ap-5-10.dvi Acta Polytechnica Vol. 50 No. 5/2010 Root Asymptotics for the Eigenfunctions of Univariate Differential Operators B. Shapiro Abstract 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. Keywords: root-counting measure, exactly solvable operator, Schrödinger equation. 1 Objective Study asymptotic properties of sequences {pn(z)}, of polynomials/entire functions in z which either 1. are polynomial/entire eigenfunctions of a uni- variate linear ordinary differential operatorwith polynomial coefficients; or 2. are polynomial solutions of more general pen- cils of such operators, e.g. homogenized spectral problems and Heine-Stieltjes spectral problems; or 3. satisfy a finite recurrence relation with (in gen- eral) varying coefficients. 2 Basic notions and examples Definition 1 An operator T = k∑ i=1 Qi(z) di dzi is called exactly solvable if degQi(z) ≤ i and there exists at least one value i such that degQi(z)= i. Obviously, T(zj)= aj z j +lower order terms, i.e. T acts by an (infinite) triangularmatrix in themono- mial basis {1, z, , z2, . . .} of C[z]. Lemma 1 For any exactly solvable T and suffi- ciently large n there exists a unique (up to a scalar) eigenpolynomial pn(z) of degree n. Typical problem. Given an exactly solvable T de- scribe the root asymptotics for the sequence of poly- nomials {pn(z)}. 2.1 Two asymptotic measures Given a polynomial family {pn(z)} where degpn(z) = n we define two basic measures: (i) asymptotic root-countingmeasure μ; (ii) asymptotic ratio measure ν. Definition 2 Associate to each pn(x) a finite prob- ability measure μn by placing the mass 1 n at every root of pn(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 μn (if it exists in the sense of weak convergence) will be called the asymptotic root-counting measure of {pn(z)}. Definition 3 Consider the ratio qn(z) = pn−1(z) pn(z) . (Assume for simplicity that pn(z) has no multiple roots and expand qn(z)= n∑ i=1 κi,n z − zi,n .) Associate to qn(z) the finite complex-valued measure by placing κi,n at zi,n. Define the asymptotic ratio measure of the sequence {pn(z)} as ν = lim n→∞ νn. Observation. Supports of μ and ν coincide but ν is often complex-valued. 2.2 Examples Below we show the root distribution for p55(z) for 4 different exactly solvable operators T1 = z(z − 1)(z − I) d3 dz3 ; T2 = (z − I)(z + I)(z − 2+3I)(z − 3 − 2I) d4 dz4 ; T3 = (z − I)(z + I)(z − 2+3I)(z − 3 − 2I) (z +3) d5 dz5 ; T4 = (z 2 +1)(z − 2+3I)(z − 3 − 2I)(z +3) (z +1+ I) d6 dz6 of the form Q(z) dk dzk where Q(z) is a monic polyno- mial of degree k +1. 77 Acta Polytechnica Vol. 50 No. 5/2010 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 2.5 3 -3 -2 -1 0 1 2 -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 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 p55(z). 2.3 Classical prototypes Theorem 1 (G. Szegö) If {pn(z)} is a family of polynomials orthogonal w.r.t a positive weight w(z) supported on [−1,1] such that ∫ 1 −1 lnw(z)dz < ∞ then the asymptotic root-counting measure has the density 1 π √ 1 − z2 , x ∈ [−1,1]. Theorem 2 (G. Szegö) If {pn(z)} is a family of polynomials orthogonal w.r.t a weight w(z) sup- ported on [−1,1] such that ∫ 1 −1 lnw(z)dz√ 1 − z2 > −∞ then the asymptotic ratio measure has the density 2 √ 1 − z2 π , z ∈ [−1,1]. 3 First results 3.1 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 Cρ(z)= ∫ C dρ(ξ) z − ξ . Example. If ρ(z) = 1 π √ 1 − z2 , z ∈ [−1,1] then Cμ = 1 √ z2 − 1 in C \ [−1,1] and Cν = 2 z + √ z2 − 1 in C \ [−1,1]. Definition 5 An exactly solvable operator T = k∑ i=1 Qi(z) di dzi is called non-degenerate if degQk(z)= k. Proposition 1 Assuming that Ψ(z) = lim n→∞ p′n(z) npn(z) exists in some open neighborhood Ω of C one gets that Ψ(z) satisfies in Ω the algebraic equation Qk(z)Ψ k(z)= 1. Theorem 3 (H. Rullg̊ard) Let Qk(z) be a monic degree k polynomial. Then there exists a unique prob- ability measure μQ such that 1) suppμQ is compact; 2) its Cauchy transform Cμ satisfies the equation Qk(z)C k μ(z)= 1 almost everywhere in C. 78 Acta Polytechnica Vol. 50 No. 5/2010 0 0.5 1 1.5 2 2.5 3 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 1.2 -1.5 -1 -0.5 0 0.5 1 Fig. 2: The measure μQ before and after the straightening transformation in the case Q(z)= (z −1)(z −3)(z − I) Theorem 4 (Main result, see Fig. 2) In the above notation 1) suppμQ is a curvilinear tree which is straight- ened out by the analytic mapping ξ(z)= ∫ z a dz k √ Qk(z) . 