Spectrum of the Lamé Operator and Application, II: When an Endpoint is a Cusp

Abstract

This article is the second part of our study of the spectrum $\sigma (L_n;\tau )$ of the Lamé operator

$$\begin{aligned} L_n=\frac{d^2}{dx^2}-n(n+1)\wp ( x+z_0;\tau )\quad \text {in}\;\;L^2(\mathbb {R}, \mathbb {C}), \end{aligned}$$

where $n\in \mathbb {N}$, $\wp (z;\tau )$ is the Weierstrass elliptic function with periods 1 and $\tau$, and $z_0\in \mathbb {C}$ is chosen such that $L_n$ has no singularities on $\mathbb {R}$. An endpoint of $\sigma (L_n;\tau )$ is called a cusp if it is an intersection point of at least three semi-arcs of $\sigma (L_n;\tau )$. We obtain a necessary and sufficient condition for the existence of cusps in terms of monodromy datas and prove that $\sigma (L_n;\tau )$ has at most one cusp for fixed $\tau$. We also consider the case $n=2$ and study the distribution of $\tau$’s such that $\sigma (L_2;\tau )$ has a cusp. For any $\gamma \in \Gamma _{0}(2)$ and the fundamental domain $\gamma (F_0)$, where $F_{0}:=\{ \tau \in \mathbb {H} |\ 0\leqslant {\text {Re}} \tau \leqslant 1, |z-\frac{1}{2}|\geqslant \frac{1}{2}\}$ is the basic fundamental domain of $\Gamma _0(2)$, we prove that there are either 0 or 3 $\tau$’s in $\gamma (F_0)$ such that $\sigma (L_2;\tau )$ has a cusp and also completely characterize those $\gamma$’s. To prove such results, we will give a complete description of the critical points of the classical modular forms $e_1(\tau ), e_2(\tau ), e_3(\tau )$, which is of independent interest.