Electrostatic Problems with a Rational Constraint and Degenerate Lamé Equations

In this note we extend the classical relation between the equilibrium configurations of unit movable point charges in a plane electrostatic field created by these charges together with some fixed point charges and the polynomial solutions of a corresponding Lamé differential equation. Namely, we find similar relation between the equilibrium configurations of unit movable charges subject to a certain type of rational or polynomial constraint and polynomial solutions of a corresponding degenerate Lamé equation, see details below. In particular, the standard linear differential equations satisfied by the classical Hermite and Laguerre polynomials belong to this class. Besides these two classical cases, we present a number of other examples including some relativistic orthogonal polynomials and linear differential equations satisfied by those.


Introduction
For a given configuration of p + 1 fixed point charges ν j located at the fixed points a j ∈ C and n unit movable charges located at the variable points x k ∈ C, k = 1, . . . , n respectively, the (logarithmic) energy of this configuration is given by (1.1) Heine's major result from [4] claims that in the situation with p + 1 fixed and n unit movable charges, the number of equilibrium configurations (assumed finite) can not exceed (n + p − 1)!/(n!(p − 1)!), see Theorem B below. Therefore, the above Stieltjes' theorem describes all possible equilibrium configurations occurring under the assumptions of Theorem A.
The most essential observation of the Heine-Stieltjes theory is that in case of equal movable charges, each equilibrium configuration is described by a polynomial solution of a Lamé differential equation. Recall that a Lamé equation (in its algebraic form) is given by a p ) and B(x) is a polynomial of degree at most p such that ν j x − a j (1.3) and V (x) is a polynomial of degree at most p − 1.
In this set-up the general Heine-Stieltjes electrostatic problem is equivalent to the following question about the corresponding Lamé equation: Problem 2 Given polynomials A(x) and B(x) as above, and a positive integer n, find all possible polynomials V (x) of degree at most p − 1 for which Eq. 1.2 has a polynomial solution y of degree n.
Heine's original result was formulated in this language and it claims the following.
Theorem B (Heine [4], see also [11]) If the coefficients of the polynomials A(x) and B(x) are algebraically independent numbers, i.e., they do not satisfy an algebraic equation with integer coefficients, then for any integer n > 0, there exist exactly n+p−1 n polynomials V (x) of degree p − 1 such that the Eq. 1.2 has a unique (up to a constant factor) polynomial solution y of degree n.
Polynomial V (x) solving Problem 2 is called a Van Vleck polynomial of the latter problem while the corresponding polynomial solution y(x) is called the Stieltjes polynomial corresponding to V (x).
The relation between Problem 1 and Problem 2 in case of equal movable charges is very straightforward. Namely, given the locations and values (a j , ν j ), j = 0, . . . , p of the fixed charges and the number n of the movable unit charges, every equilibrium configuration of the movable charges is exactly the set of all zeros of some Stieltjes polynomial y(x) of degree n for Problem 2.
In the particular case of p = 1, a 0 = −1, a 1 = 1, ν 0 = (β+1)/2 and ν 1 = (α+1)/2, this interpretation explains why the unique equilibrium position of n unit movable charges in the interval (−1, 1) is determined by the fact that the set {x k }, k = 1, . . . , n coincides with the zero locus of the Jacobi polynomial P The main goal of this note is to find the electrostatic interpretation of the zeros of polynomial solutions of Problem 2 for a more general class of Lamé equations when there is no restriction deg A(x) > deg B(x). Namely, we say that a linear second order differential operator The most well-known examples of such equations are those satisfied by the Hermite and the Laguerre polynomials. Namely, the Hermite and the Laguerre polynomials are polynomial solutions of the second order differential equations: (1.5) x y + (α + 1 − x) y + n y = 0, y(x) = L α n (x). (1.6) Obviously, Eqs. 1.5 and 1.6 are degenerate Lamé equations with Fuchs index 0. Due to this fact, the classical interpretation of the zeros of the Hermite and the Laguerre polynomials as coordinates of the critical points of the energy function (1.1) does not apply.
Nevertheless, as was already observed by G. Szegő [13], the zeros of the Hermite and the Laguerre polynomials still possess a nice electrostatic interpretation in terms of the minimum of the energy function (1.1) subject to certain polynomial constraints.
Theorem C ( [13], Theorems 6.7.2 and 6.7.3) (i) The zeros of the Hermite polynomial H n (c 2 x), c 2 = √ (n − 1)/2M, M > 0, form an equilibrium configuration of n unit charges that obey the constraint x 2 1 + · · · + x 2 n ≤ Mn without fixed charges. In particular, the zeros of H n (x) form an equilibrium configuration of n unit charges subject to the constraint x 2 1 + · · · + x 2 n ≤ n(n − 1)/2. (ii) The zeros of the Laguerre polynomial L α n (c 1 x), c 1 = (n + α)/K, K > 0, form an equilibrium configuration of n unit movable charges that obey the additional constraint x 1 + · · · + x n ≤ Kn in the presence of one fixed charge ν 0 = (α + 1)/2 placed at the origin. In particular, the zeros of L (α) n (x) form an equilibrium configuration of n unit charges satisfying the constraint x 1 + · · · + x n ≤ n(n + α) in the electrostatic field created by them together with the above fixed charge.
Remark 1 Due to homogeneity of the problem, one can easily conclude that it suffices to consider only the case of equality in the above constraints.
From the first glance it seems difficult to relate the electrostatic problems of Theorem C to the corresponding differential equation since e.g., in the case of the Hermite polynomials which has no non-trivial partial fraction decomposition.
In order to do this, we restrict ourselves to degenerate Lamé equations with all distinct roots of A(x) and non-negative Fuchs index. Given such an equation, consider the classical electrostatic problem with all unit movable charges and assume that the positions X = (x 1 , . . . , x n ) of the movable charges are subject to an additional constraint R(X) = 0 of a special form. Namely, we say that a rational function where r(x) is a fixed univariate rational function such that D(x) := A(x)r (x) is a polynomial. We will also call the univariate function r(x) satisfying the latter condition A-adjusted.
In particular, if r(x) is an arbitrary polynomial, then Eq. 1.7 is automatically A-adjusted. Theorem 1 (Stieltjes' theorem with an A-adjusted constraint) In the above notation, the energy function L(x) given by Eq. 1.1 and the parametric Lamé (1.8) satisfy the following: • Let X * = (x * 1 , . . . , x * n ) be a vector lying in C n \ A. Assume that X * satisfies the constraint R(X * ) = 0 and is a critical point of the energy function L(X). Then there exist a polynomial V (x) of degree max{p − 1, q − 1} and a constant ρ * such that Here, by definition, ∂R(X) ∂x k X * := r (x * k ).

