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

中文翻译：

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Lamé 算子的频谱和应用，II：当端点是尖点时

本文是我们研究Lamé算子$$\begin{aligned} L_n=\frac{d^2的谱$$\sigma (L_n;\tau )$$σ ( L n ; τ )的第二部分}{dx^2}-n(n+1)\wp ( x+z_0;\tau )\quad \text {in}\;\;L^2(\mathbb {R}, \mathbb {C}) , \end{aligned}$$ L n = d 2 dx 2 - n ( n + 1 ) ℘ ( x + z 0 ; τ ) in L 2 ( R , C ) ，其中 $$n\in \mathbb {N }$$ n ∈ N , $$\wp (z;\tau )$$ ℘ ( z ; τ ) 是周期为 1 和 $$\tau $$ τ 的 Weierstrass 椭圆函数，$$z_0\in \mathbb {C}$$ z 0 ∈ C 被选择为使得 $$L_n$$ L n 在 $$\mathbb {R}$$ R 上没有奇点。如果 $$\sigma (L_n;\tau )$$ σ ( L n ; τ ) 的端点是 $$\sigma (L_n;\tau ) 的至少三个半弧的交点，则称为尖点$$ σ ( L n ; τ ) 。我们根据单项数据获得了尖点存在的充要条件，并证明$$\sigma (L_n;\tau )$$ σ (L n ; τ ) 对于固定的$$\tau 至多有一个尖点$$ τ 。我们还考虑 $$n=2$$ n = 2 的情况并研究 $$\tau $$ τ 的分布使得 $$\sigma (L_2;\tau )$$ σ ( L 2 ; τ )有一个尖头。对于任何 $$\gamma \in \Gamma _{0}(2)$$ γ ∈ Γ 0 ( 2 ) 和基本域 $$\gamma (F_0)$$ γ ( F 0 ) ，其中 $$F_{ 0}:=\{ \tau \in \mathbb {H} |\ 0\leqslant {\text {Re}} \tau \leqslant 1, |z-\frac{1}{2}|\geqslant \frac{ 1}{2}\}$$ F 0 : = { τ ∈ H | 0 ⩽ Re τ ⩽ 1 , | z - 1 2 | ⩾ 1 2 } 是$$\Gamma _0(2)$$ Γ 0 ( 2 ) 的基本基本域，我们证明$$\gamma ( F_0)$$ γ ( F 0 ) 使得 $$\sigma (L_2;\tau )$$ σ ( L 2 ; τ ) 有一个尖点，也完全表征了那些 $$\gamma $$ γ 。为了证明这样的结果，我们将对经典模形式$$e_1(\tau ), e_2(\tau ), e_3(\tau )$$ e 1 ( τ ) , e 2 ( τ ) , e 3 ( τ ) ，这是独立的兴趣。