The Lamé Equation on a Circle and Applications to Singular Limit Eigenvalue Problems

Abstract

Let $n\in \{1,2,\dots \}$ , $\ell \in \{1,2\}$ and $0<k<1$ . We are concerned with the eigenvalues of the Lamé equation of the form $$\begin{array}{r}-{\varphi }^{″}+\ell (\ell +1)\{e_3+k^2{\mathrm{sn}}^2(x,k)\}\varphi =\lambda \varphi \text{for}x\in \mathbb{R}/2nK(k)\mathbb{Z},\end{array}$$ where $e_3:=-(1+k^2)/3$ is a constant, $\mathrm{sn}(x,k)$ denotes Jacobi’s elliptic function and $K(k)$ denotes the complete elliptic integral of the first kind. Using integral representations of two independent solutions, we obtain exact expressions of all eigenvalues and establish asymptotic formulas for all eigenvalues as $k\to 1$ . It is known that the Lamé equation appears as a linearized eigenvalue problem of important semilinear elliptic equations including the Allen–Cahn equation, a scalar field equation and the sine-Poisson equation. We also establish asymptotic formulas for the eigenvalues of the linearization of various boundary value problems of semilinear elliptic equations.