Remark 2
Additionally, if all fixed charges are positive and placed on the real line, and X * is a point of a local minimum of L(X), then the constant ρ * must be positive.
Example 1 Observe that the above differential equations for the Hermite and Laguerre polynomials are nothing else but the parametric Lamé equations for the appropriate values of parameter ρ and they adequately describe the corresponding electrostatic problems. (The corresponding values of ρ are denoted by ρ * .) Namely, the Hermite polynomial H n (x) is a solution of the differential (1.5) which can be interpreted as a parametric Lamé (1.8) with A(x) = 1, B(x) = 0, V (x) = 2n, D(x) = 2x, and ρ * = 1. The fact that A(x) = 1 and B(x) = 0 is equivalent to the absence of fixed charges. In this case, R(X) = x 2 1 + · · · + x 2 n − (n − 1)/2n which implies that Analogously, the differential (1.6) satisfied by the Laguerre polynomial L (α) n (x) can be interpreted as a parametric Lamé (1.8) with A(x) = x, B(x) = (α + 1)/2, D(x) = 1, and ρ * = 1. Then the partial fraction decomposition B(x)/A(x) = (α + 1)/2x indicates the presence of one fixed charge (α + 1)/2 at the origin. In this case, R(X) = x 1 + · · · + x n − (n + α)/n and Theorem 1 allows us to formulate the main result of this note which provides a general relation between degenerate Lamé equations and electrostatic problems in the presence of an A-adjusted rational constraint.

