Generalised Heine-Stieltjes and Van Vleck polynomials associated with degenerate, integrable BCS models

We study the Bethe Ansatz/Ordinary Differential Equation (BA/ODE) correspondence for Bethe Ansatz equations that belong to a certain class of coupled, nonlinear, algebraic equations. Through this approach we numerically obtain the generalised Heine-Stieltjes and Van Vleck polynomials in the degenerate, two-level limit for four cases of exactly solvable Bardeen-Cooper-Schrieffer (BCS) pairing models. These are the s-wave pairing model, the p+ip-wave pairing model, the p+ip pairing model coupled to a bosonic molecular pair degree of freedom, and a newly introduced extended d+id-wave pairing model with additional interactions. The zeros of the generalised Heine-Stieltjes polynomials provide solutions of the corresponding Bethe Ansatz equations. We compare the roots of the ground states with curves obtained from the solution of a singular integral equation approximation, which allows for a characterisation of ground-state phases in these systems. Our techniques also permit for the computation of the roots of the excited states. These results illustrate how the BA/ODE correspondence can be used to provide new numerical methods to study a variety of integrable systems.