On polynomial eigenfunctions for linear ordinary differential
operators with polynomial coefficients and related problems

KTH/Nordita/SU seminar in Theoretical Physics [before December 2013]

Wednesday 24 March 2010
from 11:00
to 12:00 at
FA31

Speaker :

B.Shapiro (Matematiska Inst, SU)

Abstract :

I present some new results on the asymptotic root
distribution of generalized polynomial eigenfunctions for a large
class of univariate linear differential operators. For example, an
operator T=Q_k(x)D^k+...+ Q_1(x)D, where D=d/dx and Q_i(x) are
polynomials is called exactly solvable if for almost any polynomial p
(x) the degree of T(p(x)) equals the degree of p(x). One can easily
show that for any such T there exists and unique ( up to a scalar
factor) eigenpolynomial p_n(x) of degree n for all sufficiently large
n. I will describe, in particular, the asymptotical
root distribution of p_n(x) when n->oo illustrated on the attached
picture.