ROOT ASYMPTOTICS FOR THE EIGENFUNCTIONS OF UNIVARIATE DIFFERENTIAL OPERATORS

This paper is a brief survey of the research conducted by the author and his collaborators in the field of root asymptotics of (mostly polynomial) eigenfunctions of linear univariate differential operators with polynomial coefficients.


Objective
Study asymptotic properties of sequences {p n (z)}, of polynomials/entire functions in z which either 1. are polynomial/entire eigenfunctions of a univariate linear ordinary differential operator with polynomial coefficients; or 2. are polynomial solutions of more general pencils of such operators, e.g.homogenized spectral problems and Heine-Stieltjes spectral problems; or 3. satisfy a finite recurrence relation with (in general) varying coefficients.

Basic notions and examples
Definition 1 An operator and there exists at least one value i such that deg Q i (z) = i.
Lemma 1 For any exactly solvable T and sufficiently large n there exists a unique (up to a scalar) eigenpolynomial p n (z) of degree n.
Typical problem.Given an exactly solvable T describe the root asymptotics for the sequence of polynomials {p n (z)}.

Two asymptotic measures
Given a polynomial family {p n (z)} where deg p n (z) = n we define two basic measures: (i) asymptotic root-counting measure μ; (ii) asymptotic ratio measure ν.
(Assume for simplicity that p n (z) has no multiple roots and expand .) Associate to q n (z) the finite complex-valued measure by placing κ i,n at z i,n .Define the asymptotic ratio measure of the sequence {p n (z)} as Observation.Supports of μ and ν coincide but ν is often complex-valued.

Examples
Below we show the root distribution for p 55 (z) for 4 different exactly solvable operators

Classical prototypes
Theorem 1 (G.Szegö) If {p n (z)} is a family of polynomials orthogonal w.r.t a positive weight w(z) then the asymptotic root-counting measure has the density then the asymptotic ratio measure has the density 3 First results

Non-degenerate exactly solvable operators
The next subsection is based on [10,2].Definition 4 The Cauchy transform of a (com-plex-valued) measure ρ satisfying C dρ(ξ) < ∞ is given by

Definition 5
An exactly solvable operator Theorem 3 (H.Rullgård) Let Q k (z) be a monic degree k polynomial.Then there exists a unique probability measure Theorem 4 (Main result, see Fig. 2) In the above notation 1) supp μ Q is a curvilinear tree which is straightened out by the analytic mapping 2) supp μ Q contains all the zeros of Q k (z) and is contained in the convex hull of those.
3) There is a natural formula for the angles between the branches, and the masses of the branches satisfy Kirchhoff law.Below we show an example of such a measure in a proper scale and with all angles between its vertices marked, see Fig.   dz k ?Some partial results in this direction can be found in [12].

Degenerate exactly solvable operators
This subsection is based on [1].
Classical examples: leading to Laguerre resp.Hermite polynomials.

Proposition 2
The union of all roots of all polynomial eigenfunctions of an exactly solvable T is unbounded if and only if T is degenerate.
Problem 2 Given a degenerate T with the family of eigenpolynomials {p n (z)} how fast does the maximum r n of the modulus of roots of p n (z) grow?

Conjecture 1
Given a degenerate T = k j=1 Q j (z) d j dz j denote by j 0 the largest j for which where c T > 0 is a positive constant and Corollary 1 (of the latter Conjecture) The Cauchy transform C(z) of the asymptotic root measure μ of the scaled eigenpolynomial q n (z) = p n (n d z) of a degenerate T satisfies the following algebraic equation for almost all complex z: where A is the set consisting of all j for which the The latter equation for the Cauchy transform (if true) leads to very detailed information about the support of the asymptotic root-counting measure for the sequence of scaled eigenpolynomials.We illustrate this in Fig. 4.

