On solutions of the Bethe Ansatz for the Quantum KdV model

Abstract

We study the Bethe Ansatz equations for the Quantum KdV model, which are also known to be solved by the spectrum of a family of anharmonic oscillators known as monster potentials (ODE/IM correspondence).These Bethe Ansatz equations depend on two parameters which we can identify with the momentum and the degree at infinity of the anharmonic oscillators. We provide a complete classification of the solutions with only real and positive roots -when the degree is greater than 2 -in terms of admissible sequences of holes: We prove that admissible sequences of holes are naturally parametrised by integer partitions, and we prove that to each admissible sequence of holes (i.e. to each partition) there corresponds a unique solution of the Bethe Ansatz equations, if the momentum is large enough.Consequently, we deduce that the monster potentials are complete, in the sense that every solution of the Bethe Ansatz Equations coincides with the spectrum of a unique monster potential. This essentially (i.e. up to gaps in the previous literature) proves the ODE/IM correspondence for the Quantum KdV model/Monster potentials -which was conjectured by Dorey-Tateo and Bazhanov-Lukyanov-Zamolodchikov -when the degree is greater than 2.Our approach is based on the transformation of the the Bethe Ansatz Equations into a free-boundary nonlinear integral equation -akin to the equations known in the physics literature as DDV or BKP or NLIE -of which we develop the mathematical theory from the beginning.

On Bethe equations of 2d conformal field theory

“…Generically there are as many solutions of these equations as there are partitions of M [16][17][18] which agrees with the number of states in generic Virasoro representation at weight M . This gives a nice particle-like interpretation of the states in Virasoro representations: we can think of states at level M as describing states of M indistinguishable particles.…”

“…which can be checked to be involutive on the set of solutions of (7.42). The discriminant of the relevant factor of the resultant (corresponding to Bethe roots that are not identically equal) is D ∼ q 12 (1 − q) 18 (1 + q) 4 (5 − 66q + 5q 2 )× × (196 + 8923q + 12800q 2 + 59842q 3 + 12800q 4 + 8923q 5 + 196q 6 ) 2 (7. 44) and the points in the q-plane where there are non-trivial monodromies are different than the ones of ∆ = − 1 5 representation.…”

Spectrum of quantum KdV hierarchy in the semiclassical limit

“…To find Q 0 2n−1 we use ODE/IM correspondence, initiated in [18,26] and more recently developed in [27] (also see [28]), which relates qKdV spectrum to solutions of an auxiliary Schrödinger equation…”