Root-counting measures of Jacobi polynomials
and topological types and critical
of related quadratic differentials
Two main topics of this paper are asymptotic distributions of zeros of Jacobi polynomials and topology of critical trajectories of related quadratic differentials. First, we will discuss recent developments and some new results concerning the limit of the root-counting measures of these polynomials. In particular, we will show that the support of the limit measure sits on the critical trajectories of a quadratic differential of the form . Then we will give a complete classification, in terms of complex parameters , , and , of possible topological types of critical geodesics for the quadratic differential of this type.
Key words and phrases:Jacobi polynomials, asymptotic root-counting measure, quadratic differentials, critical trajectories
2010 Mathematics Subject Classification:30C15, 31A35, 34E05
1. Introduction: From Jacobi polynomials to quadratic differentials
Two main themes of this work are asymptotic behavior of zeros of certain polynomials and topological properties of related quadratic differentials. The study of asymptotic root distributions of hypergeometric, Jacobi, and Laguerre polynomials with variable real parameters, which grow linearly with degree, became a rather hot topic in recent publications, which attracted attention of many authors , , , , , , , , . In this paper, we survey some known results in this area and present some new results keeping focus on Jacobi polynomials.
Recall that the Jacobi polynomial of degree with complex parameters is defined by
where with a non-negative integer and an arbitrary complex number . Equivalently, can be defined by the well-known Rodrigues formula:
The following statement, which can be found, for instance, in [24, Proposition 2], gives an important characterization of Jacobi polynomials as solutions of second order differential equation.
For arbitrary fixed complex numbers and , the differential equation
with a spectral parameter has a non-trivial polynomial solution of degree if and only if . This polynomial solution is unique (up to a constant factor) and coincides with .
Working with root distributions of polynomials, it is convenient to use root-counting measures and their Cauchy transforms, which are defined as follows.
For a polynomial of degree with (not necessarily distinct) roots , its root-counting measure is defined as
where is the Dirac measure supported at .
Given a finite complex-valued Borel measure compactly supported in its Cauchy transform is defined as
and its logarithmic potential is defined as
In case when is the root-counting measure of a polynomial , we will write instead of . It follows from Definitions 1 and 2 that the Cauchy transform of the root-counting measure of a monic polynomial of degree coincides with the normalized logarithmic derivative of ; i.e.,
and its logarithmic potential is given by the formula:
Let be a sequence of Jacobi polynomials and let be the corresponding sequence of their root-counting measures. The main question we are going to address in this paper is the following:
Assuming that the sequence weakly converges to a measure compactly supported in , what can be said about properties of the support of the measure and about its Cauchy transform ?
Regarding the Cauchy transform , our main result in this direction is the following theorem.
Suppose that a sequence of Jacobi polynomials satisfies conditions:
(a) the limits and exist, and ;
(b) the sequence of the root-counting measures converges weakly to a probability measure , which is compactly supported in .
Then the Cauchy transform of the limit measure satisfies almost everywhere in the quadratic equation:
The proof of Theorem 1 given in Section 2 consists of several steps. Our arguments in Section 2 are similar to the arguments used in a number of earlier papers on root asymptotics of orthogonal polynomials.
Equation (1.4) of Theorem 1 implies that the support of the limit measure has a remarkable structure described by Theorem 2 below. And this is exactly the point where quadratic differentials, which are the second main theme of this paper, enter into the play.
In notation of Theorem 1, the support of consists of finitely many trajectories of the quadratic differential
and their end points.
Thus, to understand geometrical structure of the support of we have to study geometry of critical trajectories, or more generally critical geodesics of the quadratic differential of Theorem 1. We will consider a slightly more general family of quadratic differentials depending on three complex parameters , , where
It is well-known that quadratic differentials appear in many areas of mathematics and mathematical physics such as moduli spaces of curves, univalent functions, asymptotic theory of linear ordinary differential equations, spectral theory of Schrödinger equations, orthogonal polynomials, etc. Postponing necessary definitions and basic properties of quadratic differentials till Section 3, we recall here that any meromorphic quadratic differential defines the so-called -metric and therefore it defines -geodesics in appropriate classes of curves. Motivated by the fact that the family of quadratic differentials (1.5) naturally appears in the study of the root asymptotics for sequences of Jacobi polynomials and is one of very few examples allowing detailed and explicit investigation in terms of its coefficients, we will consider the following two basic questions:
How many simple critical -geodesics may exist for a quadratic differential of the form (1.5)?