2) suppμQ contains all the zeros of Qk(z) and is contained in the convex hull of those. 3) There is a natural formula for the angles be- tween the branches, and the masses of the branches satisfy Kirchhoff law. Belowwe showan example of such ameasure in a proper scale and with all angles between its vertices marked, see Fig. 3. 0 0.5 1 1.5 -1.5 -1 -0.5 0 60 150 150 120 150 90 150 150 120 90 150 60 Fig. 3: Example of μQ with angles Problem 1 Is it true that the support of themeasure μQ is a subset of the Stokes lines of the corresponding operator Q dk dzk ? Somepartial results in this direction canbe found in [12]. 3.2 Degenerate exactly solvable operators This subsection is based on [1]. Definition 6 An exactly solvable T of order k is called degenerate iff degQk < k. Classical examples: T = z d2 dz2 + (az + b) d dz , T = d2 dz2 +(az + b) d dz leading to Laguerre resp. Hermite polynomials. Proposition 2 The union of all roots of all polyno- mial eigenfunctions of an exactly solvable T is un- bounded if and only if T is degenerate. Problem 2 Given a degenerate T with the family of eigenpolynomials {pn(z)} how fast does the maxi- mum rn of the modulus of roots of pn(z) grow? Conjecture 1 Given a degenerate T = k∑ j=1 Qj(z) dj dzj denote by j0 the largest j for which degQj(z)= j. Then lim n→∞ rn nd = cT where cT > 0 is a positive constant and d := max j∈[j0+1,k] ( j − j0 j − degQj ) . Corollary 1 (of the latter Conjecture) The Cauchy transform C(z) of the asymptotic root measure μ of the scaled eigenpolynomial qn(z) = pn(n dz) of a degenerate T satisfies the following al- gebraic equation for almost all complex z: zj0Cj0(z)+ ∑ j∈A αj,degQj z degQj Cj(z)= 1, where A is the set consisting of all j for which the maximum d := max j∈[j0+1,k] ( j − j0 j − degQj ) is attained, i.e. A = {j : (j − j0)/(j − degQj)= d}. 79 Acta Polytechnica Vol. 50 No. 5/2010 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 -1 -0.5 0 0.5 1 Fig. 4: Examples of the root distributions of scaled eigenpolynomials to degenerate exactly solvable operators The latter equation for the Cauchy transform (if true) leads to very detailed information about the support of the asymptotic root-countingmeasure for the sequence of scaled eigenpolynomials. We illus- trate this in Fig. 4. 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 asymp- totic root-counting measure, which makes the situa- tion somewhat unsatisfactory. To make the whole symbol of an operator impor- tant we consider (following the classical pattern of e.g. W. Wasow,M. Fedoryuk) the homogenized spec- tral problem of the form Tλ = k∑ i=0 Qi(z)λ k−i d i dzi , where each Qi(x)= aiiz i +ai,i−1z i−1+ . . . is a poly- nomial of degree i. Definition 7 A non-degenerate T is called of gen- eral type iff degQk(z)= k and k∑ i=0 aiiλ k−i =0 has k distinct zeros. Proposition 3 If T is of general type then 1) for all sufficiently large n there exist exactly k distinct values λn,j , j = 1, . . . , k of the spectral pa- rameter λ such that the operator Tλ has a polynomial eigenfunction pn,j(z) of degree n. 2) Asymptotically λn,j ∼ nλj where λ1, . . . , λk is the set of roots of the algebraic equation k∑ i=0 ai,ix k−i =0. Conjecture 2 If T is of general type and all λ1, . . . , λk have distinct arguments then for each j = 1, . . . , k ∃! probability measure μj with compact sup- port whose Cauchy transform Cj(z) satisfies almost everywhere in C k∑ i=1 Qi(z)(λj Cj(z)) i =0. Conjecture 3 Cj(z)= lim n→∞ p′n,j(z) λn,j pn,j(z) outside the support of μj which is the union of finitelymany seg- ments of analytic curves forming a curvilinear tree. Observation. Near ∞ ∈ CP1 the Cauchy transforms λ1C1(z), . . . , λkCk(z) are independent sections of the symbol equationof Tλ consideredas abranchedcover over CP1. Problem 3 Find an explicit description of (the sup- port) of the measures μi. Is there any relation of these measures to the periods of the plane curve k∑ i=1 Qi(z)y i =0? 