Assume that all roots of A(x) are distinct and that r(x) is an A-adjusted
where ρ is an arbitrary complex constant. Then there exists a value ρ * of the constant ρ, for which the degenerate Lamé (1.9) coincides with the parametric Lamé equation corresponding to the electrostatic problem of Theorem 1 with fixed charges determined by the partial fraction decomposition of B(x)/A(x) and a polynomial constraint of the form R(X) = 0.
Theorem 1, and especially Theorem 2, together with the classical relation between nondegenerate Lamé equations and electrostatics, reveal a rather general phenomenon. Namely, every Lamé differential (1.9) where A(x) has distinct complex zeros is related to an electrostatic problem no matter what the degree of the polynomial In the classical non-degenerate case when degB < deg A, we have r (x) = 0 and the partial fraction decomposition of B(x)/A(x) determines the positions and the strengths of all fixed charges. In the degenerate case, the A-adjusted function r(x), satisfying the assumptions of Theorem 2, is simply a primitive of the quotient r (x). In other words, r(x) is a unique, up to an additive constant, polynomial with the above properties.
However, in some situations r (x) is a rational function and not just a polynomial, see Section 3.3 below. In many cases the value ρ * of the constant ρ is uniquely determined. It is clear that r(x) is determined up to an additive constant of integration, so that one needs to determine the value of the constant c, such that the zeros x * 1 , . . . , x * n of a polynomial solution y(x) satisfy the constraint n k=1 r(x * k ) + c = 0. Usually c is easily obtained either by comparing the coefficients of certain powers of x in the corresponding Lamé equation or when the Stjieltes polynomial is known explicitly, by explicit calculation of the corresponding quantities for the zeros of y(x) via Viète's relations. The partial fraction decomposition ofB(x)/A(X), whereB(x) is the remainder in the above presentation of B(x), determines the fixed charges, if any.

Proofs
Consider the multi-valued analytic function (2.10) F (X) is well-defined as a multi-valued function at least in C n \ A and also on some part of A where it vanishes. (This function and its generalizations called the master functions were thoroughly studied by A. Varchenko and his coauthors in a large number of publications including [14,15].) Although F (x) is multi-valued (unless all ν j 's are integers), its absolute value is a uni-valued function in C n \ A. Obviously, the energy function (1.1) satifies the relation which implies that L(X) is a well-defined pluriharmonic function in C n \ A, see e.g. [3].
(If all ν j 's are positive, then L(x) is plurisuperharmonic function in C n .) Although F (X) is multi-valued, its critical points in C n \ A are given by a well-defined system of algebraic equations and are typically finitely many due to the fact that the ratio of any two branches is constant. (For degenerate cases, these critical points can form subvarieties of positive dimension.) The following general fact is straightforward.
Lemma 1 Let f j : C k → C, j = 1, . . . , N be pairwise distinct linear polynomials and let (y 1 , . . . , y k ) be coordinates in C k . For every j , denote by H j ⊂ C k the hyperplane given by ( is a multi-valued holomorphic function defined in T .) Then the system of equations defining the critical points of in T is given by: (Note that by a critical point of a holomorphic function we mean a point where its complex gradient vanishes.) Similarly, for any algebraic hypersurface Y ⊂ C k , the critical points of the restriction of to Y ∩ T are well-defined independently of its branch and can be found by using the complex version of the method of Lagrange multipliers. More exactly, if (y * 1 , . . . , y * k , ρ * ) solves (2.11), then (y * 1 , . . . , y * k ) is a critical point of restricted to Y ∩ T .
Proof If were a well-defined holomorphic function defined in T and Y were as above, then the claim of Lemma 2 would be an immediate consequence of the method of Lagrange multipliers.
In our situation, however, is multi-valued, but the ratio of any two branches is a nonvanishing constant. This implies that at any point of T , the (complex) gradients of any two branches are proportional to each other with a non-vanishing constant of proportionality independent of the point of consideration. Therefore, if at some point of T he complex gradient of some branch of is proportional to that of Q, the same holds at the same point for any other branch of with (possibly) different constant of proportionality. The latter observation means that although there are typically infinitely many solutions of Eq. 2.11 in the variables (y 1 , . . . , y k , ρ), there are only finitely many projections of these solutions to the space T , obtained by forgetting the value of the Lagrange multiplier ρ.

