Spectrum of the Lamé operator and application, I: Deformation along Reτ=12

Abstract In this paper, we study the spectrum of the Lame operator L = d 2 d x 2 − 6 ℘ ( x + z 0 ; τ ) in L 2 ( R , C ) , where ℘ ( z ; τ ) is the Weierstrass elliptic function with periods 1 and τ, and z 0 ∈ C is chosen such that L has no singularities on R . (i) We completely determine the explicit location of intersection points of spectral arcs. (ii) We give a complete picture of the deformation of the spectrum as τ = 1 2 + i b and b > 0 varies. In particular, we show that the spectrum has exactly 9 different types of graphs for different b's, and we also give the explicit range of b for each type of graphs. This solves open problems raised in [17] . (iii) As an application of the spectrum and the deep connection of the Lame equation with the mean field equation from [4] , we prove the existence of τ = 1 2 + i b such that the mean field equation △ u + e u = 16 π δ 0 on the rhombus torus E τ : = C / ( Z + Z τ ) has no even axisymmetric solutions but does have 2 even not-axisymmetric solutions. This gives the first positive answer to a long-standing open problem.