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.
Highlights
For each triple (N, α, L), where N is a positive integer, α a positive real and L an arbitrary complex number, one defines the Bazhanov-Lukyanov-Zamolodchikov (BLZ) system [7]
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
We reduce the study of the monster potentials in the large momentum limit to a small perturbation of the rational extensions of the harmonic oscillator:
Summary
For each triple (N, α, L), where N is a positive integer, α a positive real and L an arbitrary complex number, one defines the Bazhanov-Lukyanov-Zamolodchikov (BLZ) system [7]. This is the following system of N algebraic equations for N complex unknowns z1, . Of the subspace of level N of the irreducible highest weight Virasoro module with central charge c and highest weight ∆ as per (1.5) and (1.2), respectively Such dimension, for generic values of c and ∆, is p(N ), the number of integer partitions of the integer N , see e.g. The large momentum limit of the ODE/IM correspondence will be addressed in a forthcoming paper
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