Trajectories of quadratic differentials for Jacobi polynomials with complex parameters

Trajectories of quadratic differentials for Jacobi polynomials with complex parameters

Andrei Martínez-Finkelshtein, Pedro Martínez-González,
and Faouzi Thabet
Abstract

Motivated by the study of the asymptotic behavior of Jacobi polynomials with and we establish the global structure of trajectories of the related rational quadratic differential on . As a consequence, the asymptotic zero distribution (limit of the root-counting measures of ) is described. The support of this measure is formed by an open arc in the complex plan (critical trajectory of the aforementioned quadratic differential) that can be characterized by the symmetry property of its equilibrium measure in a certain external field.

AMS Mathematics Subject Classifications (2010): 33C45; 42C05; 30C15; 30C85; 30F15; 30F30; 31A15

Keywords: Asymptotics, zeros, orthogonal polynomials, equilibrium measure, -property, quadratic differential, Riemann surface.

1 Introduction

The motivation of this work is the large-degree analysis of the behavior of the Jacobi polynomials , when the parameters , are complex and depend on the degree linearly. Recall that these polynomials can be given explicitly by (see [14, 21])

or, equivalently, by the well-known Rodrigues formula

(1.1)

Clearly, polynomials are entire functions of the complex parameters .

If we fix and allow , the zeros of will cluster on and distribute there according to the well-known arcsine law. A non-trivial asymptotic behavior can be obtained in the case of varying coefficients and . Namely, we will consider sequences

(1.2)

where both and are fixed. The case can be studied by the already standard techniques from the potential theory [4] or by the saddle point method applied to their integral representation, see e.g. [3]. The general situation was analyzed in [8, 11, 12].

In this paper we are interested in the situation when at least one of the parameters, or , is non-real, see e.g. Figure 1. To be more precise, we assume that

(1.3)

Clearly, results for the case and can be easily deduced by reversing the roles of and .

\setkeys

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Figure 1: Zeros of the polynomial for and .

The key feature of the Jacobi polynomials that we use in their asymptotic analysis is their orthogonality property. It is well known that when , the Jacobi polynomials are orthogonal on with respect to the weight function . But as it was shown in [9], for general we can associate with a complex, non-hermitian orthogonality, where the integration goes along some contour in the complex plane. This is the key, at least in theory, to the study of the limit root location, as well as of the so-called weak (or -th root) asymptotics of these polynomials (via the Gonchar-Rakhmanov-Stahl (GRS) theory [5, 19]), and of their strong uniform asymptotics on the whole plane (by means of the Riemann-Hilbert (RH) steepest descent method of Deift-Zhou [2]).

The GRS method, in its general form, allows to reduce the description of the cluster set of the zeros of the sequence (1.2)–(1.3) to the problem of finding in a given homotopic class of curves the one with the so-called -property in the associated external field (see the definition (4.5) in Section 4), which in our case is

(1.4)

The most general known existence theorem for the -curves is contained in [16] under the assumption that external field is harmonic in a complement to a finite set (see also [10] for the polynomial external fields). This is the case of in (1.4) for , see [11] for the analysis of the weak asymptotics (via the GRS theory) and [8, 12] for the strong uniform asymptotics on (by the RH technique).

However, is not harmonic (it is not even single valued) in if we assume (1.3), which adds a new essential feature to the problem. In this case, the explicit construction of the curve with the -property is a consequence of the analysis of the structure of trajectories of the following quadratic differential on the Riemann sphere :

where

Although the local structure of such trajectories is well known, the global topology of the so-called critical graph is usually much more difficult to analyze. Thus, one of the central results of this paper is this description, carried out in Section 3, Theorem 3.1. As a result, we claim that for every pair of parameters satisfying (1.3) there exists an analytic Jordan arc , homotopic in the punctured plane to a Jordan arc connecting both zeros of the polynomial in , and given by the equation

This curve is the limiting set for the zeros of the Jacobi polynomials. Namely, with each we associate its normalized zero-counting measure , such that for any compact set in ,

(1.5)

Here the zeros are counted with their multiplicities.

