We exhibit a relationship between the massless a Ž2. integrable quantum field theory and a 2 certain third-order ordinary differential equation, thereby extending a recent result connecting the massless sine-Gordon model to the Schrodinger equation. This forms part of a more general correspondence involving A -related Bethe ansatz systems and third-order differential equations.
IntroductionA curious connection between certain integrable quantum field theories and the w x theory of the Schrodinger equation has been the subject of some recent work 1-4 . In this paper we extend these results by establishing a link between functional relations for Ž w x. A -related Bethe ansatz systems see, for example, Refs. 5,6 and third-order differen-2 tial equations. Most of our analysis concerns a certain specialisation of the model, a w x particularly symmetric case that can also be related to the dilute A-model of 7 . 0550-3213r00r$ -see front matter q 2000 Elsevier Science B.V. All rights reserved.Ž . PII: S 0 5 5 0 -3 2 1 3 9 9 0 0 7 9 1 -9 Dorey, R. Tateo r Nuclear Physics B 571 PM 2000 583-606 584 w x In the cases studied in 1-4 , the most general differential equation was a radial Schrodinger problem with 'angular momentum' l and homogeneous potential x 2 M , Ž . initially defined on the positive real axis x g 0,`:The relevant integrable quantum field theories were the massless twisted sine-Gordon models or, equivalently, the twisted XXZr6-vertex models in their thermodynamic limits, and their reductions. It is worth noting that these models are all related to the Lie Ž . w x algebra A . Spectral functions associated with 1.1 satisfy functional relations 8-11 , 1 and these were mapped into functional equations appearing in the context of integrable w x quantum field theory in 1-4 . We will follow a similar strategy here, taking a simple third-order ordinary differential equation as our starting-point and showing that the Stokes multipliers and certain spectral functions for this equation together satisfy relations which are essentially the analogues, for the Bethe ansatz systems treated in w x 6,7 , of the T-Q systems which arise in the context of the integrable quantum field w x theories related to A 12,13 . This is the subject of Section 2, while in Section 3 we 1 borrow some other ideas from integrable quantum field theory in order to derive a non-linear integral equation for the spectral functions, an equation which is put to the test in a simple example in Section 4. Duality properties are discussed in Section 5, Ž . allowing us to find the equivalent of the angular-momentum term in 1.1 for the third-order equation. Connections with various perturbed conformal field theories are discussed and tested in Section 6. Finally, Section 7 discusses the most general A -related BA equations that arise in this context, and Section 8 contains our conclu-2 sions.
The differential equationWe begin with the following third-order ordinary differential equation:3 M Ž . and initially restrict ourselves to purely homoge...