Spectrum of the Lamé operator along Reτ = 1/2: The genus 3 case

In this paper, we study the spectrum $\sigma \left(L\right)$ of the Lamé operator$L=\frac{d^2}{d{x}^2}-12\wp (x+z_0;\tau )\text{in}{L}^2(R,C),$ where $\wp (z;\tau )$ is the Weierstrass elliptic function with periods 1 and τ, and $z_0\in C$ is chosen such that L has no singularities on $R$. We prove that a point $\lambda \in \sigma \left(L\right)$ is an intersection point of different spectral arcs but not a zero of the spectral polynomial if and only if λ is a zero of the following cubic polynomial:$\frac{4}{15}{\lambda }^3+\frac{8}{5}{\eta }_1{\lambda }^2-3g_2\lambda +9g_3-6{\eta }_1{g}_2=0.$ We also study the deformation of the spectrum as $\tau =\frac{1}{2}+ib$ with $b>0$ varying. We discover 10 different types of graphs for the spectrum as b varies around the double zeros of the spectral polynomial.