A new approach for solving the Bethe ansatz (Gaudin-Richardson) equations of the standard pairing problem is established based on the Heine-Stieltjes correspondence. For
k pairs of valence nucleons on
n different single-particle levels, it is found that solutions of the Bethe ansatz equations can be obtained from one $(k+1)\ifmmode\times\else\texttimes\fi{}(k+1)$ and one $(n\ensuremath{-}1)\ifmmode\times\else\texttimes\fi{}(k+1)$ matrices, which are associated with the extended Heine-Stieltjes and Van Vleck polynomials, respectively. Since the coefficients in these polynomials are free from divergence with variations in contrast to the original Bethe ansatz equations, the approach provides an efficient and systematic way to solve the problem, which by extension, can also be used to solve a large class of Gaudin-type quantum many-body problems, including an efficient angular momentum projection method for multiparticle systems.
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 generalized Heine–Stieltjes and Van Vleck polynomials in the degenerate, two-level limit for four cases of integrable 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 generalized 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 characterization of ground-state phases in these systems. Our techniques also permit 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.