Lamé function

The Lamé equation

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 \operatorname{sn}^2x , where \operatorname{sn} is the elliptic sine function, 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 to t with t=\operatorname{sn} x, 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 (observe we now write \Lambda, not A as above)

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

where k is the elliptic modulus of the Jacobian elliptic functions and \kappa and \Omega are constants. For \Omega = 0 the equation becomes the Lamé equation with \Lambda = A. For \Omega = 0, k = 0, \kappa = 2h, \Lambda -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.

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

Asymptotic expansions of periodic ellipsoidal wave functions, and therewith also of Lamé functions, for large values of \kappa have been obtained by Müller. The asymptotic expansion obtained by him for the eigenvalues \Lambda is, with q approximately an odd integer (and to be determined more precisely by boundary conditions – see below),

: \begin{align} \Lambda(q) = {} & q\kappa - \frac{1}{2^3}(1+k^2)(q^2+1) - \frac{q}{2^6\kappa}{(1+k^2)^2(q^2+3)\[6pt] &- 4k^2(q^2+5)} {} -\frac{1}{2^{10}\kappa^2} \Big{(1+k^2)^3(5q^4+34q^2+9)\ & - 4k^2(1+k^2)(5q^4+34q^2+9) {} - 384\Omega^2k^4(q^2+1)\Big} - \cdots , \end{align}

(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 q and \kappa (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 K(k) is the quarter period given by a complete elliptic integral)

: \operatorname{Ec}(2K) = \operatorname{Ec}(0) = 0,;; \operatorname{Es}(2K) = \operatorname{Es}(0) = 0,

as well as (the prime meaning derivative)

: (\operatorname{Ec})^'_{2K} = (\operatorname{Ec})^'0 = 0, ;; (\operatorname{Es})^'{2K} = (\operatorname{Es})^'_0 = 0,

defining respectively the ellipsoidal wave functions

: \operatorname{Ec}^{q_0}_n, \operatorname{Es}^{q_0+1}_n, \operatorname{Ec}^{q_0-1}_n, \operatorname{Es}^{q_0}_n

of periods 4K, 2K, 2K, 4K, and for q_0=1,3,5, \ldots one obtains

: \begin{align} && q-q_0 = \mp 2\sqrt{\frac{2}{\pi}} \left( \frac{1+k}{1-k}\right)^{-\kappa/k} \left(\frac{8\kappa} {1-k^2}\right)^{q_0/2}\frac{1}{[(q_0-1)/2]!} \times \qquad\qquad\qquad \[6pt] && \times \left[ 1

• \frac{3(q^2_0+1)(1+k^2)}{2^5\kappa} + \cdots \right]. \qquad\qquad \quad \end{align} Here the upper sign refers to the solutions \operatorname{Ec} and the lower to the solutions \operatorname{Es}. Finally expanding \Lambda(q) about q_0, one obtains

: \begin{align} \Lambda_{\pm}(q) \simeq {} & \Lambda(q_0) + (q-q_0)\left(\frac{\partial \Lambda}{\partial q} \right)_{q_0} + \cdots \[6pt] = {} & \Lambda(q_0) +(q-q_0)\kappa \left[1 - \frac{q_0(1+k^2)}{2^2\kappa} - \frac{1}{2^6\kappa^2}{3(1+k^2)^2(q^2_0+1) -4k^2(q^2_0+2q_0+5)}+ \cdots\right] \[6pt] \simeq {} & \Lambda(q_0) \mp 2\kappa\sqrt{\frac{2}{\pi}} \left( \frac{1+k}{1-k} \right)^{-\kappa/k} \left( \frac{8\kappa}{1-k^2}\right)^{q_0/2} \frac{1}{[(q_0-1)/2]!} \Big[ 1 - \frac{1}{2^5\kappa}(1+k^2)(3q^2_0+8q_0+3) \[6pt] & {} + \frac{1}{3.2^{11}\kappa^2}{3(1+k^2)^2(9q^4_0+8q^3_0-78q^2_0-88q_0-87) \[6pt] & {} + 128k^2(2q^3_0+9q^2_0+10q_0+15)} - \cdots\Big]. \end{align}

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).

Notes

References

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• {{Citation | author2-link = Wilhelm Magnus | author4-link = Francesco Tricomi
• {{Citation | author-link = Gabriel Lamé
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References

1. 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
2. (1992). "Solitons, bounces and sphalerons on a circle". Elsevier BV.
3. W. Müller, Harald J.. (1966). "Asymptotic Expansions of Ellipsoidal Wave Functions and their Characteristic Numbers". Wiley.
4. Müller, Harald J. W.. (1966). "Asymptotic Expansions of Ellipsoidal Wave Functions in Terms of Hermite Functions". Wiley.
5. Müller, Harald J. W.. (1966). "On Asymptotic Expansions of Ellipsoidal Wave Functions". Wiley.
6. Ince, E. L.. (1940). "VII—Further Investigations into the Periodic Lamé Functions". Cambridge University Press (CUP).