The ODE/IM (ordinary differential equation / integrable model) correspondence relates the spectral data of one-dimensional Schrödinger equations to the functional relations of two-dimensional quantum integrable models. Closed TBA (thermodynamic Bethe ansatz) or NLIE (nonlinear integral equation) systems are known for the ground state, including general polynomial potentials with a regular singularity (Ito-Shu 2020). The excited states of the integrable model correspond, on the differential-equation side, to 'monster' potentials: fine-tuned potentials carrying additional trivial-monodromy singularities whose positions obey Bethe-like algebraic equations (Bazhanov-Lukyanov-Zamolodchikov). Although the higher-state correspondence has been proven at the level of Bethe-ansatz equations, QQ-systems, and spectral determinants (Masoero-Raimondo; Conti-Masoero, 2020-2026), no TBA or NLIE system describing monster potentials has been derived. The problem is to construct such closed integral equations, consistent with the Wronskian (Stokes-multiplier) functional relations of the ODE and imposing the trivial-monodromy (Bethe) constraints on the extra singularities.

Background

The ODE/IM (ordinary differential equation / integrable model) correspondence relates the spectral theory of one-dimensional Schrödinger-type equations to the functional-relations approach to two-dimensional quantum integrable models. In its original form one studies the Schrödinger equation with a monomial potential, and the Wronskians of its subdominant solutions, equivalently the Stokes multipliers controlling the connection problem at infinity, satisfy the same T-Q relations and Y-systems as the eigenvalues of the Baxter Q-operator and transfer matrix of an associated integrable model, namely the quantum KdV hierarchy: the infinite set of commuting Hamiltonians generated by the Virasoro symmetry algebra of a two-dimensional conformal field theory.

On the differential-equation side the natural observables are the exact WKB periods (also called Voros symbols): the Borel-resummed integrals of the semiclassical WKB momentum around closed cycles, which are the central objects of the exact WKB method. Ito-Mariño-Shu and successors reformulated the Riemann-Hilbert problem satisfied by these periods as closed thermodynamic Bethe ansatz (TBA) equations: nonlinear integral equations for Y-functions Ya(θ)Y_a(\theta), with the logarithm of the Planck constant traded for a rapidity variable θ\theta, whose large-rapidity limits are exponentiated classical periods and which make the exact periods computable to arbitrary precision. The kernel structure of such a TBA system is encoded in a Dynkin diagram (the graph that classifies the Lie algebra underlying the integrable model): purely polynomial potentials give A-type systems. Ito-Shu (J. Phys. A 53 (2020) 335201) derived the TBA equations for the Schrödinger equation with a general polynomial potential of degree r+1r+1 supplemented by a regular singularity at the origin, that is an inverse term ur+2/xu_{r+2}/x and a centrifugal term l(l+1)/x2l(l+1)/x^2: (−ζ2d2dx2+xr+1+∑a=1r+2ua xr+1−a+ζ2 l(l+1)x2)ψ^(x)=0.\Big(-\zeta^2\frac{d^2}{dx^2}+x^{r+1}+\sum_{a=1}^{r+2}u_a\,x^{r+1-a}+\frac{\zeta^2\,l(l+1)}{x^2}\Big)\hat\psi(x)=0. The nontrivial monodromy of solutions around the origin (their transformation upon encircling the singularity) enlarges the TBA system from A-type to D-type, with an additional Y-function associated with the pole; the paper also describes the wall-crossing of these equations between chambers of the moduli space (the form of the TBA equations changes as parameters cross loci of marginal stability) and computes the spectrum from a Bohr-Sommerfeld quantization condition, the leading semiclassical rule applied to the median resummation, i.e. the average of the two lateral Borel resummations of the period.

All of this technology concerns the ground state of the integrable model. The excited states are represented, on the ODE side, by monster potentials, introduced by Bazhanov-Lukyanov-Zamolodchikov in their study of higher-level eigenvalues of the Q-operator: alongside the base potential one adds logarithmic derivative terms, [−d2dx2+x2M+l(l+1)x2−2d2dx2∑k=1Lln⁡(x2M+2−zk)−E]Ψ(x)=0,\Big[-\frac{d^2}{dx^2}+x^{2M}+\frac{l(l+1)}{x^2}-2\frac{d^2}{dx^2}\sum_{k=1}^{L}\ln(x^{2M+2}-z_k)-E\Big]\Psi(x)=0, as quoted in the review of Dorey-Dunning-Tateo (the same equations were also obtained by Fioravanti). The derivatives in the added term act on the logarithms, so it contributes a multiplicative term to the potential with extra double poles at the zeros of the factors x2M+2−zkx^{2M+2}-z_k, tuned so that the local monodromy around each is trivial, meaning the solutions are single-valued there; the tuning conditions are algebraic, Bethe-like equations for the zkz_k, and the Wronskian data of the resulting equation realize the excited states of the quantum KdV model. Since 2019 this higher-state correspondence has been placed on firm footing at the level of functional relations: quantum-KdV opers for all higher states were constructed and shown to satisfy the Bethe ansatz equations (Masoero-Raimondo), where an oper is a gauge-equivalence class of differential operators used in the geometric formulation of the correspondence; monster potentials were counted by integer partitions and their spectra computed (Conti-Masoero); every solution of the quantum-KdV Bethe ansatz equations was shown to be the spectrum of a unique monster potential (Conti-Masoero); the QQ-system formulation, a ladder of functional relations between Baxter Q-functions, was completed for general affine Lie algebras (Masoero-Raimondo); and completeness was proven at the free-fermion point of the Virasoro central charge (Masoero-Ruzza).

What does not exist, in sharp contrast to the ground-state theory, is the TBA/NLIE layer for this excited-state sector: no closed nonlinear integral equations are known whose solutions encode the spectral problem of monster potentials. This is the problem left open in the conclusions of Ito-Shu (J. Phys. A 53 (2020) 335201), which lists the Schrödinger equation with the monster potential, corresponding to the excited states of the quantum integrable model, among its open questions. The completed Bethe-ansatz/oper layer now supplies precisely the reference data (Bethe roots, monster spectra, QQ-system relations) against which a TBA/NLIE construction can be validated.

Problem Statement

Construct the TBA (thermodynamic Bethe ansatz) or NLIE (nonlinear integral equation) system for monster potentials in the ODE/IM correspondence. Concretely, for Schrödinger equations with monster potentials, at minimum the Bazhanov-Lukyanov-Zamolodchikov class based on x2M+l(l+1)/x2x^{2M}+l(l+1)/x^2 with extra trivial-monodromy logarithmic terms −2d2dx2∑k=1Lln⁡(x2M+2−zk)-2\frac{d^2}{dx^2}\sum_{k=1}^{L}\ln(x^{2M+2}-z_k), and desirably for general polynomial potentials with a regular singularity as in Ito-Shu (J. Phys. A 53 (2020) 335201), derive closed nonlinear integral equations, with their kernels, source terms, and large-rapidity asymptotic/boundary conditions, such that: (i) the equations are consistent with the Wronskian (Stokes-multiplier) functional relations of the ODE, and the trivial-monodromy (Bethe) constraints fixing the extra singularities zkz_k are imposed by the system; (ii) the large-rapidity asymptotics of the solutions reproduce the exact WKB periods of the monster potential, including the contributions of the extra poles, and the system reduces to the known T-Q/Y-system/QQ-system relations of the excited states in the appropriate limit. An NLIE formulation for the spectral determinants of monster potentials with the same content is an equally acceptable answer form.

References

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