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ALGEBROGEOMETRIC ASPECTS OF HEINESTIELTJES THEORY
, 2008
"... The goal of the paper is to develop a HeineStieltjes theory for univariate linear differential operators of higher order. Namely, for a given linear ordinary differential operator d(z) = Pk di i=1 Qi(z) dzi with polynomial coefficients set r = maxi=1,...,k(deg Qi(z) − i). If d(z) satisfies the co ..."
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Cited by 15 (2 self)
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The goal of the paper is to develop a HeineStieltjes theory for univariate linear differential operators of higher order. Namely, for a given linear ordinary differential operator d(z) = Pk di i=1 Qi(z) dzi with polynomial coefficients set r = maxi=1,...,k(deg Qi(z) − i). If d(z) satisfies the conditions: i) r ≥ 0 and ii) deg Qk(z) = k + r we call it a nondegenerate higher Lamé operator. Following the classical approach of E. Heine and T. Stieltjes, see [18], [41] we study the multiparameter spectral problem of finding all polynomials V (z) of degree at most r such that the equation: d(z)S(z) + V (z)S(z) = 0 has for a given positive integer n a polynomial solution S(z) of degree n. We show that under some mild nondegeneracy assumptions there exist exactly n+r ´ such polynomials Vn,i(z) whose corresponding eigenpolynomials Sn,i(z)
ON HIGHER HEINESTIELTJES POLYNOMIALS
"... Abstract. Take a linear ordinary differential operator d(z) = P k i=1 Qi(z) di dz i with polynomial coefficients and set r = maxi=1,...,k(deg Qi(z) − i). If d(z) satisfies the conditions: i) r ≥ 0 and ii) deg Qk(z) = k + r we call it a nondegenerate higher Lamé operator. Following the classical e ..."
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Cited by 6 (2 self)
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Abstract. Take a linear ordinary differential operator d(z) = P k i=1 Qi(z) di dz i with polynomial coefficients and set r = maxi=1,...,k(deg Qi(z) − i). If d(z) satisfies the conditions: i) r ≥ 0 and ii) deg Qk(z) = k + r we call it a nondegenerate higher Lamé operator. Following the classical examples of E. Heine and T. Stieltjes we initiated in [6] the study of the following multiparameter spectral problem: for each positive integer n find polynomials V (z) of degree at most r such that the equation: d(z)S(z) + V (z)S(z) = 0 has a polynomial solution S(z) of degree n. We have shown that under some mild nondegeneracy assumptions on T there exist exactly `n+r ´ spectral polyn nomials Vn,i(z) of degree r and their corresponding eigenpolynomials Sn,i(z) of degree n. Localization results of [6] provide the existence of abundance of converging as n → ∞ sequences of normalized spectral polynomials { e Vn,in (z)} where e Vn,in (z) is the monic polynomial proportional to Vn,in (z). Below we calculate for any such converging sequence { e Vn,in (z)} the asymptotic rootcounting measure of the corresponding family {Sn,in(z)} of eigenpolynomials. We also conjecture that the sequence of sets of all normalized spectral polynomials { e Vn,i(z)} having eigenpolynomials S(z) of degree n converges as n → ∞ to the standard measure in the space of monic polynomials of degree r which depends only on the leading coefficient Qk(z). 1.
ON SPECTRAL POLYNOMIALS OF THE HEUN EQUATION. II.
, 2009
"... The wellknown Heun equation has the form j Q(z) d2 ff d + P(z) + V (z) S(z) = 0, dz2 dz where Q(z) is a cubic complex polynomial, P(z) and V (z) are polynomials of degree at most 2 and 1 respectively. One of the classical problems about the Heun equation suggested by E. Heine and T. Stieltjes in t ..."
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The wellknown Heun equation has the form j Q(z) d2 ff d + P(z) + V (z) S(z) = 0, dz2 dz where Q(z) is a cubic complex polynomial, P(z) and V (z) are polynomials of degree at most 2 and 1 respectively. One of the classical problems about the Heun equation suggested by E. Heine and T. Stieltjes in the late 19th century is for a given positive integer n to find all possible polynomials V (z) such that the above equation has a polynomial solution S(z) of degree n. Below we prove a conjecture of the second author, see [17] claiming that the union of the roots of such V (z)’s for a given n tends when n → ∞ to a certain compact connecting the three roots of Q(z) which is given by a condition that a certain
ROOT ASYMPTOTICS FOR THE EIGENFUNCTIONS OF UNIVARIATE DIFFERENTIAL OPERATORS
"... Abstract. The present paper is a short survey of the research conducted by the present author and his coauthors in the field of root asymptotics of (mostly polynomial) eigenfunctions of linear univariate differential operators with polynomial coefficients. 1. Objective Study asymptotic properties of ..."
