Spectrum of the Lam\'{e} operator along $\mathrm{Re}\tau={1}/{2}:$ The genus $3$ case

In this paper, we study the spectrum $\sigma(L)$ of the Lam\'{e} operator \begin{equation*}L=\frac{d^2}{dx^2}-12\wp(x+z_0;\tau)\quad \text{in}\;\;L^2(\mathbb{R}, \mathbb{C}), \end{equation*} where $\wp(z;\tau)$ is the Weierstrass elliptic function with periods $1$ and $\tau$, and $z_0\in\mathbb{C}$ is chosen such that $L$ has no singularities on $\mathbb{R}$. We prove that a point $\lambda\in \sigma(L)$ is an intersection point of different spectral arcs but not a zero of the spectral polynomial if and only if $\lambda$ is a zero of the following cubic polynomial: \begin{equation*} \frac{4}{15} \lambda^3+\frac{8}{5}\eta_1 \lambda^2-3g_2 \lambda+9g_3-6\eta_1 g_2=0. \end{equation*} We also study the deformation of the spectrum as $\tau=\frac{1}{2}+ib$ with $b>0$ varying. We discover $7$ different types of graphs for the spectrum as $b$ varies around the double zeros of the spectral polynomial.