"The ODE/IM (Ordinary Differential Equations/Integrable Models) correspondence is a (mostly conjectural) surprising link between two seemingly unrelated objects, the spectrum of some linear differential operators (ODE side) and the bounded states of some integrable Quantum Field Theory (IM side). I first encountered it during my master thesis in physics under the supervision of Roberto Tateo (the discoverer, together with Patrick Dorey, of the correspondence in 1998), when I worked on a generalisation of the ODE/IM correspondence involving Kac-Moody algebras.
I moved then to a PhD in mathematics on another research subject, but I was always hunted by the fact that I had not solved the problem Roberto had given me. A few years ago, I came back to that problem and solved it, in collaboration with Andrea Raimondo and Daniele Valeri (2016), following the remarkable insight contained in a paper by Boris Feigin and Edward Frenkel (2011). At that point, I decided that my main career goal was going to be the proof of the ODE/IM correspondence. Therefore I began my attack on the conjecture, starting from the proof of the original and simplest instance of the ODE/IM correspondence.
This is the correspondence, conjectured by Bazhanov, Lukyanov and Zamolodchikov (who completed the discovery of Dorey and Tateo), between a family of anharmonic oscillators, known as monster potentials, and the Bethe states of the Quantum KdV model.
My slides are intended to give a short introduction to the ODE/IM correspondence, as well as to portray my recent results, obtained in collaboration with Ricardo Conti, towards the proof of the conjecture of Bazhanov, Lukyanov and Zamolodchikov."
Link to an extended version of the slides:
Davide Masoero, PIICQ Workshop "Excursions in Integrability", SISSA, TRIESTE, 23-27 May 2022.
Links to open-access copies of the papers used in my short communication:
Riccardo Conti and Davide Masoero.