Lamé polynomials, hyperelliptic reductions and Lamé band structure

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21 Citations (Scopus)

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 partly 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 earlier literature is corrected.

Original languageEnglish (US)
Pages (from-to)1115-1153
Number of pages39
JournalPhilosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences
Volume366
Issue number1867
DOIs
StatePublished - Mar 28 2008

Fingerprint

Band Structure
Band structure
polynomials
Polynomials
Dispersion Relation
Polynomial
Elliptic Curves
curves
Space Curve
Hyperelliptic Curves
Periodic Potential
Hermite
Differential equations
Moduli Space
integers
Tables
Parameter Space
Table
Genus
coverings

Keywords

  • Band structure
  • Dispersion relation
  • Hermite-Krichever Ansatz
  • Hyperelliptic reduction
  • Lamé equation
  • Lamé polynomial

ASJC Scopus subject areas

  • General

Cite this

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abstract = "The band structure of the Lam{\'e} equation, viewed as a one-dimensional Schr{\"o}dinger 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 partly incorrect. The Hermite-Krichever Ansatz, which expresses Lam{\'e} 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 Lam{\'e} equation parameters take complex values, are investigated. If the Lam{\'e} 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 Lam{\'e} polynomials, i.e. band-edge solutions, are given. A table in the earlier literature is corrected.",
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AB - 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 partly 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 earlier literature is corrected.

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