In Section 5 we show that the sequence converges (as ) in the weak-* topology to a measure , supported on , absolutely continuous with respect to the linear Lebesgue measure on , and given by the formula

see Theorem 5.1. In Section 4 we establish that this is the equilibrium measure on in an external field, characterized by the above-mentioned -property (4.5).

It is worth noticing that the fact has an alternative interpretation from the point of view of the hypergeometric differential equation corresponding to : for each , is a discrete critical measures in the external field , and is the continuous critical measure in the same field. A general convergence theorem of this kind was proved in [13]. However, the reduction of the case analyzed here to the results of [13] is not direct. In particular, the existence and uniqueness of continuous critical measures for external fields with complex parameters is in general an open problem.

Our final remark is that using the construction of the measure and the steepest descent method for the Riemann–Hilbert characterization of the Jacobi polynomials [8, 9] the strong asymptotic formula can be proved. For instance (see (4.2) below),

where we take the holomorphic branch of the square root in such that

Then function

is holomorphic in the same domain. If denote the two zeros of , let

Then there is a sequence such that

locally uniformly in . Constants are chosen to match the leading term of .

This result (as well as its analogues on the limiting curve and at ) is established following almost literally the arguments of [12], and we refer the interested reader to that paper for details.

2 Critical points of

A rational quadratic differential on the Riemann sphere is a form , where is a rational function of a local coordinate . If is a conformal change of variables then

represents in the local parameter . The critical points of are its zeros and poles; all other points of are called regular points. We refer the reader to [6, 15, 20, 22] for further definitions and properties of quadratic differentials.

In this section we focus on a specific rational quadratic differential on the Riemann sphere ,

(2.1)

with

(2.2)

It depends on two parameters, and , for which (1.3) holds. Since

it is sufficient to restrict our attention to the following case:

(2.3)

for any other combination of the parameters with and we can readily derive the conclusions by combining the mappings

The quadratic differential (2.1) has five critical points on ; three of them at and . Since

(2.4)

under assumptions (2.3) these are double poles of . The other two critical points are the zeros of , that we describe next.

Let . Fixed , we denote by

(2.5)

the branch of this function, as a function of , in the cut plane , such that for . Equivalently, is a conformal mapping of onto the upper half plane with a slit:

(2.6)

With this notation, the zeros of are

(2.7)

respectively. Since and it is obvious that for and satisfying (2.3), and are simple and different from . Furthermore, the following assertions hold:

Lemma 2.1.

Under the assumptions (2.3), and . In particular, with ,

(2.8)

where (resp., ) denotes the boundary values of from the upper (resp., lower) half plane.

Proof.

The polynomial in (2.2) can be rewritten as

(2.9)

it is a quadratic polynomial in , whose discriminant is

In particular, if , then , so that with such the identity can hold only for . This proves that under assumptions (2.3) the roots of cannot belong to . Furthermore, if has a real root (hence, ), by (2.9),

and the assumption implies that .

From the results of [11] (actually, it is straightforward to check) we know that function

decreases monotonically from to as traverses from to , while

increases monotonically on , and decreases monotonically on . Since is locally conformal, (2.8) follows from the correspondence of boundary points.

Finally, by (2.7),

By (2.6), the right hand side belongs , so that for any pair satisfying (2.3),

Assuming that for certain satisfying (2.3), the root , and thus, , it follows that

or equivalently,

(2.10)

However, and since and ,

which yields a contradiction with (2.10). This proves that for satisfying (2.3), , and thus, . ∎

3 Domain configuration of

Recall that the horizontal trajectories (or just trajectories) of are the loci of the equation

while the vertical or orthogonal trajectories are obtained by replacing by in the equation above. The trajectories and the orthogonal trajectories of produce a transversal foliation of the Riemann sphere .

A trajectory of starting and ending at (if exists) is called finite critical or short; if it starts at one of the zeros but tends to either pole, we call it infinite critical trajectory of . In a slight abuse of terminology, we say that such an infinite critical trajectory, if it exists, joins the zero with the corresponding pole. Since has only three poles, Jenkins’ three pole Theorem [7] asserts that it cannot have any recurrent trajectory.

The set of both finite and infinite critical trajectories of together with their limit points (critical points of ) is the critical graph of .