For given , , describe topology of all simple critical -geodesics.
A complete description of topological structure of trajectories of quadratic differentials (1.5) which, in particular, answers questions 1) and 2), is given by lengthy Theorem 5 stated in Section 9.
The rest of the paper consists of two parts and is structured as follows. The first part, which is the area of expertise of the first author, includes Sections 2, 4, and 5. Section 2 contains the proof of Theorem 1 and related results. The material presented in Section 4 is mostly borrowed from a recent paper  of the first author. It contains some general results connecting signed measures, whose Cauchy transforms satisfy quadratic equations, and related quadratic differentials in . In particular, these results imply Theorem 2 as a special case. In Section 5, we formulate a number of general conjectures about the type of convergence of root-counting measures of polynomial solutions of a special class of linear differential equations with polynomial coefficients, which includes Riemann’s differential equation.
Remaining sections constitute the second part, which is the area of expertise of the second author. In Section 3, we recall basic information about quadratic differentials, their critical trajectories and geodesics. This information is needed for presentation of our results in Sections 6–10. In Section 6, we describe possible domain configurations for the quadratic differentials (1.5). Then, in Section 7, we describe possible topological types of the structure of critical trajectories of quadratic differentials of the form (1.5). Finally in Sections 8–10, we identify sets of parameters corresponding to each topological type. The latter allows us to answer some related questions.
We note here that our main proofs presented in Sections 6–10 are geometrical based on general facts of the theory of quadratic differentials. Thus, our methods can be easily adapted to study trajectory structure of many quadratic differentials other then quadratic differential (1.5).
Section 11 is our Figures Zoo, it contains many figures illustrating our results presented in Sections 6–10.
Acknowledgements. The authors want to acknowledge the hospitality of the Mittag-Leffler Institute in Spring 2011 where this project was initiated. The first author is also sincerely grateful to R. Bøgvad, A. Kuijlaars, A. Martínez-Finkelshtein, and A. Vasiliev for many useful discussions.
2. Proof of Theorem 1
If a sequence of Borel probability measures in weakly converges to a probability measure with a compact support, then the sequence of its Cauchy transforms converges to in . Moreover there exists a subsequence of which converges to pointwise almost everywhere.
The next result is recently obtained by the first author jointly with R.Bøgvad and D. Khavinsion, see Theorem 1 of  and has an independent interest.
Let be any sequence of polynomials satisfying the following conditions:
1. as ,
2. almost all roots of all lie in a bounded convex open when . (More exactly, if denotes the number of roots of counted with multiplicities which are located in then ), then for any ,
where is the number of roots of counted with multiplicities which are located inside , the latter set being the -neighborhood of in .
Let be any sequence of polynomials satisfying the following conditions:
1. as ,
2. the sequence (resp. ) of the root-counting measures of (resp. ) weakly converges to compactly supported measures (resp ).
Then and satisfy the inequality with equality on the unbounded component of . Here (resp. ) is the logarithmic potential of the limiting measure (resp. ).
Without loss of generality, we can assume that all are monic. Let be a compact convex set containing almost all the zeros of the sequences and , i.e., . By (1.3) we have
with convergence in . Hence by (1.2),
Now, if is a positive compactly supported test function, then
where denotes Lebesgue measure in the complex plane. Since is locally integrable, the function is continuous, and hence bounded by a constant for all in . Since asymptotically almost all zeros of belong to , the last expression in (2.2) tends to when . This proves that .
In the complement of , is harmonic and is subharmonic, hence is a negative subharmonic function. Moreover, in the complement of , converges to the Cauchy transform of a.e. in . Since is a nonconstant holomorphic function in the unbounded component of , it follows from (2.1) that there. ∎
Notice that Lemma 2 implies the following interesting fact.
In notation of Lemma 2, if has Lebesque area 0 and the complement is path-connected, then . In particular, in this case the whole sequence weakly converges to .
In general, however as shown by a trivial example of the sequence . Also even if exists the limit does not have to exist for the whole sequence. An example of this kind is the sequence where and , .