Homogenized spectral problem for non-degenerate T
This section is based on [6].An observant reader has noticed that so far only the leading coefficient of an exactly solvable operator effected the asymptotic root-counting measure, which makes the situation somewhat unsatisfactory.
To make the whole symbol of an operator important we consider (following the classical pattern of e.g.W. Wasow, M. Fedoryuk) the homogenized spectral problem of the form where each Proposition 3 If T is of general type then 1) for all sufficiently large n there exist exactly k distinct values λ n,j , j = 1, . . ., k of the spectral parameter λ such that the operator T λ has a polynomial eigenfunction p n,j (z) of degree n.
2) Asymptotically λ n,j ∼ nλ j where λ 1 , . . ., λ k is the set of roots of the algebraic equation Conjecture 2 If T is of general type and all λ 1 , . . ., λ k have distinct arguments then for each j = 1, . . ., k ∃! probability measure μ j with compact support whose Cauchy transform C j (z) satisfies almost everywhere in outside the support of μ j which is the union of finitely many segments of analytic curves forming a curvilinear tree.
Observation.Near ∞ ∈ CP 1 the Cauchy transforms λ 1 C 1 (z), . . ., λ k C k (z) are independent sections of the symbol equation of T λ considered as a branched cover over CP 1 .
Problem 3 Find an explicit description of (the support) of the measures μ i .Is there any relation of these measures to the periods of the plane curve

Heine-Stieltjes theory
This section is based on [11].Take an arbitrary uni- has at least two distinct roots we call T a general Lame-type operator.Consider the following multi-parameter spectral problem.For a given non-negative integer n find all polynomials V (z) of degree at most r such that the equation has a polynomial solution p(z) of degree n. (Classically, p(z) is called a Stieltjes polynomial and V (z) is called a Van Vleck polynomial.) Proposition 4 Under the above assumptions for any sufficiently large n there exist exactly n + r r degree n Stieltjes polynomials p n,j (z) and corresponding Van Vleck polynomials V n,j (z).
Proposition 5 If a sequence { V n,jn (z)}, n = 1, . . ., of scaled Van Vleck polynomials converges to some polynomial V (z) then the sequence of finite measures μ n,j of the corresponding family of eigenpolynomials {p n,jn (z)} converges to a measure μ V satisfying the properties: a) supp μ V is a forest of curvilinear trees; b) the union of the leaves of supp μ V coincides with the union of all zeros of Q k (z) and those of V (z).c) supp μ V is straightened out by the transformation given by .
Explanations to Fig. 6 and 7.In Fig. 6 we give two examples of different Van Vleck polynomials V (z) and the corresponding Stieltjes polynomials p(z).
The average size dots are the 4 roots of the polynomial unique large dot is the only root of V (z) (which is linear in this case).Small dots show the roots of p(z).
In Fig. 7 we show the union of all roots of p(z) of degree 25 for the same problem.This section is based on [4].Consider a finite recurrence of length (k + 1) given by p n+1 (z) = Q 1 (z)p n (z) + . . .+ Q k (z)p n−k+1 (z), with polynomial or rational coefficients {Q 1 (z), . . ., Q k (z)} uniquely determined by the initial k-tuple {p 0 (z), . . ., p k (z)}.Theorem 6 There exists a finite subset Θ ⊂ C depending on the initial k-tuple and a curve Σ depending on the recurrence such that the asymptotic ratio exists and satisfies the symbol equation Here Σ is the so-called Stokes discriminant of ( * ) which is the set of all z for which the equation ( * ) has at most two roots with the same and maximal absolute value.

1 0 1 2Fig. 1 : 1 .
Fig. 1: Roots of p55(z) for the above T 's Explanations to Fig. 1.The larger dots show the roots of the corresponding Q(z) and the smaller dots are the fifty five roots of the corresponding p 55 (z).

Fig. 3 :
Fig. 3: Example of μQ with anglesProblem 1 Is it true that the support of the measure μ Q is a subset of the Stokes lines of the corresponding operator Q d k dz k ?Some partial results in this direction can be found in[12].

Fig. 4 :
Fig. 4: Examples of the root distributions of scaled eigenpolynomials to degenerate exactly solvable operators

Fig. 5 :Fig. 6 :
Fig.5: Three root-counting measures and their union for a homogenized spectral problem with an operator of order 3

Fig. 9 :
Fig. 9: Zeros of p31(z) satisfying the recurrence relation (z + 1)pn(z) = (z 2 + 1)pn−1(z) + (z − 5I)pn−2(z) + (z 3 − 1 − I)pn−3(z) Definition 2 Associate to each p n (x) a finite prob- n (x).(If some root is multiple we place at this point the mass equal to its multiplicity divided by n.)The limit μ = lim n μ n (if it exists in the sense of weak convergence) will be called the asymptotic root-counting measure of {p n (z)}.Definition 3 Consider the ratio q n