According to [6, Theorem 3.5] (see also [20, §10]), the complement of the closure of in consists of a finite number of domains called the domain configuration of . Among the possible types of domains there are the so-called circle and strip domains. A circle domain of is a maximal simply connected domain swept out by regular closed trajectories of surrounding a double pole that is the only singularity of in . A strip domain or a digon of is a maximal simply connected domain swept out by regular trajectories of , each diverging to a double pole in both directions; these double poles must represent distinct boundary points of (see [18]).

The main result of this section is the following theorem, which describes the critical graph as well as the domain configuration of (see Figure 2).

Theorem 3.1.

Let and . Then there exists a short trajectory of , joining and . This trajectory is unique, homotopic in the punctured plane to a Jordan arc connecting in .

Furthermore, the structure of the critical graph of is as follows:

  • the short trajectory of , joining and ;

  • the unique finite critical trajectory of emanating from and forming a closed loop, encircling ;

  • the infinite critical trajectory , emanating from and diverging towards ;

  • the infinite critical trajectory , emanating from and diverging towards .

splits into two connected domains: the bounded circle domain with center at , and an unbounded strip domain , whose boundary points are and .

In other words, we claim that the critical graph of is made of 2 short and 2 infinite critical trajectories. Recall that it is sufficient to analyze the case when satisfy assumptions (2.3).

\setkeys

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Figure 2: Typical structure of the critical graph for the trajectories of under the assumptions (2.3). These trajectories are depicted for and .

In order to prove Theorem 3.1 we start from the local structure of the trajectories of at its critical points (see e.g. [6, 15, 20, 22]). Recall that at any regular point the trajectories are locally simple analytic arcs passing through this point, and through every regular point of passes a uniquely determined horizontal and uniquely determined vertical trajectory, mutually orthogonal at this point [20, Theorem 5.5]. Furthermore, there are trajectories emanating from under equal angles .

By (2.4) we conclude that the trajectories are closed Jordan curves in a neighborhood of , and the radial or the log-spiral form at and . The radial structure at occurs if , and at infinity, when .

Let be a Jordan arc in joining and . Then in we can fix a single-valued branch of by requiring that

(3.1)

Clearly, conditions

(3.2)

determine uniquely the homotopy class of in the punctured plane . We have,

Proposition 3.2.

Let satisfy assumptions (2.3), and let be a Jordan arc in joining and , and is its single-valued branch in fixed by the condition (3.1). Then

(3.3)

where is the boundary value on one of the sides of .

Moreover, the integral in the left hand side of (3.3) takes the value if and only if is such that conditions (3.2) are satisfied.

Proof.

By the properties of the square root, the integral in the left hand side of (3.3) can be written as

which can be calculated using the residues of the integrand at and . Thus,

As it will be seen in Section 5, the short trajectory , joining the zeros , beings the carrier of the asymptotic zero distribution of the Jacobi polynomials, must satisfy

By the proof of Proposition 3.2, this is equivalent to conditions (3.2). So, we need to establish the homotopic class of curves for which conditions (3.2) are satisfied. According to Proposition 3.4 below, there cannot exist a trajectory passing through either pole and joining both zeros . This shows that the homotopic class of curves within the domain given by assumptions (2.3) remains invariant, and it is sufficient to analyze the limit case , , for which, by Lemma 2.1, . By (3.1)–(3.2),

which shows that cuts at some point . We conclude that

Proposition 3.3.

Under assumptions (2.3), Jordan arcs joining and , and such that conditions (3.2) are satisfied, are homotopic in the punctured plane to a Jordan arc connecting in .

Another tool needed to finish the proof of Theorem 3.1 is the following result:

Proposition 3.4.

Under assumptions (2.3),

  1. There cannot exist two infinite critical trajectories emanating from the zeros of and diverging to the pole at .

  2. There cannot exist two infinite critical trajectories emanating from the same zero of and diverging to .

Its proof is based on the so-called Teichmüller lemma (see [20, Theorem 14.1]) and follows literally the arguments that have been used in [1, Lemma 4]. We omit repeating them here for the sake of brevity.