Luckily, the latter phenomenon can never occur for sequences of Jacobi polynomials, see Proposition 3 below. (Apparently it can not occur for a much more general class of polynomial sequences introduced in § 5.)
If the sequence of the root-counting measures of a sequence of Jacobi polynomials weakly converges to a measure compactly supported in and the sequence of the root-counting measures of a sequence weakly converges to a measure compactly supported in then one of the following alternatives holds:
(i) the sequences and (and, therefore, the sequences and ) are bounded;
(ii) the sequence is unbounded and the sequence is bounded, in which case where is the unit point mass at (or, equivalently, );
(iii) both sets and are unbounded, in which case, there exists at least one and a subsequence such that and where is the unit point mass at (or, equivalently, ).
Indeed, assume that the alternative (i) does not hold. Then there is a subsequence such that at least one of is unbounded along this subsequence. By our assumptions and weakly. Hence, by Lemma 1, there exists a subsequence of indices along which pointwise converges to and pointwise converges to a.e. in . Consider the sequence of differential equations satisfied by and divided termwise by :
If for a subsequence of indices, while stays bounded, then the Cauchy transform of the limiting (along this subsequence) measure must vanish identically in order for (2.3) to hold in the limit . But is obviously impossible.
On the other hand, if for a subsequence of indices, while stays bounded, then the limit of (2.3) when coincides with implying . Thus in Case (ii), the sequence converges to .
Now assume, that or a subsequence of indices, both and tend to . Then dividing (2.3) by and letting we conclude that the sequence must be bounded. Therefore there exists its subsequence which converges to some . Taking the limit along this subsequence, we obtain
This is true for all for which the Cauchy transform converges, i.e. almost everywhere outside the support of . Using the main results of [7, 8] claiming that the support of consists of piecewise smooth compact curves and/or isolated points together with the fact that must have a discontinuity along every curve in its support, we conclude that the support of is the point . Thus in Case (iii), the sequence converges to . ∎
The next statement provides more information about Case (i) of Lemma 3.
Assume that the sequence of the root-counting measures for a sequence of Jacobi polynomials weakly converges to a compactly supported measure in Assume additionally that and with . Then, for any positive integer the sequence of the root-counting measures for the sequence of the -th derivatives converges to the same measure .
Observe that if an arbitrary polynomial sequence of increasing degrees has almost all roots in a convex bounded set , then, by Proposition 2, almost all roots of are in , for any . Therefore, if the sequence of the root-counting measures of weakly converges to a compactly supported measure then there exists at least one weakly converging subsequence of . Additionally, by the Gauss-Lucas Theorem, the support of its limiting measure belongs to the (closure of the) convex hull of the support of . Thus the weak convergence of implies the existence of a weakly converging subsequence .
Proposition 3 is obvious in Cases (ii) and (iii) of Lemma 3. Let us concentrate on the remaning Case (i). Our assumptions imply that along a subsequence of the sequence of Cauchy transforms of polynomials converges pointwise almost everywhere. We first show that the above sequence can not converge to on a set of positive measure.
Indeed, the differential equation satisfied by after its division by is given by (2.3). Since the sequences and converge and , equation (2.3) shows that cannot converge to on a set of positive measure. Analogously, we see that cannot converge to 0 on a set of positive measure either. Indeed, differentiating (2.3), we get that satisfies the equation
Using the same analysis as for , we can conclude that the limit along a subsequence exists pointwise and is non-vanishing almost everywhere.
Denote the logarithmic potentials of the root-counting measures associated to and by and respectively. Denote their limits by and (where apriori is a limit only along some subsequence). With a slight abuse of notation, the following holds
due to the above claim about . But since by Lemma 2, we see that and, in particular exists as a limit over the whole sequence. Hence the asymptotic root-counting measures of and actually coincide. Similar arguments apply to higher derivatives of the sequence . ∎
Proof of Theorem 1.
