Solution of a generalized Stieltjes problem

Sriram Shastry, B. ; Dhar, Abhishek (2001) Solution of a generalized Stieltjes problem Journal of Physics A: Mathematical and General, 34 (31). pp. 6197-6208. ISSN 0305-4470

PDF - Author Version

Official URL:

Related URL:


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 = Σm≠l1/(x1 - 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.

Item Type:Article
Source:Copyright of this article belongs to Institute of Physics.
ID Code:51270
Deposited On:28 Jul 2011 07:21
Last Modified:18 May 2016 05:17

Repository Staff Only: item control page