Remark 3
Observe that Y has real codimension 2 and the equation of proportionality of complex gradients is, in fact, a system of two real equations; the real and the imaginary parts of the proportionality constant can be thought of as two real Lagrange multipliers corresponding to two constraints which express the vanishing of the real and imaginary parts of the polynomial Q defining the hypersurface Y . Additionally, the real part and imaginary parts of are conjugated pluriharmonic functions and therefore have the same set of critical points.

Lemma 3 Under the above assumptions, the set of critical points of L(X) in C n \ A as well as the set of critical points of the restriction of L(X) to any Y \ A, where Y is an algebraic hypersurface in C n coincides with those of F (X).
Notice that in Lemma 3, the meaning of a critical point of the real-valued function L(X) and the meaning of a critical point the multi-valued holomorphic function F (X) are different. In the former case we require that the real gradient of L(X) with respect to the real and imaginary parts of the complex coordinates vanishes while in the latter case we require that the complex gradient of F (X) vanishes. Lemma 3 holds due to the fact that L(X) is a pluriharmonic function closely related to F (X).
Proof As a warm-up exercise let us prove that the sets of critical points of L(X) and F (X) in C n \ A coincide. Notice that F (X) is non-vanishing in C n \ A which means that no branch of F (X) vanishes there. As was mentioned above, where Log F (X) is a multi-valued holomorphic logarithm function in C n \ A which is well-defined due to the fact that F (X) is non-vanishing. Similarly to the case of F (X), vanishing of the complex gradient of Log F (X) is given by a well-defined system of algebraic equations which coincides with that for F (X).
Let us now express the real gradient of L(X) using the complex gradient of Log F (X). Consider the decomposition of the complex variable x k into its real and imaginary parts, i.e. x k = u k + I · v k , where I = √ −1. Recall also that due to the Cauchy-Riemann equations, for any (locally) holomorphic function W (x 1 , . . . , The real gradient grad R L is given by F (X)) .
Denoting G := − Log F (X) and using the latter relations, we get Thus grad R L interpreted as a complex vector in coordinates (x 1 , . . . , x n ) is given by Therefore, grad R L = 0 if and only if grad C Log F (X) = 0, which is equivalent to grad C F (X) = 0. Let us finally discuss the situation when one restricts L(X) to an algebraic hypersurface Y . If p ∈ Y is a singular point of Y , then both L(X) and Log F (X) have critical points at p. If p ∈ Y is a nonsingular point, then introduce new complex coordinates (x 1 , . . . ,x n−1 ,x n ) adjusted to the tangent plane to Y at p. Namely, the origin with respect these coordinates is placed at p, the hyperplane spanned by (x 1 , . . . ,x n−1 ) coincides with the tangent plane to Y at p, andx n spans the same line as grad C Q(p), where Q is the polynomial defining Y . Formulas (2.12) and Eq. 2.13 are valid with respect to the new coordinate system as well. The condition that L(X)| Y has a critical point at p means that grad R L(p) lies in the complex line spanned byx n which is equivalent to the vanishing of Analogously, the condition that G(X)| Y has a critical point at p means that grad C G(p) spans in the complex line spanned byx n which is equivalent to the vanishing of n − 1 complex quantities ∂G(0) ∂x 1 , . . . , ∂G(0) ∂x n−1 . But vanishing of the former (2n − 2) real quantities is obviously equivalent to the vanishing of the latter (n − 1) complex quantities.
Proof of Theorem 1 We want to find all the critical points of L(X) subject to the restriction R(X) = 0, that is for X lying in \ A. (Recall that L(X) is defined in C n \ A and is the hypersurface in C n given by R(X) = 0.) Using Lemma 3, we need to find the critical points of the multi-valued holomorphic function F (X) restricted to . System (2.11) of Lemma 2 provides the corresponding equations defining the critical points. In the particular case of F (X) and R(X) as above, this system is composed by the equations and R(X) = r(x 1 ) + · · · + r(x n ) = 0. This system contains n + 1 equations in the (n + 1) variables x 1 , . . . , x n and ρ. Abusing our notation, assume for the moment that X = (x 1 , . . . , x n ) is not the set of variables for C n , but some concrete complex vector in C n \ A solving the system (2.14). Introducing the polynomials y(x) = n j =1 (x − x j ), y k (x) = y(x)/(x − x k ) and taking into account the obvious relations y k (x k ) = y (x k ), 2y k (x k ) = y (x k ), we conclude that the first n equations of the system (2.14) are equivalent to the system given by (2.15) Indeed, multiplying the k-th equation of Eq. 2.14 by A(x k )y k (x k ), we obtain exactly the k-th equation of Eq. 2.15.
Under our assumptions, is a polynomial in x k of degree q. Since the first term in Eq. 2.15 is a polynomial of degree n + p − 1 and the second one is a polynomial of degree n + q − 1, then by the fundamental theorem of algebra, there exists a polynomial V (x), of degree max{p − 1, q − 1}, such that Notice that the condition that R(X) is a rational symmetric function of a special form was imposed to guarantee that the parametric Lamé equation admits a polynomial solution.
Proof of Theorem 2 Indeed, observe that the above conditions show that the degenerate Lamé (1.9) takes the form According to the second statement of Theorem 1, the existence of a pair (V (x), y(x)) yields that X * = (x * 1 , . . . , x * n ) is a critical point of the restriction of the energy function L(X) to the hypersurface given by R(X) = 0. Here V (x) is a Van Vleck polynomial satisfying the is a Stieltjes polynomial which satisfies the latter differential equation with X * = (x * 1 , . . . , x * n ) obeying the restriction R(X * ) = 0.