The polynomial satisfies the equation (2.3). By Proposition 3 we know that, under the assumptions of Theorem 1, if converges to a.e. in then the sequence also converges to the same a.e. in . Therefore, the expression converges to a.e. in . Thus (which is well-defined a.e. in ) should satisfy the equation
where and . ∎
Apparently the condition that the sequences and are bounded should be enough for the conclusion of Theorem 1. (The existence of the limits and should follow automatically with some weak additional restriction.) Indeed, since the sequences and are bounded, we can find at least one subsequence of indices along which both sequences of quotients converge. Assume that we have two possible distinct (pairs of) limits and along different subsequences. But then the same complex-analytic function should satisfy a.e. two different algebraic equations of the form (1.4) which is impossible at least for generic and .
3. Preliminaries on quadratic differentials
In this section, we recall some definitions and results of the theory of quadratic differentials on the complex sphere . Most of these results remain true for quadratic differentials defined on any compact Riemann surface. But for the purposes of this paper, we will focus on results concerning the domain structure and properties of geodesics of quadratic differentials defined on . For more information on quadratic differentials in general, the interested reader may consult classical monographs of Jenkins  and Strebel  and papers  and .
A quadratic differential on a domain is a differential form with meromorphic and with conformal transformation rule
where is a conformal map from onto a domain . Then zeros and poles of are critical points of , in particular, zeros and simple poles are finite critical points of . Below we will use the following notations. By , , and we denote, respectively, the set of all poles, set of all finite critical points, and set of all infinite critical points of . Also, we will use the following notations: , , .
A trajectory (respectively, orthogonal trajectory) of is a closed analytic Jordan curve or maximal open analytic arc such that
A trajectory is called critical if at least one of its end points is a finite critical point of . By a closed critical trajectory we understand a critical trajectory together with its end points and (not necessarily distinct), assuming that these end points exist.
Let denote the closure of the set of points of all critical trajectories of . Then, by Jenkins’ Basic Structure Theorem [21, Theorem 3.5], the set consists of a finite number of circle, ring, strip and end domains. The collection of all these domains together with so-called density domains constitute the so-called domain configuration of . Here, we give definitions of circle domains and strip domains only; these two types will appear in our classification of possible domain configurations in Section 5. Fig. 1–4 show several domain configurations with circle and strip domains. For the definitions of other domains, we refer to [21, Ch. 3].
We recall that a circle domain of is a simply connected domain with the following properties:
contains exactly one critical point , which is a second order pole,
the domain is swept out by trajectories of each of which is a Jordan curve separating from the boundary ,
contains at least one finite critical point.
Similarly, a strip domain of is a simply connected domain with the following properties:
contains no critical points of ,
contains exactly two boundary points and belonging to the set (these boundary points may be situated at the same point of ),
the points and divide into two boundary arcs each of which contains at least one finite critical point,
is swept out by trajectories of each of which is a Jordan arc connecting points and .
As we mentioned in the Introduction, every quadratic differential defines the so-called (singular) -metric with the differential element . If is a rectifiable arc in then its -length is defined by
According to their -lengthes, trajectories of can be of two types. A trajectory is called finite if its -length is finite, otherwise is called infinite. In particular, a critical trajectory is finite if and only if it has two end points each of which is a finite critical point.
An important property of quadratic differentials is that transformation rule (8.1) respects trajectories and orthogonal trajectories and their -lengthes, as well as it respects critical points together with their multiplicities and trajectory structure nearby.
A locally rectifiable (in the spherical metric) curve is called a -geodesic if it is locally shortest in the -metric.
Next, given a quadratic differential , we will discuss geodesics in homotopic classes. For any two points , let denote the set of all homotopic classes of Jordan arcs joining and . Here the letter stands for ”Jordan”. It is well-known that there is a countable number of such homotopic classes. Thus, we may write .
Every class can be extended to a larger class by adding non-Jordan continuous curves joining and , each of which is homotopic on to some curve in the following sense.
There is a continuous function from the square to such that
, for all ,
For every fixed , the curve is in the class .
The following proposition is a special case of a well-known result about geodesics, see e.g. [33, Theorem 18.2.1].
For every , there is a unique curve , called -geodesic in , such that for all , . This geodesic is not necessarily a Jordan arc.
A -geodesic from to is called simple if and is a Jordan arc on joining and . A -geodesic is called critical if both its end points belong to the set of finite critical points of .
Let be a quadratic differential on . Then for any two points and every continuous rectifiable curve on joining the points and there is a unique shortest curve belonging to the homotopic class of .