5 Heine-Stieltjes theory This section is based on [11]. Take an arbitrary uni- variate linear differential operator T = k∑ i=0 Qi(z) di dzi with polynomial coefficients and set r =max i (degQi(z) − i). Definition 8 If r ≥ 0, degQk(z)= k+r and Qk(z) has at least two distinct roots we call T a general Lame-type operator. 80 Acta Polytechnica Vol. 50 No. 5/2010 0 1 2 3 4 5 6 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 -1 0 1 2 3 4 0 1 2 3 4 5 6 -1 0 1 2 3 4 Fig. 5: Three root-counting measures and their union for a homogenized spectral problem with an operator of order 3 0 0.5 1 1.5 2 2.5 3 -3 -2 -1 0 1 2 0 0.5 1 1.5 2 2.5 3 -3 -2 -1 0 1 2 Fig. 6: Examples of μQ’s for T =(z 2 +1)(z +2I −3)(z −3I − 2) d3 dz3 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 T(p(z))+ V (z)p(z)= 0, has a polynomial solution p(z) of degree n. (Classi- cally, 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 pn,j(z) and corre- sponding Van Vleck polynomials Vn,j(z). Proposition 5 If a sequence {Ṽn,jn(z)}, n =1, . . . , of scaled Van Vleck polynomials converges to some polynomial Ṽ (z) then the sequence of finite measures μn,j of the corresponding family of eigenpolynomials {pn,jn(z)} converges to a measure μṼ satisfying the properties: a) suppμ Ṽ is a forest of curvilinear trees; b) the union of the leaves of suppμ Ṽ coincides with the union of all zeros of Qk(z) and those of Ṽ (z). c) suppμ Ṽ is straightened out by the transforma- tion given by ∫ z a Ṽ (z)dz Qk(z) . 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 poly- nomial Q(z) = (z2 +1)(z +2I − 3)(z − 3I − 2), the 81 Acta Polytechnica Vol. 50 No. 5/2010 unique largedot is the only root of V (z) (which is lin- ear 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. 0 0.5 1 1.5 2 2.5 3 -3 -2 -1 0 1 2 Fig. 7: Union of μQ’s for the above T 6 Schrödinger operator with polynomial potential This section is based on [7, 8]. Consider the opera- tor H = − d2 dz2 + P(z) where P(z) = z2l + 2l−1∑ i=0 aiz i is a monic polynomial of even degree with real coef- ficients. It is well-known that the classical spectral problem H(y)= λy (1) where y belongs to L2(R) has a discrete and simple spectrum 0 < λ0 < λ1 < λ2 < . . . < λn < . . . Denote by φ0(z), φ1(z), . . . , φn(z), . . . the sequenceof the cor- responding eigenfunctions. These eigenfunctions are real entire functions of order l +1 and φn(z) has ex- actly n real zeros. Set ψn(z)= φn( 2l √ λnz) which we call the scaled n-th eigenfunction. TheStokes graph ofanycomplexpolynomial P(z) is the following object. Each root of P(z) is called a turning point. A (local) Stokes line of P(z) is amax- imal segment of the real analytic curve containing at most two turning points (finite or infinite) which solves the equation: � ξz0(z) = 0 where (2) ξz0(z) = ∫ z z0 √ P(u)du =0, with respect to z, where z0 is one of the turning points of P(z). The Stokes graph STP of the polyno- mial P(z) is the union of all its local Stokes curves. A local Stokes line connecting twofinite turning points, i.e. two roots of P(z) is called short. (The Stokes graph ST(P) of a generic P(z) has no short Stokes lines.) Proposition 6 For a given positive integer l the Stokes graph ST(z2l − 1) consists of 1) l short Stokes lines for l odd and l − 1 short Stokes lines for l even connecting all pairs of the roots of z2l − 1 which are symmetric w.r.t the imaginary axis; 2) for l odd each root of z2l − 1 is connected