Hermite Polynomials in Disguise
A straightforward change of variables in the differential (1.5) satisfied by the Hermite polynomials H n (x) implies that for any m ∈ N, the polynomial y(x) = y m,n (x) := H n (x m ) solves the degenerate Lamé differential equation

Laguerre Polynomials in Disguise
A procedure similar to that in the previous example shows that for any m ∈ N, the polynomial y(x) = y m,n (x) := L α n (x m ) solves the differential equation For any fixed m, the zeros of L α n (x m ) are located at the intersections of the m rays emanating from to origin with the slopes e 2πi/j , j = 0, . . . , m − 1 and the n circles with the radii ( k ) 1/m , k = 1, . . . , n, where k are the zeros of L α n (x). Theorem 2 implies that the coordinates of these mn zeros form a critical point of the logarithmic energy of the electrostatic field generated by the moving charges together with the charge (1 + αm)/2 at the origin, where the mn movable charges x k are subject to the constraint mn k=1 x m k = mn(n + α).

Laguerre Polynomials and Electrostatic Problem with a Rational Constraint
Substituting x → x + 1/x in L α n (x), we conclude that the polynomial of degree 2n Y (x) = x n L α n (x + 1/x) solves the differential equation The partial fraction decomposition ofB(x)/A(x) is given bỹ Theorems 1 and 2 imply that the 2n zeros of x n L α n (x + 1/x) are the coordinates of a critical point of the logarithmic energy of the electrostatic field generated by the movable charges together with the following five fixed charges. One charge equal to −n + (1 − a)/2 is placed at the origin; two charges equal to −1/2 are placed at ±1, and two charges (a + 1)/2 are placed at ±i.
Since the relation x k +1/x k = k associates each zero k of L α n (x) to two zeros of x n L α n (x + 1/x), we conclude that the critical points are either positive reals or belong to the semicircle {x ∈ C : |x| = 1, (x) > 0}.

