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.