Root asymptotics of spectral polynomials for the Lame operator

Details Root asymptotics of spectral polynomials for the Lame operator Root asymptotics of spectral polynomials for the Lame operator

2007-01-30

arxiv.org/abs/math/0701883v2

Commun. Math. Phys. 282 (2008), 323-337

Mathematics

Classical Analysis and ODEs

final version, to appear in Commun. Math. Phys.; 13 pages, 3 figures, LaTeX2e

Scientific paper

10.1007/s00220-008-0551-0

The study of polynomial solutions to the classical Lam\'e equation in its algebraic form, or equivalently, of double-periodic solutions of its Weierstrass form has a long history. Such solutions appear at integer values of the spectral parameter and their respective eigenvalues serve as the ends of bands in the boundary value problem for the corresponding Schr\"odinger equation with finite gap potential given by the Weierstrass $\wp$-function on the real line. In this paper we establish several natural (and equivalent) formulas in terms of hypergeometric and elliptic type integrals for the density of the appropriately scaled asymptotic distribution of these eigenvalues when the integer-valued spectral parameter tends to infinity. We also show that this density satisfies a Heun differential equation with four singularities.

Affiliated with

Also associated with

No associations

Rating

Root asymptotics of spectral polynomials for the Lame operator does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Root asymptotics of spectral polynomials for the Lame operator, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Root asymptotics of spectral polynomials for the Lame operator will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-470527