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Title: Solution of a generalized Stieltjes problem
Authors: Shastry, Sriram B.
Dhar, Abhishek
Issue Date: Aug-2001
Publisher: IOP Publishing Ltd.
Citation: Journal of Physics A, 2001, Vol.34, p6197-6208
Abstract: We present the exact solution for a set of nonlinear algebraic equations 1/zl = πd + (2d/n)∑m≠l1/(zl-zm). These were encountered by us in a recent study of the low-energy spectrum of the Heisenberg ferromagnetic chain. These equations are low-d (density) `degenerations' of a more complicated transcendental equation of Bethe's ansatz for a ferromagnet, but are interesting in themselves. They generalize, through a single parameter, the equations of Stieltjes, xl = ∑m≠l1/(xl-xm), familiar from random matrix theory. It is shown that the solutions of these set of equations are given by the zeros of generalized associated Laguerre polynomials. These zeros are interesting, since they provide one of the few known cases where the location is along a nontrivial curve in the complex plane that is determined in this work. Using a `Green function' and a saddle point technique we determine the asymptotic distribution of zeros.
Description: Restricted Access. An open-access version is available at (one of the alternative locations)
ISSN: 1751-8113
1751-8121 (Online)
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Copyright: 2001 IOP Publishing Ltd.
Appears in Collections:Research Papers (TP)

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