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Abstract. The present paper is a short survey of the research conducted by the present author and his coauthors in the field of root asymptotics of (mostly polynomial) eigenfunctions of linear univariate differential operators with polynomial coefficients. 1. Objective Study asymptotic properties of functional sequences {pn(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 HeineStieltjes spectral problems; or (3) satisfy a finite recurrence relation with (in general) varying coefficients. 2. Basic notions and examples Definition 1. An operator T = ∑k di i=1 Qi(z) dzi is called exactly solvable if deg Qi(z) ≤ i and there exists at least one value i such that deg Qi(z) = i. Obviously, T(z j) = ajz j + lower order terms, i.e. T acts by an (infinite) triangular matrix in the monomial basis {1, z, , z 2,....} of C[z]. Lemma 1. For any exactly solvable T and sufficiently large n there exists and unique (up to a scalar) eigenpolynomial pn(z) of degree n. Typical problem. Given an exactly solvable T describe the root asymptotics for the sequence of polynomials {pn(z)}. 2.1. Two asymptotic measures. Given a polynomial family {pn(z)} where deg pn(z) = n we define two basic measures: (i) asymptotic rootcounting measure µ; (ii) asymptotic ratio measure ν. Definition 2. Associate to each pn(x) a finite probability 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 = limn µn (if it exists in the sense of the weak convergence) will be called the asymptotic rootcounting measure of {pn(z)}.
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"... Abstract: We define twistorial topological strings by considering tt ∗ geometry of the 4d N = 2 supersymmetric theories on the NekrasovShatashvili 1 2 Ω background, which leads to quantization of the associated hyperKähler geometries. We show that in one limit it reduces to the refined topological ..."
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Abstract: We define twistorial topological strings by considering tt ∗ geometry of the 4d N = 2 supersymmetric theories on the NekrasovShatashvili 1 2 Ω background, which leads to quantization of the associated hyperKähler geometries. We show that in one limit it reduces to the refined topological string amplitude. In another limit it is a solution to a quantum RiemannHilbert problem involving quantum KontsevichSoibelman operators. In a further limit it encodes the hyperKähler integrable systems studied by GMN. In the context of AGT conjecture, this perspective leads to a twistorial extension of Toda. The 2d index of the 1 2 Ω theory leads to the recently introduced index for N = 2 theories in 4d. The twistorial topological string can alternatively be viewed, using the work of NekrasovWitten, as studying the vacuum geometry of 4d N = 2 supersymmetric theories on T 2 × I where I is an interval with specific boundary conditions at the two ends. ar
AATLDE, Boris Shapiro, B2
"... Aspects of the asymptotic theory of linear ordinary differential equations ..."
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Aspects of the asymptotic theory of linear ordinary differential equations
Polynomial Solutions of the Heun Equation
, 2011
"... We review properties of certain types of polynomial solutions of the Heun equation. Two aspects are particularly concerned, the interlacing property of spectral and Stieltjes polynomials in the case of real roots of these polynomials and asymptotic root distribution when complex roots are present. ..."
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We review properties of certain types of polynomial solutions of the Heun equation. Two aspects are particularly concerned, the interlacing property of spectral and Stieltjes polynomials in the case of real roots of these polynomials and asymptotic root distribution when complex roots are present.
SCURVES IN POLYNOMIAL EXTERNAL FIELDS
"... Abstract. Curves in the complex plane that satisfy the Sproperty were first introduced by Stahl and they were further studied by Gonchar and Rakhmanov in the 1980s. Rakhmanov recently showed the existence of curves with the Sproperty in a harmonic external field by means of a maxmin variational p ..."
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Abstract. Curves in the complex plane that satisfy the Sproperty were first introduced by Stahl and they were further studied by Gonchar and Rakhmanov in the 1980s. Rakhmanov recently showed the existence of curves with the Sproperty in a harmonic external field by means of a maxmin variational problem in logarithmic potential theory. This is done in a fairly general setting, which however does not include the important special case of an external field ReV where V is a polynomial of degree ≥ 2. In this paper we give a detailed proof of the existence of a curve with the Sproperty in the external field ReV within the collection of all curves that connect two or more preassigned directions at infinity in which ReV → +∞. Our method of proof is very much based on the works of Rakhmanov on the maxmin variational problem and of Mart́ınezFinkelshtein and Rakhmanov on critical measures. Contents