Let us establish the structure of the critical graph . Under the assumptions (2.3), is the center of a circle domain , whose boundary, , is made of critical trajectories. Since , we conclude that is made of short critical trajectories. Hence, a priori there are two possibilities:

  1. either is made of two short trajectories, both connecting and , or

  2. is a single closed critical trajectory passing either through or .

For a fixed let be the closure of the domain defined by the conditions (2.3) in the -plane. Observe that the origin does not belong to the image of by the mapping (2.5)–(2.6), which means that are simple in the whole . A consequence of this fact and of Proposition 3.4 is that the homotopic class in of the curves comprising the critical graph is invariant for . For the structure is well-known (see e.g. [11]): , and is comprised of the interval and of two loops, one emanating from and encircling , and another one emanating from and encircling . In other words, it corresponds to the condition (b) above. Hence, we may discard the possibility (a) for the whole set of parameters satisfying the assumptions (2.3).

In the case (b), let be the zero of on the boundary of . Then the third trajectory, emanating from the same zero, cannot diverge to or : it would oblige two critical trajectories, coming from the other zero of , to diverge to the same pole, contradicting Proposition 3.4.

Thus, we conclude that there exists a short trajectory, , connecting and . Since we have discarded the case (a) mentioned above, this settles automatically the rest of the structure of the critical graph .

Finally, the fact that it is the zero on the boundary of (and in consequence, that is connected with both and by critical trajectories) can be established by the deformation arguments, like in the proof of Proposition 3.3.

The distinguished short trajectory plays an essential role in what follows. For the rest of the paper we use a notation for the holomorphic branch of in :

(3.4)

Since by assumptions (2.3), , we have that the complement of in is a connected domain whose boundary points are and (see Figure 2). Let us show that it is actually a strip domain, as claimed.

We introduce in the following analytic function,

(3.5)

Let be the orthogonal trajectory of emanating from that is the analytic continuation of the horizontal trajectory that joins and . Function in (3.5) is defined in such a way that

Proposition 3.5.

Under assumptions (2.3), function is a conformal mapping of the domain onto the vertical strip .

Proof.

We fix the orientation of the critical graph as follows: both and are emanating from , is entering , and is oriented clockwise. This orientation induces the and (that is the right and left) sides of each curve, that we indicate with superscripts. For convenience, we reproduce again the Figure 2 in Figure 3, indicating now the corresponding sides of the curves.

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Figure 3: Sides of the curves forming the critical graph with the orientation indicated in the text.

Since is made of trajectories, maps each of these curves onto a vertical line. Using (3.2), (3.4), and operating as in the proof of Proposition 3.2, we have

(3.6)

Thus,

and in consequence, establishes a bijection of the boundary

of the strip domain , oriented from 1 to , and the imaginary axis , oriented from to . By orientation preservation, lies in the right half-plane.

More precisely, let be a simple Jordan arc, from to , and intersecting only once, in . Using again the arguments from the proof of Proposition 3.2,

Thus, under assumptions (2.3),

which shows that the other boundary of the strip domain is mapped by onto the vertical line .

In the next section we will need one more technical result, related to the domain configuration of . Let be a Jordan curve joining and , lying entirely (except for its endpoints) in , passing through in such a way that , and otherwise disjoint with the critical graph . We denote the open arc of joining with , and by the open arc of joining with .

Lemma 3.6.

With the notations above,

Proof.

First, observe that by (3.2),

where we choose an appropriate branch of the logarithm. This shows that the inequalities hold in a neighborhood of . On the other hand, assume there is a point , , such that

(3.7)

By assumptions, . Let be the horizontal trajectory of passing through ; it must intersect at least one of the vertical trajectories of emanating from . Hence, deforming the path from to into the union of an arc and an arc from we run into contradiction with (3.7). ∎

4 An equilibrium problem for the logarithmic potential

On the short trajectory we define the following measure, absolutely continuous with respect to the arc-length measure:

(4.1)

with defined in (3.4), and the boundary values are with respect to the chosen orientation of . Since is a horizontal trajectory of , and using (3.6) we conclude that is a positive probability measure defined on this arc. Straightforward calculations using residues, similar to those performed in (3.6), show that

(4.2)

For measure on , its logarithmic potential is defined by

By (4.2), there exists a constant such that for ,

(4.3)

where

(4.4)

is a multivalued analytic function in with a single-valued real part.

