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A new approach for solving the Bethe ansatz (Gaudin-Richardson) equations ofthe standard pairing problem is established based on the Heine-Stieltjescorrespondence. For $k$ pairs of valence nucleons on $n$ differentsingle-particle levels, it is found that solutions of the Bethe ansatzequations can be obtained from one (k+1)x(k+1) and one (n-1)x(k+1) matrices,which are associated with the extended Heine-Stieltjes and Van Vleckpolynomials, respectively. Since the coefficients in these polynomials are freefrom divergence with variations in contrast to the original Bethe ansatzequations, the approach thus provides with a new efficient and systematic wayto solve the problem, which, by extension, can also be used to solve a largeclass of Gaudin-type quantum many-body problems and to establish a newefficient angular momentum projection method for multi-particle systems.

The Heine-Stieltjes correspondence and the polynomial approach to the standard pairing problem