In mathematics, a Lamé function, or ellipsoidal harmonic function, is a solution of Lamé's equation, a second-order ordinary differential equation. It was introduced in the paper . Lamé's equation appears in the method of separation of variables applied to the Laplace equation in elliptic coordinates. In some special cases solutions can be expressed in terms of polynomials called Lamé polynomials.

The Lamé equation

Lamé's equation is

where A and B is the Weierstrass elliptic function, where \operatorname{sn} is the elliptic sine function for an integer n the elliptic modulus, in which case the solutions extend to meromorphic functions defined on the whole complex plane. For other values of B the solutions have branch points.

, Lamé's equation can also be rewritten in algebraic form as

which after a change of variable becomes a special case of Heun's equation.

A more general form of Lamé's equation is the ellipsoidal equation or ellipsoidal wave equation as above)

the equation reduces to the Mathieu equation

The Weierstrassian form of Lamé's equation is quite unsuitable for calculation (as Arscott also remarks, p.191). The most suitable form of the equation is that in Jacobian form, as above. The algebraic and trigonometric forms are also cumbersome to use. Lamé equations arise in quantum mechanics as equations of small fluctuations about classical solutions—called periodic instantons, bounces or bubbles—of Schrödinger equations for various periodic and anharmonic potentials.

Asymptotic expansions

have been obtained by Müller. The asymptotic expansion approximately an odd integer (and to be determined more precisely by boundary conditions – see below),

(another (fifth) term not given here has been calculated by Müller, the first three terms have also been obtained by Ince (as in the corresponding calculations for Mathieu functions, and oblate spheroidal wave functions and prolate spheroidal wave functions is the quarter period given by a complete elliptic integral)

as well as (the prime meaning derivative)

defining respectively the ellipsoidal wave functions

one obtains

one obtains

In the limit of the Mathieu equation (to which the Lamé equation can be reduced) these expressions reduce to the corresponding expressions of the Mathieu case (as shown by Müller).

Floquet theory

The Floquet theory of the Lamé equation describes solutions that are quasiperiodic with respect to a real period. is the complete elliptic integral of the first kind.

The equation may be written in Schrödinger form is an elliptic function.

, called the Floquet multiplier..

is the trace of the monodromy matrix.

-line for which the solutions stay bounded. Then the equation is described as finite-gap. In this case the two-fold covering becomes algebraic, and the spectral curve is a hyperelliptic curve, called the Lamé spectral polynomial.

Notes

References

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• . Available at Gallica.