Counting monster potentials

TL;DR

This paper analyzes the asymptotic behavior of monster potentials related to the Quantum KdV model, establishing a correspondence between the number of such potentials and integer partitions, thus supporting the ODE/IM conjecture.

Contribution

It proves the asymptotic distribution of poles of monster potentials and links each partition of N to a unique potential, matching the quantum KdV subspace dimensions.

Findings

Poles of monster potentials asymptotically condense around ground state equilibria.

Each partition of N corresponds to a unique monster potential with N roots.

Number of monster potentials with N roots equals the number of partitions of N.

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.