Explicit solution of the Lamé equation for n=1

Question

The Jacobi form of Lamé equation is given by \begin{equation} \left(\frac{d^2 }{du^2} - n(n+1)k^2 \operatorname{sn}^{2}(u)-E\right)\Psi (u) = 0, \end{equation} where $k\in(0, 1)$ is parameter related to the elliptic modulus of the Jacobi elliptic functions, and the linearly independent solutions are given as $$\Psi^ ±(u) = \frac{H(u ± α)}{ Θ(u)Θ(α)} e^{∓uZ(α)}$$ where where $dn^2α = E − k^ 2$, $dn(·)$ is the delta amplitude function, and $H(u)$, $Θ(u)$ and $Z(u)$ denote the Jacobi Eta, Theta and Zeta functions.

I am only interested in Lame equation in the case $n=1$; (Can the solution be reduced?) Since the solution as above is not easy to work with, I am wondering if it can be represented explicitly, in a nicer algebraic form? Although the problem is well studied, I find that the relevant literature is somewhat limited.

Answer 1

A simpler formula is obtained using the Weierstrass notation for elliptic functions: Lame equation with $n=1$ in this form is $$w''=(2\wp(z)+\lambda)w,$$ and the general solution is $$w_{1,2}(z)=e^{\mp z\zeta(a)}\frac{\sigma(z\pm a)}{\sigma(z)},$$ where $a$ is a solution of $\lambda=\wp(a)$. Here $\wp,\zeta,\sigma$ are Weierstrass functions.

Of course, there is no "algebraic form" since these solutions are not algebraic. You can write the Lame equation in algebraic form, then explicit formula for the solutions will involve exponential and an Abelian integral, see references below.

Speaking of the literature, it is abundant; the most common sources are:

G. H. Halphen, Traite de fonctions elliptiques et leur applications, Paris, Gauthier-Villars, 1886-1888 (3 volumes). This is the most comprehensive source.

E. T. Whittaker and G. N. Watson, A course of modern analysis, Cambridge, 1927.

E. G. C. Poole, Introduction to the theory of linear differential equations, Dover, 1960.