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.
The Lamé equation
Lamé's equation is
where A and B are constants, and is the Weierstrass elliptic function. The most important case is when , where is the elliptic sine function, and for an integer n and 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 to with , 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 which can be written (observe we now write , not as above)
where is the elliptic modulus of the Jacobian elliptic functions and and are constants. For the equation becomes the Lamé equation with . For 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 of periodic ellipsoidal wave functions, and therewith also of Lamé functions, for large values of have been obtained by Müller. The asymptotic expansion obtained by him for the eigenvalues is, with 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). Observe terms are alternately even and odd in and (as in the corresponding calculations for Mathieu functions, and oblate spheroidal wave functions and prolate spheroidal wave functions). With the following boundary conditions (in which is the quarter period given by a complete elliptic integral)
as well as (the prime meaning derivative)
defining respectively the ellipsoidal wave functions
of periods and for one obtains
Here the upper sign refers to the solutions and the lower to the solutions . Finally expanding about 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).
- H. J. W. Müller-Kirsten, Introduction to Quantum Mechanics: Schrödinger Equation and Path Integral, 2nd ed. World Scientific, 2012, ISBN 978-981-4397-73-5; J.-Q. Liang, H. J. W. Müller-Kirsten and D. H. Tchrakian, Phys. Lett. B 282 (1992) 105.
- 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.
- E. L. Ince, Proc. Roy. Soc. Edin. A60 (1939) 47, 83.
- 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 (ed.), Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
- Rozov, N. Kh. (2001) , "Lamé function", in Hazewinkel, Michiel (ed.), Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, 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. (eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, MR 2723248
- Müller-Kirsten, Harald J. W. (2012), Introduction to Quantum Mechanics: Schrödinger Equation and Path Integral, 2nd ed., World Scientific