Schrödinger-Type Equations
Now we consider another type of analogs of the equation satisfied by the Hermite polynomials. We say that Eq. 1.4 is of Schrödinger-type if A(x) is a non-vanishing constant (which we can always assume equals to 1). In this case, there are no fixed charges and the system of equations defining the equilibrium (usually called the Bethe ansatz) is given by for an appropriate constant C. (Notice that the critical points of the Vandemonde function and its generalizations to several types of hypersurfaces have been studied in [7][8][9].) Some interesting examples of Schrödinger-type equations are satisfied by the Laguerre polynomials of certain degrees with special values of parameter α. More precisely, for a given m ∈ N, consider the equation (3.17) One expects that eventual polynomial solutions of Eq. 3.17 must have degree mn. Observe that a basis of linearly independent solutions of Eq. 3.17 is given by 1 F 1 ( −mn/(m + 1), 1 − 1/(m + 1), x m+1 ) and x · 1 F 1 ( (1 − mn)/(m + 1), 1 + 1/(m + 1), x m+1 ), is the basic hypergeometric function given by Here (t) j is the standard Pochhammer symbol defined by (t) 0 := 1 and (t) j := t (t + 1) · · · (t + j − 1), j being a positive integer. It is clear that 1 F 1 (a, b, x) reduces to a polynomial if and only if a is a nonpositive integer, i.e., a = −N with N ∈ N. Moreover for b > 0, these polynomials coincide with the Laguerre polynomials since L α N (x) is a constant multiple of 1 F 1 (−N, α + 1, x). Therefore, Eq Applying Theorem 2 to Eq. 3.17 we conclude the following. Since A(x) ≡ 1, there are no fixed charges. There are Nm movable charges x k in the complex plane which obey the constraint mn k=1 x m+1 where C = (m+1)N (N +1)/(m+1)). In one of the situations N = md −1, or equivalently N = (mn − 1)/(m + 1) and in the other case N = md − 1 = mn/(m + 1). In the first case, one charge is at the origin and the remaining nm − 1 = N(m + 1) = (m + 1)(md − 1) are placed on the m + 1 rays e 2jπi/(m+1) , j = 0, . . . , m. In the second case, there are only nm = N(m + 1) = dm(m + 1) charges placed on the latter m + 1 rays. The partial fraction decomposition of B(z)/A(z) is given by

Relativistic Hermite Polynomials
Placing two fixed equal negative charges −(N + n − 1)/2 at ±i √ N and n unit movable charges on the real line, we obtain after a straightforward calculation, that the zeros of H N n (x) coincide with the unique equilibrium configuration of n movable unit charges where the energy attains its minimum. In fact, this minimum is global.
Let us briefly discuss how this equilibrium configuration depends on the positive parameter N . One can deduce the following.
• When N is a small positive number, then the fixed negative charges are located close to the origin and their total strength equals −(n − 1). Therefore, they attract all movable charges to the origin. • When N grows, then all movable charges move away from the origin because the force of attraction of the fixed negative charges decreases. This can be proved rigorously via a refinement of Sturm's comparison theorem obtained in [1]. • When N → +∞, the force of attraction of the fixed negative charges decreases because they move away from the real line, but at the same time their strength increases in such a way that the location of each movable charge has a limit, see Fig. 1. The conclusion based on the above observations is as follows. Since H N n (x) converges locally uniformly to the Hermite polynomial H n (x) as N goes to infinity, the zeros of H N n (x) converge to those of H n (x). Therefore, when N increases, the negative charges at ±i √ N increase in absolute value. The corresponding equilibrium configuration formed by the coordinates of the zeros of H N n (x) is such that the latter zeros move monotonically to those of H n (x). What is special about this phenomenon is that the influence of the increasing negative charges at ±i √ N do not "disappear at infinity" as one can suspect, but it transforms into a constraint.