Let us define

Equation (4.3) can be rewritten as

Since is a trajectory of , we see that

Let be a Jordan curve joining and , lying entirely (except for its endpoints) in , passing through in such a way that , and otherwise disjoint with the critical graph . From Lemma 3.6 we conclude that

This property characterizes the fact that is actually the equilibrium measure of in the external field , and is the corresponding equilibrium constant (see [5, 17]). Furthermore, for defined in (4.4) the trivial identity on111Here we understand by the open arc without its endpoints . yields the so-called -property in the external field : for every ,

(4.5)

where are the normals to .

5 Relation to the asymptotics of Jacobi polynomials with varying parameters

Let us return to the Jacobi polynomials considered in Section 1, and consider the case of varying coefficients and and study the asymptotic behavior of the zeros of the sequences of polynomials given in (1.2), where the constants and satisfy the assumptions (1.3). As it was mentioned, it is sufficient to restrict our attention to the case (2.3).

Our main goal now is to study the convergence of the sequence of the zero counting measures (1.5) in the weak- topology and, if the limit exists, to find it explicitly.

The main result of this section is the following theorem:

Theorem 5.1.

Let the sequence of generalized Jacobi polynomials in (1.2) be such that the pair satisfies assumptions (2.3). Then there is a unique measure such that

The measure is supported on the short trajectory , is absolutely continuous with respect to the linear Lebesgue measure on , and is given by the formula (4.1).

\setkeys

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Figure 4: Critical graph of , with and (Figure 2), and the zeros of the corresponding polynomial (Figure 1) superimposed.

The main property satisfied by polynomials is the non-hermitian orthogonality conditions. Integrating by parts successively the Rodrigues formula (1.1), it is straightforward to obtain the following result, proved in [9]:

Proposition 5.2.

Under assumptions (2.3), let be a Jordan curve joining and , and lying entirely (except for its endpoints) in . Then, for all sufficiently large ,

Here the integral is understood in terms of the analytic continuation of any branch of the integrand along  .

The main tools for the study of the weak asymptotic behavior of polynomials satisfying a non-hermitian orthogonality have been developed in the seminal works of Stahl [19] and Gonchar and Rakhmanov [5]. They showed that when the complex analytic weight function depends on the degree of the polynomial, the limit zero distribution is characterized by an equilibrium problem on a compact set in the presence of an external field and satisfying the -property described in Section 4. In fact, Theorem 5.1 is a direct consequence of Proposition 5.2, the properties of established in Section 4, and the original work [5] (see also [11]).

Finally, as it was mentioned in the Introduction, measure and the structure of the trajectories of are also the main ingredients of the steepest descent method for the Riemann–Hilbert characterization of the Jacobi polynomials. The analysis follows almost literally the calculations of [12], so we refer the reader to that paper for the details.

Acknowledgments

The first and second authors (AMF and PMG) were partially supported by MICINN of Spain and by the European Regional Development Fund (ERDF) under grants MTM2011-28952-C02-01 and MTM2014-53963-P, by Junta de Andalucía (the research group FQM-229), and by Campus de Excelencia Internacional del Mar (CEIMAR) of the University of Almería. Additionally, AMF was supported by Junta de Andalucía through the Excellence Grant P11-FQM-7276. Part of this work was carried out during the visit of AMF to the Department of Mathematics of the Vanderbilt University. He acknowledges the hospitality of the hosting department, as well as a partial support of the Spanish Ministry of Education, Culture and Sports through the travel grant PRX14/00037.

We also wish to thank the anonymous referee for very useful remarks.


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A. Martínez-Finkelshtein (andrei@ual.es)
Department of Mathematics
University of Almería, Spain, and
Instituto Carlos I de Física Teórica y Computacional
Granada University, Spain

P. Martínez-González (pmartine@ual.es)
Department of Mathematics
University of Almería, Spain

F. Thabet (faouzithabet@yahoo.fr)
ISSAT, University of Gabes,
Gabés, Tunisia



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