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

This article is the second part of our study of the spectrum σ (L n ; τ) of the Lamé operator L n = d 2 d x 2 - n (n + 1) ℘ (x + z 0 ; τ) in L 2 (R , C) , where n ∈ N , ℘ (z ; τ) is the Weierstrass elliptic function with periods 1 and τ , and z 0 ∈ C is chosen such that L n has no singularities on R . An endpoint of σ (L n ; τ) is called a cusp if it is an intersection point of at least three semi-arcs of σ (L n ; τ) . We obtain a necessary and sufficient condition for the existence of cusps in terms of monodromy datas and prove that σ (L n ; τ) has at most one cusp for fixed τ . We also consider the case n = 2 and study the distribution of τ 's such that σ (L 2 ; τ) has a cusp. For any γ ∈ Γ 0 (2) and the fundamental domain γ (F 0) , where F 0 : = { τ ∈ H | 0 ⩽ Re τ ⩽ 1 , | z - 1 2 | ⩾ 1 2 } is the basic fundamental domain of Γ 0 (2) , we prove that there are either 0 or 3 τ 's in γ (F 0) such that σ (L 2 ; τ) has a cusp and also completely characterize those γ 's. To prove such results, we will give a complete description of the critical points of the classical modular forms e 1 (τ) , e 2 (τ) , e 3 (τ) , which is of independent interest. [ABSTRACT FROM AUTHOR]