by 2 infinite Stokes lines to ∞. More exactly, the 2 infinite Stokes lines passing through the root e πik l , k =0, . . . ,2l − 1 are tangent at ∞ to the Stokes rays having the nearest slope to πik l ; 3) for l even each root of z2l − 1 except for ±i is connected to ∞ by 2 infinite Stokes lines with the same property as above. The roots ±i have 3 infinite Stokes lines each. Theorem 5 For any monic polynomial PC(z) of even degree the sequence of meromorphic func- tions {Cn(z)} = { ψ′n(z) nψn(z) } converges to C(z) = −Kl √ z2l − 1 uniformly on any compact set lying in the domain C \ U Cl, where Kl = √ πΓ ( 3l+1 2l ) Γ ( 2l+1 2l ) . (Here by − √ z2l − 1 we mean the branch which is negative for positive z > 1. Also U Cl is a certain subset of local Stokes lines marked by bold on Fig. 8.) �� �� �� Fig. 8: Stokes lines of z2l − 1 for l =1,2,3 82 Acta Polytechnica Vol. 50 No. 5/2010 7 Finite recurrences This section is based on [4]. Consider a finite recur- rence of length (k +1) given by pn+1(z)= Q1(z)pn(z)+ . . . + Qk(z)pn−k+1(z), with polynomial or rational coefficients {Q1(z), . . . , Qk(z)} uniquely determined by the ini- tial k-tuple {p0(z), . . . , pk(z)}. Theorem 6 There exists a finite subset Θ ⊂ C de- pending on the initial k-tuple and a curve Σ depend- ing on the recurrence such that the asymptotic ratio Ψ(z) = lim n→∞ pn+1(z) pn(z) exists and satisfies the symbol equation Ψk(z)= Q1(z)Ψ k−1(z)+ . . . + Qk(z) (∗) in C \ (Σ ∪ Θ). Here Σ is the so-called Stokes dis- criminant of (∗) which is the set of all z for which the equation (∗) has at most two roots with the same and maximal absolute value. -3 -2 -1 0 1 -3 -2 -1 0 1 2 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)+(z3 − 1 − I)pn−3(z) Acknowledgement I want to thank my coauthors T. Bergkvist, J. Borcea, R. Bøgvad, A. Eremenko, A. Gabrielov, G. Masson, H. Rullg̊ard for the pleasure of working with them and for the numerous insights and results we obtained together. I want to thank the organizers of the miniconference ‘Analytic and AlgebraicMeth- ods in Quantum Mechanics, V’ for the financial sup- port and a great pleasure of visiting Prague in May 2009, where these results were presented. References [1] Bergkvist, T.: On asymptotics of polynomial eigenfunctions for exactly-solvable differential operators, J. Approx. Theory 149(2), (2007), 151–187. [2] Berqkvist, T., Rullg̊ard, H.: On polynomial eigenfunctions for a class of differential opera- tors, Math. Res. Lett., 9 (2002), 153–171. [3] Berqkvist, T., Rullg̊ard, H., Shapiro, B.: On Bochner-Krall orthogonal polynomial systems, Math. Scand., 94 (2004), 148–154. [4] Borcea, J., Bøgvad, R., Shapiro, B.: On ratio- nal approximation of algebraic functions, Adv. Math. 204 (2006), 448–480. [5] Borcea, J., Shapiro, B.: Root asymptotics of spectral polynomials for the Lamé operator, Comm. Math. Phys, 282 (2008), 323–337. [6] Borcea, J., Bøgvad, R., Shapiro, B.: Homoge- nized spectral pencils for exactly solvable oper- ators: asymptoticsofpolynomial eigenfunctions, Publ. RIMS, 45 (2009), 525–568. [7] Gabrielov, A., Eremenko, A., Shapiro, B.: Ze- ros of eigenfunctions of someanharmonicoscilla- tors,Annales de l’InstitutFourier,58(2) (2008), 603–624. [8] Gabrielov, A., Eremenko, A., Shapiro, B.: High energy eigenfunctions of one-dimensional Schrödinger operators with polynomial poten- tials, Comput. Methods Funct. Theory, 8(2) (2008), 513–529. [9] Holst, T., Shapiro,B.: OnhigherHeine-Stieltjes polynomials, to appear in Isr. J. Math. [10] Masson, G., Shapiro, B.: On polynomial eigen- functionsofahypergeometric-typeoperator,Ex- per. Math., 10 (2001), 609–618. [11] Shapiro,B.: Algebro-geometric aspects ofHeine- Stieltjes theory, submitted. [12] Shapiro, B., Takemura, K., Tater, M.: On spec- tral polynomials of the Heun equation. II, sub- mitted. Prof. Boris Shapiro E-mail: shapiro@math.su.se Department of Mathematics Stockholm University SE-106 91, Stockholm, Sweden 83