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On Asymptotics of Polynomial Eigenfunctions for Exactly-Solvable Differential Operators
Stockholm University, Faculty of Science, Department of Mathematics.
2007 (English)In: Journal of Approximation Theory, ISSN 0021-9045, E-ISSN 1096-0430, Vol. 149, no 2, 151-187 p.Article in journal (Refereed) Published
Abstract [en]

In this paper we study the class of differential operators with polynomial coefficients Qj in one complex variable satisfying the condition degQjj with equality for at least one j. We show that if degQk<k then the root with the largest modulus of the nth degree eigenpolynomial pn of T tends to infinity when n→∞, as opposed to the case when degQk=k, which we have treated previously in [T. Bergkvist, H. RullgÅrd, On polynomial eigenfunctions for a class of differential operators, Math. Res. Lett. 9 (2002) 153–171]. Moreover, we present an explicit conjecture and partial results on the growth of the largest modulus of the roots of pn. Based on this conjecture we deduce the algebraic equation satisfied by the Cauchy transform of the asymptotic root measure of the appropriately scaled eigenpolynomials, for which the union of all roots is conjecturally contained in a compact set.

Place, publisher, year, edition, pages
2007. Vol. 149, no 2, 151-187 p.
Keyword [en]
Exactly-solvable operators, Asymptotic zero distribution, Root growth, Polynomial eigenfunctions, Eigenpolynomials
National Category
Mathematics
Identifiers
URN: urn:nbn:se:su:diva-24129DOI: 10.1016/j.jat.2007.04.010ISI: 000252160800004OAI: oai:DiVA.org:su-24129DiVA: diva2:196823
Available from: 2007-03-09 Created: 2007-02-22 Last updated: 2011-02-16Bibliographically approved
In thesis
1. Asymptotics of Eigenpolynomials of Exactly-Solvable Operators
Open this publication in new window or tab >>Asymptotics of Eigenpolynomials of Exactly-Solvable Operators
2007 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

The main topic of this doctoral thesis is asymptotic properties of zeros in polynomial families arising as eigenfunctions to exactly-solvable differential operators. The study was initially inspired by a number of striking results from computer experiments performed by G. Masson and B. Shapiro for a more restrictive class of operators. Our research is also motivated by a classical question going back to S. Bochner on a general classification of differential operators possessing an infinite sequence of orthogonal eigenpolynomials. In general however, the sequence of eigenpolynomials of an exactly-solvable operator is not an orthogonal system and it can therefore not be studied by means of the extensive theory known for such systems. Our study can thus be considered as the first steps to a natural generalization of the asymptotic behaviour of the roots of classical orthogonal polynomials. Exactly-solvable operators split into two major classes: non-degenerate and degenerate. We prove that in the former case, as the degree tends to infinity, the zeros of the eigenpolynomial are distributed according to a certain probability measure which is compactly supported on a tree and which depends only on the leading term of the operator. Computer experiments indicate the existence of a limiting root measure in the degenerate case too, but that it is compactly supported (conjecturally on a tree) only after an appropriate scaling which is conjectured (and partially proved) in this thesis. One of the main technical tools in this thesis is the Cauchy transform of a probability measure, which in the considered situation satisfies an algebraic equation. Due to the connection between the asymptotic root measure and its Cauchy transform it is therefore possible to obtain detailed information on the limiting zero distribution.

Place, publisher, year, edition, pages
Stockholm: Matematiska institutionen, 2007. 5 p.
Keyword
exactly-solvable, eigenpolynomials, zero distribution, asymptotics
National Category
Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:su:diva-6657 (URN)91-7155-380-0 (ISBN)
Public defence
2007-03-30, sal 14, hus 5, Kräftriket, Stockholm, 13:00
Opponent
Supervisors
Available from: 2007-03-09 Created: 2007-02-22Bibliographically approved

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