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On higher heine-stieltjes polynomials
Stockholm University, Faculty of Science, Department of Mathematics.
Stockholm University, Faculty of Science, Department of Mathematics.
2011 (English)In: Israel Journal of Mathematics, ISSN 0021-2172, E-ISSN 1565-8511, Vol. 183, no 1, p. 321-345Article in journal (Refereed) Published
##### Abstract [en]

Take a linear ordinary differential operator d(z) = Pk i=1 Qi(z) di dzi with polynomial coefficients and set r = maxi=1,...,k(degQi(z) − i). If d(z) satisfies the conditions: i) r   0 and ii) degQk(z) = k + r we call it a non-degenerate higher Lam´e 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 non-degeneracy assumptions on T there exist exactly `n+r n ´ spectral polynomials 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 ! 1 sequences of normalized spectral polynomials {eVn,in (z)} where eVn,in (z) is the monic polynomial proportional to Vn,in(z). Below we calculate for any such converging sequence {eVn,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 {eVn,i(z)} having eigenpolynomials S(z) of degree n converges as n ! 1to the standard measure in the space of monic polynomials of degree r which depends only on the leading coefficient Qk(z).

##### Place, publisher, year, edition, pages
2011. Vol. 183, no 1, p. 321-345
##### Keyword [en]
Strebel differential, Heine-Stieltjes polynomials
##### National Category
Mathematical Analysis
Mathematics
##### Identifiers
ISI: 000291804000011OAI: oai:DiVA.org:su-49776DiVA, id: diva2:379423
Available from: 2010-12-31 Created: 2010-12-17 Last updated: 2017-12-11Bibliographically approved

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Shapiro, BorisHolst, Thomas
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