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Mathematical Physics

arXiv:math-ph/0309005 (math-ph)
[Submitted on 1 Sep 2003 (v1), last revised 17 Jul 2004 (this version, v4)]

Title:Lamé polynomials, hyperelliptic reductions and Lamé band structure

Authors:Robert S. Maier (University of Arizona)
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Abstract: The band structure of the Lamé equation, viewed as a one-dimensional Schrödinger equation with a periodic potential, is studied. At integer values of the degree parameter l, the dispersion relation is reduced to the l=1 dispersion relation, and a previously published l=2 dispersion relation is shown to be partially incorrect. The Hermite-Krichever Ansatz, which expresses Lamé equation solutions in terms of l=1 solutions, is the chief tool. It is based on a projection from a genus-l hyperelliptic curve, which parametrizes solutions, to an elliptic curve. A general formula for this covering is derived, and is used to reduce certain hyperelliptic integrals to elliptic ones. Degeneracies between band edges, which can occur if the Lamé equation parameters take complex values, are investigated. If the Lamé equation is viewed as a differential equation on an elliptic curve, a formula is conjectured for the number of points in elliptic moduli space (elliptic curve parameter space) at which degeneracies occur. Tables of spectral polynomials and Lamé polynomials, i.e., band edge solutions, are given. A table in the older literature is corrected.
Comments: 38 pages, 1 figure; final revisions
Subjects: Mathematical Physics (math-ph); Classical Analysis and ODEs (math.CA); Exactly Solvable and Integrable Systems (nlin.SI)
MSC classes: 33E10 (Primary); 34L40, 14K25 (Secondary)
Cite as: arXiv:math-ph/0309005
  (or arXiv:math-ph/0309005v4 for this version)
  https://doi.org/10.48550/arXiv.math-ph/0309005
Journal reference: Philos. Trans. Roy. Soc. London Ser. A 366 (2008), 1115-1153
Related DOI: https://doi.org/10.1098/rsta.2007.2063

Submission history

From: Robert Maier [view email]
[v1] Mon, 1 Sep 2003 23:28:45 UTC (60 KB)
[v2] Thu, 17 Jun 2004 00:42:47 UTC (66 KB)
[v3] Sun, 27 Jun 2004 01:28:52 UTC (67 KB)
[v4] Sat, 17 Jul 2004 01:57:03 UTC (67 KB)
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