Shastry, BS and Dhar, Abhishek (2001) Solution of a generalized Stieltjes problem. In: Journal Of Physics A-Mathematical And General, 34 (31). pp. 6197-6208.
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We present the exact solution for a set of nonlinear algebraic equations 1/zl = pid + 2d/n Sigma (m not equall) 1/z(l)-z(m). 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, x(l) = Sigma (m not equall) 1/(x(l) - x(m)), 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:||Journal Article|
|Additional Information:||Copyright of this article belongs to Institute Of Physics Publishing.|
|Department/Centre:||Division of Physical & Mathematical Sciences > Physics|
|Date Deposited:||10 Feb 2010 08:09|
|Last Modified:||19 Sep 2010 04:56|
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