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Stockholm University, Faculty of Science, Department of Mathematics.
Department of Theoretical Physics, Nuclear Physics Institute, Academy of Sci- ences, 250 68 ˇ Reˇz near Prague, Czech Republic.
Department of Mathematical Sciences, Yokohama City University, 22-2 Seto, Kanazawa- ku, Yokohama 236-0027, Japan. .
2012 (English)In: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, Vol. 311, no 2, 277-300 p.Article in journal (Refereed) Published
Abstract [en]

The well-known Heun equation has the form



dz2 + P(z)



+ V (z)ffS(z) = 0,

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 19-th 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 ! 1 to a certain compact

connecting the three roots of Q(z) which is given by a condition that a certain

natural abelian integral is real-valued, see Theorem 2.


Place, publisher, year, edition, pages
2012. Vol. 311, no 2, 277-300 p.
Keyword [en]
Heun equation, Stokes lines
National Category
Mathematical Analysis
Research subject
URN: urn:nbn:se:su:diva-49774DOI: 10.1007/s00220-012-1466-3ISI: 000302243700001OAI: diva2:379415
3Available from: 2010-12-31 Created: 2010-12-17 Last updated: 2012-06-28Bibliographically approved

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