Lamé function

The Lamé Equation

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 (Gabriel Lamé 1837). 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.

Lamé's equation is

$\frac{d^2y}{dx^2} + (A+B\weierp(x))y = 0,$

where A and B are constants, and $\wp$ is the Weierstrass elliptic function. The most important case is when $B\weierp(x) = - \kappa^2 sn^2x$ and $\kappa^2 = n(n+1)k^2$ for an integer n and $k$ 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.

By changing the independent variable, Lamé's equation can also be rewritten in algebraic form as

$\frac{d^2y}{dt^2} +\frac{1}{2}\left(\frac{1}{t-e_1}+\frac{1}{t-e_2}+\frac{1}{t-e_3}\right)\frac{dy}{dt} - \frac{A+Bt}{4(t-e_1)(t-e_2)(t-e_3)}y = 0,$

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 which can be written

$\frac{d^2y}{dx^2} + (A - \kappa^2 sn^2x - \Omega^2k^4 sn^4x)y = 0,$

where $k$ is the elliptic modulus of the Jacobian elliptic functions and $\kappa$ and $\Omega$ are constants. For $\Omega = 0, k = 0, \kappa = 2h, A -2h^2 = \lambda, x= z \pm \frac{\pi}{2}$ the equation reduces to the Mathieu equation

$\frac{d^2y}{dz^2} + (\lambda - 2h^2\cos 2z)y = 0.$

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.^[1]

Asymptotic Expansions

Asymptotic expansions of ellipsoidal wave functions, and therewith also of Lamé functions, have been obtained by Müller.^[2]

References

1. H.J.W. Müller-Kirsten, Introduction to Quantum Mechanics: Schrödinger Equation and Path Integral, 2nd ed. World Scientific, 2012.
2. H.J.W. Müller, Asymptotic Expansions of Ellipsoidal Wave Functions and their Characteristic Numbers, Math. Nachr. 31 (1966) 89 - 101; Asymptotic Expansions of Ellipsoidal Wave Functions in Terms of Hermite Functions, Math. Nachr. 32 (1966) 49 - 62; On Asymptotic Expansions of Ellipsoidal Wave Functions, Math. Nachr. 32 (1966) 157 - 172.

• Arscott, F. M. (1964), Periodic Differential Equations, Oxford: Pergamon Press, pp. 191–236.
• Erdélyi, Arthur; Magnus, Wilhelm; Oberhettinger, Fritz; Tricomi, Francesco G. (1955), Higher transcendental functions (PDF), Bateman Manuscript Project, Vol. III, New York–Toronto–London: McGraw-Hill, pp. XVII + 292, MR 0066496, Zbl 0064.06302.
• Lamé, G. (1837), "Sur les surfaces isothermes dans les corps homogènes en équilibre de température", Journal de mathématiques pures et appliquées 2: 147–188. Available at Gallica.
• Rozov, N. Kh. (2001), "Lamé equation", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
• Rozov, N. Kh. (2001), "Lamé function", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
• Volkmer, H. (2010), "Lamé function", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W., NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0521192255, MR 2723248
• Müller-Kirsten, Harald J. W. (2012), Introduction to Quantum Mechanics: Schrödinger Equation and Path Integral, 2nd ed., World Scientific