Furthermore, is a geodesic in this class.
Let . A geodesic ray from is a maximal simple rectifiable arc with such that for every , , the arc is a geodesic from to .
Let be a circle domain of centered at and let be a geodesic ray from such that for some .
Then either enters into through the point and then approaches to staying in or is an arc of some critical trajectory .
Let be a second order pole of and let be the homotopic class of closed curves on separating from . Then there is exactly one real , , such that the quadratic differential has a circle domain, say , centered at . Furthermore, the boundary is the only critical -geodesic (non-Jordan in general) in the class .
In particular, may contain at most one critical geodesic loop.
We will need some simple mapping properties of the canonical mapping related to the quadratic differential , which is defined by
with some and some fixed branch of the radical. A simply connected domain without critical points of is called a -rectangle if the boundary of consists of two arcs of trajectories of separated by two arcs of orthogonal trajectories of this quadratic differential. As well a canonical mapping maps any -rectangle conformally onto a geometrical rectangle in the plane with two sides parallel to the horizontal axis.
4. Cauchy transforms satisfying quadratic equations and quadratic differentials
Below we relate the question for which triples of polynomials the equation
with admits a compactly supported signed measure whose Cauchy transform satisfies (4.1) almost everywhere in to a certain problem about rational quadratic differentials. We call such measure a motherbody measure for (4.1).
For a given quadratic differential on a compact surface denote by the union of all its critical trajectories and critical points. (In general, can be very complicated. In particular, it can be dense in some subdomains of .) We denote by (the closure of) the set of finite critical trajectories of (4.2). (One can show that is an imbedded (multi)graph in . Here by a multigraph on a surface we mean a graph with possibly multiple edges and loops.) Finally, denote by the subgraph of consisting of (the closure of) the set of finite critical trajectories whose both ends are zeros of .
A non-critical trajectory of a meromorphic is called closed if such that for all . The least such is called the period of . A quadratic differential on a compact Riemann surface without boundary is called Strebel if the set of its closed trajectories covers up to a set of Lebesgue measure zero.
Going back to Cauchy transforms, we formulate the following necessary condition of the existence of a motherbody measure for (4.1).
(i) any connected smooth curve in the support of coincides with a horizontal trajectory of the quadratic differential
(ii) the support of includes all branching points of (4.1).
Remark. Observe that if and are coprime, the set of all branching points coincides with the set of all zeros of . In particular, in this case part (ii) of Proposition 6 implies that the set for the differential should contain all zeros of .
The fact that every curve in should coincide with some horizontal trajectory of (4.2) is well-known and follows from the Plemelj-Sokhotsky’s formula. It is based on the local observation that if a real measure is supported on a smooth curve , then the tangent to at any point should be perpendicular to where and are the one-sided limits of when , see e.g. . (Here stands for the usual complex conjugation.) Solutions of (4.1) are given by
their difference being
Since the tangent line to the support of the real motherbody measure satisfying (4.1) at its arbitrary smooth point , is orthogonal to it is exactly given by the condition . The latter condition defines the horizontal trajectory of at .
Finally the observation that should contain all branching points of (4.1) follows immediately from the fact that is a well-defined univalued function in . ∎
Proposition 6 allows us, under mild nondegeneracy assumptions, to formulate necessary and sufficient conditions for the existence of a motherbody measure for (4.1) which however are difficult to verify. Namely, let with affine coordinates be the algebraic curve given by (the projectivization of) equation (4.1). has bidegree and is hyperelliptic. Let be the projection of on the -plane along the -coordinate. From (4.1) we observe that induces a branched double covering of by . If and are coprime and if , the set of all branching points of coincides with the set of all zeros of . (If then is also a branching pont of of multiplicity .) We need the following lemma.
If and are coprime, then at each pole of (4.1) i.e. at each zero of , only one of two branches of goes to . Additionally the residue of this branch at this zero equals that of .
Indeed if and are coprime, then no zero of can be a branching point of (4.1) since . Therefore only one of two branches of goes to at . More exactly, the branch attains a finite value at while the branch goes to where we use the agreement that . Now consider the residue of the branch at . Since residues depend continuously on the coefficients it suffices to consider only the case when is a simple zero of . Further if is a simple zero of then