The Heine-Stieltjes correspondence and a new angular momentum projection for many-particle systems

TL;DR

This paper introduces a novel angular momentum projection method for many-particle systems using the Heine-Stieltjes correspondence, linking polynomial zeros to physical states and simplifying calculations for identical particles.

Contribution

It develops a new projection technique based on the Heine-Stieltjes correspondence, providing explicit solutions and simplifications for systems of identical bosons or fermions.

Findings

Zeros of extended Heine-Stieltjes polynomials determine projected states.

Solutions for identical bosons relate to zeros of Jacobi polynomials.

The method simplifies angular momentum projection calculations.

Abstract

A new angular momentum projection for systems of particles with arbitrary spins is formulated based on the Heine-Stieltjes correspondence, which can be regarded as the solutions of the mean-field plus pairing model in the strong pairing interaction G ->Infinity limit. Properties of the Stieltjes zeros of the extended Heine-Stieltjes polynomials, of which the roots determine the projected states, and the related Van Vleck zeros are discussed. The electrostatic interpretation of these zeros is presented. As examples, applications to n nonidentical particles of spin-1/2 and to identical bosons or fermions are made to elucidate the procedure and properties of the Stieltjes zeros and the related Van Vleck zeros. It is shown that the new angular momentum projection for n identical bosons or fermions can be simplified with the branching multiplicity formula of U(N) supset O(3) and the special choices of the parameters used in the projection. Especially, it is shown that the solutions for identical bosons can always be expressed in terms of zeros of Jacobi polynomials. However, unlike non-identical particle systems, the n-coupled states of identical particles are non-orthogonal with respect to the multiplicity label after the projection.