ON SPECTRAL POLYNOMIALS OF THE HEUN EQUATION. II.
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
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  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.
Heun equation, Stokes lines
Research subject Mathematics
IdentifiersURN: urn:nbn:se:su:diva-49774DOI: 10.1007/s00220-012-1466-3ISI: 000302243700001OAI: oai:DiVA.org:su-49774DiVA: diva2:379415