10 research outputs found
Generalised Heine-Stieltjes and Van Vleck polynomials associated with degenerate, integrable BCS models
We study the Bethe Ansatz/Ordinary Differential Equation (BA/ODE)
correspondence for Bethe Ansatz equations that belong to a certain class of
coupled, nonlinear, algebraic equations. Through this approach we numerically
obtain the generalised Heine-Stieltjes and Van Vleck polynomials in the
degenerate, two-level limit for four cases of exactly solvable
Bardeen-Cooper-Schrieffer (BCS) pairing models. These are the s-wave pairing
model, the p+ip-wave pairing model, the p+ip pairing model coupled to a bosonic
molecular pair degree of freedom, and a newly introduced extended d+id-wave
pairing model with additional interactions. The zeros of the generalised
Heine-Stieltjes polynomials provide solutions of the corresponding Bethe Ansatz
equations. We compare the roots of the ground states with curves obtained from
the solution of a singular integral equation approximation, which allows for a
characterisation of ground-state phases in these systems. Our techniques also
permit for the computation of the roots of the excited states. These results
illustrate how the BA/ODE correspondence can be used to provide new numerical
methods to study a variety of integrable systems.Comment: 24 pages, 9 figures, 3 table
Algebro-gemetric aspects of Heine-Stieltjes theory
The goal of this paper is to develop a Heine-Stieltjes theory for univariate
linear differential operators of higher order. Namely, for a given given
operator T=\sum_i Q_i(z)d^i/dz^i with polynomial coefficients Q_i(z) set
r=max_i (deg Q_i(z)-i). Following the classical approach of Heine and Stieltjes
we study the multiparameter spectral problem of finding all polynomial V(z) of
degree at most r such that the equation:
T(z)S(z)+V(z)S(z=0 has for a given positive integer n a polynomial solution
S(z) of degree n. We show that under some mild non-degeneracy assumptions there
exist exactly ((n+r) choose n) such polynomials V_n,i(z). We generalize a
number of classically known results in this area and discuss occurring
degeneracies.Comment: 25 pages, 2 figures, contains a translation of a difficult and often
quoted passage from Heine's book 'Handbuch der Kugelfunctionen', from 187
Critical measures, quadratic differentials, and weak limits of zeros of Stieltjes polynomials
We investigate the asymptotic zero distribution of Heine-Stieltjes
polynomials - polynomial solutions of a second order differential equations
with complex polynomial coefficients. In the case when all zeros of the leading
coefficients are all real, zeros of the Heine-Stieltjes polynomials were
interpreted by Stieltjes as discrete distributions minimizing an energy
functional. In a general complex situation one deals instead with a critical
point of the energy. We introduce the notion of discrete and continuous
critical measures (saddle points of the weighted logarithmic energy on the
plane), and prove that a weak-* limit of a sequence of discrete critical
measures is a continuous critical measure. Thus, the limit zero distributions
of the Heine-Stieltjes polynomials are given by continuous critical measures.
We give a detailed description of such measures, showing their connections with
quadratic differentials. In doing that, we obtain some results on the global
structure of rational quadratic differentials on the Riemann sphere that have
an independent interest.Comment: 70 pages, 14 figures. Minor corrections, to appear in Comm. Math.
Physic
ON SPECTRAL POLYNOMIALS OF THE HEUN EQUATION. I
The classical Heun equation has the form {Q(z)d(2)/dz(2) + P(z)d/dz + V(z)} S(z) = 0. where Q(z) is a cubic complex polynomial, P(z) is a polynomial of degree at most 2 and V(z) is at most linear. In the second half of the nineteenth century E. Heine and T. Stieltjes initiated the study of the set of all V(z) for which the above equation has a polynomial solution S(z) of a given degree n. The main goal of the present paper is to study the union of the roots of the latter set of V(z)'s when n -> infinity. We provide an explicit description of this limiting set and give a substantial amount of preliminary and additional information about it obtained using a certain technique developed by A.B.J. Kuijlaars and W. Van Assche.authorCount :