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Root asymptotics of spectral polynomials for the Lame operator
Stockholm University, Faculty of Science, Department of Mathematics.
Stockholm University, Faculty of Science, Department of Mathematics.
2008 (English)In: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, Vol. 282, no 2, 323-337 p.Article in journal (Refereed) Published
Abstract [en]

The study of polynomial solutions to the classical Lamé 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 Schrödinger equation with finite gap potential given by the Weierstrass }-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.

Place, publisher, year, edition, pages
2008. Vol. 282, no 2, 323-337 p.
Keyword [en]
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URN: urn:nbn:se:su:diva-21677DOI: 10.1007/s00220-008-0551-0ISI: 000257647600002OAI: diva2:188204
Available from: 2010-12-31 Created: 2008-11-11 Last updated: 2011-06-20Bibliographically approved

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