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On Roots of Eigenpolynomials for Degenerate Exactly-Solvable Differential Operators
Stockholm University, Faculty of Science, Department of Mathematics.
Article in journal (Refereed) Submitted
URN: urn:nbn:se:su:diva-24130OAI: diva2:196824
Part of urn:nbn:se:su:diva-6657Available from: 2007-03-09 Created: 2007-02-22Bibliographically 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.
exactly-solvable, eigenpolynomials, zero distribution, asymptotics
National Category
Research subject
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
Available from: 2007-03-09 Created: 2007-02-22Bibliographically approved

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