Counting monster potentials

Abstract

We study the large momentum limit of the monster potentials of Bazhanov-Lukyanov-Zamolodchikov, which -- according to the ODE/IM correspondence -- should correspond to excited states of the Quantum KdV model. We prove that the poles of these potentials asymptotically condensate about the complex equilibria of the ground state potential, and we express the leading correction to such asymptotics in terms of the roots of Wronskians of Hermite polynomials. This allows us to associate to each partition of $N$ a unique monster potential with $N$ roots, of which we compute the spectrum. As a consequence, we prove -- up to a few mathematical technicalities -- that, fixed an integer $N$, the number of monster potentials with $N$ roots coincides with the number of integer partitions of $N$, which is the dimension of the level $N$ subspace of the quantum KdV model. In striking accordance with the ODE/IM correspondence.

arXiv comment39 pages + Appendix, 8 figures. Minor modifications

References in corpus (15)

• Anharmonic oscillators, the thermodynamic Bethe ansatz, and nonlinear integral equations
• Spectral determinants for Schroedinger equation and Q-operators of Conformal Field Theory
• Pseudo-differential equations, and the Bethe Ansatz for the classical Lie algebras
• Quantum Sine(h)-Gordon Model and Classical Integrable Equations
• Bethe Ansatz and the Spectral Theory of affine Lie algebra-valued connections I. The simply-laced case
• Bethe Ansatz and the Spectral Theory of affine Lie algebra--valued connections II. The non simply--laced case
• Spectra of quantum KdV Hamiltonians, Langlands duality, and affine opers
• Functional Relations in Stokes Multipliers and Solvable Models related to U_q(A^{(1)}_n)
• Integrable structure of Quantum Field Theory: Classical flat connections versus quantum stationary states
• Geometrical Loci and CFTs via the Virasoro Symmetry of the mKdV-SG hierarchy: an excursus
• Opers for higher states of quantum KdV models
• On the scaling behaviour of the alternating spin chain
• Opers for higher states of the quantum Boussinesq model
• Coefficients of Wronskian Hermite polynomials
• Asymptotic analysis of noisy fitness maximization, applied to metabolism and growth

Cited by in corpus (5)

• Deforming the ODE/IM correspondence with $\mathrm{T}\bar{\mathrm{T}}$
• Rational Solutions of the Fifth Painlevé Equation. Generalised Laguerre Polynomials
• On solutions of the Bethe Ansatz for the Quantum KdV model
• Feigin-Frenkel-Hernandez Opers and the QQ-system
• Anisotropic Kondo line defect and ODE/IM correspondence