzza (2026) proved the completeness conjecture outright at the free-fermion point $\alpha=1$ ($c=-2$) for all momenta, using Laguerre Wronskians and Crum-Darboux transformations (iterated Darboux transformations that add apparent poles while preserving the spectrum). What survives is the unconditional degenerate-partition analysis for general $\alpha$ and the completeness for the remaining regimes ($0<\alpha<1$, generic momentum), which is the subject of this problem.

Problem Statement

Complete the rigorous establishment of the BLZ conjectures for the quantum KdV model/monster potentials through the large-momentum perturbation program of Conti-Masoero. Concretely, prove the following three linked deliverables, which jointly discharge one objective - an unconditional proof of the bijection between monster potentials and quantum KdV states:

1. Degenerate perturbation series. Prove Conjecture 5.8 (equivalently Conjecture 5.9) of Conti-Masoero, JHEP 02 (2021) 059, in full generality: for every $\alpha>0$ and every partition $\nu$ of every $N\ge1$, the perturbation system (5.25) of that paper admits a unique algebraic Puiseux-series solution with leading term $P^{[\nu]}(t)$, unique up to permutations of equal roots. The cases still open are: (i) the completely degenerate partitions $\nu=(d,d-1,\dots,1)$ with $d\ge4$, where $P^{[\nu]}=t^N$ and the zero-root multiplicity is $N=d(d+1)/2\ge10$; (ii) an analytic proof of the invertibility of the block matrices $\tilde{J}^{[\nu]}$, $A^{[\nu]}$, $A^{[\nu]}-B^{[\nu]}(\tilde{J}^{[\nu]})^{-1}C^{[\nu]}$ and $\tilde{A}^{[\nu]}-\tilde{B}^{[\nu]}(\tilde{J}^{[\nu]})^{-1}\tilde{C}^{[\nu]}$ assumed in Proposition 8.1 for the partially degenerate case, extended to zero-root multiplicity greater than 6; and (iii) the special value $\alpha=5/2$, at which the cubic perturbation coefficient $\kappa=(5-2\alpha)/(3(2\alpha+2)^{1/4})$ vanishes and the perturbation ansatz must be amended. The solution may assume the Felder-Hemery-Veselov simplicity conjecture, as the source does, or handle exceptional root multiplicities directly.

2. Counting. Deduce, as a corollary of deliverable 1 via the source's reduction of the counting statement to Conjecture 5.8 (equivalently 5.9), the unconditional Weak BLZ conjecture for all $N$: for generic $\alpha$ and $L$, the BLZ system has exactly $p(N)\cdot N!$ solutions (equivalently, there are exactly $p(N)$ monster potentials modulo permutations of the roots) - currently rigorous for non-degenerate partitions (after Felder-Veselov-Nekrasov) and, for zero-root multiplicity at most 6, conditional on the block-matrix invertibility hypothesis of Proposition 8.1 of the source.

3. Completeness in the remaining regimes. Establish the bijection between monster potentials and quantum KdV states (the full BLZ conjecture) in the regimes not already settled - for $0<\alpha<1$ and for generic momentum, and without the Conjecture-5.9 hypothesis - including the input on the integrable side that remains to be supplied beyond the real-positive-root classification of Conti-Masoero (2023): the parametrisation of the Bethe states of quantum KdV by partitions and the identification of their hole-number sets. Supplying complete proofs of the excited-state radial-spectrum asymptotics (formula (5.23) of the source, derived there only briefly) is part of this deliverable's foundation.

Two further problems posed by the same source - the global structure (monodromy, branch points, irreducibility) of the $p(N)$-sheeted Riemann surface of higher-state potentials, and the extension of the large-momentum analysis to the massive ODE/IM correspondence and to higher-rank Kac-Moody algebras - are recorded as adjacent open problems and are not part of this problem's completion boundary.

References

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2. Riccardo Conti, Davide Masoero, 'On Solutions of the Bethe Ansatz for the Quantum KdV Model', Commun. Math. Phys. 402 (2023) 335-390. DOI: 10.1007/s00220-023-04728-3; arXiv:2112.14625; https://doi.org/10.1007/s00220-023-04728-3
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