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 Lame 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.