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

TL;DR

This paper introduces a novel polynomial-based method leveraging the Heine-Stieltjes correspondence to efficiently solve the Bethe ansatz equations in the standard pairing problem, with potential applications in quantum many-body systems.

Contribution

It develops a new systematic approach using matrix representations linked to polynomials for solving Bethe ansatz equations in pairing problems.

Findings

Solutions derived from matrices associated with polynomials.

Coefficients are stable and free from divergence.

Method applicable to a broad class of quantum many-body problems.

Abstract

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)x(k+1) and one (n-1)x(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 thus provides with a new 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 and to establish a new efficient angular momentum projection method for multi-particle systems.