### 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 language | English (US) |
---|---|

Pages (from-to) | 1115-1153 |

Number of pages | 39 |

Journal | Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences |

Volume | 366 |

Issue number | 1867 |

DOIs | |

State | Published - Mar 28 2008 |

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### Keywords

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

### ASJC Scopus subject areas

- General

### Cite this

**Lamé polynomials, hyperelliptic reductions and Lamé band structure.** / Maier, Robert S.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - Lamé polynomials, hyperelliptic reductions and Lamé band structure

AU - Maier, Robert S

PY - 2008/3/28

Y1 - 2008/3/28

N2 - 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.

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.

KW - Band structure

KW - Dispersion relation

KW - Hermite-Krichever Ansatz

KW - Hyperelliptic reduction

KW - Lamé equation

KW - Lamé polynomial

UR - http://www.scopus.com/inward/record.url?scp=38849138020&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=38849138020&partnerID=8YFLogxK

U2 - 10.1098/rsta.2007.2063

DO - 10.1098/rsta.2007.2063

M3 - Article

C2 - 17588866

AN - SCOPUS:38849138020

VL - 366

SP - 1115

EP - 1153

JO - Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences

JF - Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences

SN - 0962-8428

IS - 1867

ER -