Nodal statistics of Heine-Stieltjes and Van Vleck polynomials

• Main Author: Kritchevski, Evgenij
• Format: Others
• Language: en
• Published: McGill University 2004
• Subjects: Mathematics.
• Online Access: http://digitool.Library.McGill.CA:80/R/?func=dbin-jump-full&object_id=81348

We study the polynomial solutions of the Lame differential equation Azy'' z+2Bz y'z+C zyz=0 where A(z), B( z), C(z) ∈ C [z] are polynomials of degree N + 1, N and N - 1 respectively. We review classical results concerning the location of the zeros of y(z) and C(z) and their electrostatic interpretation. Asymptotic distribution of the zeros as the degree K of y(z) approaches infinity is then discussed. We also derive numerical methods that allow us to compute solutions of high degree K and present a variety of new fine experimental results such as the asymptotic nearest neighbor spacing distribution and the description of complex configurations.