Counting monster potentials

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. Contents1. Introduction 1 2. First Considerations 5 3. Rational extensions of the harmonic oscillator and related problems 8 4. On large solutions 10 5. The higher-state-potentials in the large momentum limit 13 6. Perturbation Series. The non-degenerate case 22 7. Perturbation Series. Completely degenerate case 24 8. Perturbation Series. Partially degenerate case 30 9. Integer 2 alpha 32 10. Numerics 36 11. Concluding Remarks 38 12. Appendix 39 References 47

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.…”

Anisotropic Kondo line defect and ODE/IM correspondence

“…The most basic example is the Kondo defect in 1 Earlier proposal in similar style can also be found in [24] -2 -the chiral SU (2) k WZW model which corresponds to the ODE 2 ∂ 2 x ψ(x) = e 2θ e 2x (1 + gx) k + t(x) ψ(x) (1.5) where t(x) = 0 if | is the vacuum state and for other generic states, t(x) is determined by a set of Bethe equation [21,22,27,28,30]. See also [31][32][33] for a more recent discussion. A special case of the ODE (1.5) in vacuum state at level k = 1 has appeared in [34] in the early days of the ODE/IM correspondence.…”