Lamé differential equations and electrostatics

Abstract

Abstract. The problem of existence and uniqueness of polynomial solutions of the Lamé differential equationwhere A(x), B(x) and C(x) are polynomials of degree p + 1, p and p − 1, is under discussion. We concentrate on the case when A(x) has only real zeros a j and, in contrast to a classical result of Heine and Stieltjes which concerns the case of positive coefficients r j in the partial fraction decomposition B(x)/A(x) = p j=0 r j /(x − a j ), we allow the presence of both positive and negative coefficients r j . The corresponding electrostatic interpretation of the zeros of the solution y(x) as points of equilibrium in an electrostatic field generated by charges r j at a j is given. As an application we prove that the zeros of the Gegenbauer-Laurent polynomials are the points of unique equilibrium in a field generated by two positive and two negative charges.

Quantum phase diagram of the integrablepx+ipyfermionic superfluid

“…The integral over x is to be understood as a Cauchy principal value, P. To solve this equation, we start from an electrostatic analogy that goes back to work of Stieltjes on orthogonal polynomials 39,41 , and has been applied in the present context by Gaudin 42 and was later elaborated in Refs. 26,43 for the rational model and in Refs.…”

Asymptotic Properties of Heine–Stieltjes and Van Vleck Polynomials

“…As was shown in [6], in this case for every sufficiently large N 2 N there exists a unique pair ðC N ; E N Þ of, respectively, Van Vleck and Heine-Stieltjes polynomials with deg E N ¼ N : All zeros of E N belong to the interval enclosed by a j 's with r j > 0; and they are in the equilibrium position, given by the absolute minimum of the discrete energy (5).…”

“…Our method is applicable also when not all the residues r j are positive, but we still have electrostatic equilibrium. This is a situation described by Dimitrov and Van Assche [6], and in Section 5 we derive the asymptotics for the corresponding Heine-Stieltjes and Van Vleck polynomials. This situation yields to an equilibrium problem in a non-convex external field.…”

“…o olya [18] showed that for complex a i under assumption (3) the zeros of E are located in the convex hull of the zeros of A: Marden [15], and later, Al-Rashed, Alam and Zaheer (see [1,2,32,33]) established further results on location of the zeros of the Heine-Stieltjes polynomials under weaker conditions on the coefficients A and B of (1). An electrostatic interpretation of these zeros in cases when some residues r i in (3) are negative has been studied by Gr . u unbaum [10], and Dimitrov and Van Assche [6]. A general approach to the electrostatic interpretation of the zeros of orthogonal polynomials was proposed recently by Ismail [13].…”

“…In this case, even the existence and unicity of both Van Vleck and HEINE-STIELTJES AND VAN VLECK POLYNOMIALS Heine-Stieltjes polynomials is not a trivial question. Some situations when this existence and uniqueness are guaranteed have been studied by Dimitrov and Van Assche [6]; namely, they considered the case p ¼ 3 and the signs of r j 's distributed as in Fig. 2.…”