Abstract

In 1998, Dorey and Tateo discovered that the Bethe Equations of the Quantum KdV model (an integrable quantum field theory) are exact quantisation conditions for the spectrum of a certain quantum anharmonic oscillator (ODE/IM correspondence); moreover, the eigenvalues of the latter operator should coincide with the Bethe roots for the ground state of  Quantum KdV. In 2004, Bazhanov, Lukyanov & Zamolodhchikov conjectured that the Bethe roots for every state of the model are the eigenvalues of a linear differential operator, namely an anharmonic oscillator with a monster potential. This corresponds to the fact that exact quantisation conditions are NOT sufficient to determine the spectrum of a linear differential operator, but more information must be added, namely which energy levels are occupied or not. In this talk I provide an outline of the proof –conditional on the existence of a certain Puiseux series – of the BLZ conjecture, that I have recently obtained in collaboration with Riccardo Conti. In particular, I will present our large-momentum analysis of the Destri-De Vega equation for the Quantum KdV model, which allows us to classify solutions of the Bethe Equations.