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

Abstract

In this paper, we study the spectrum $\sigma (L)$ of the Lamé operator \begin{equation*}L=\frac{d^2}{dx^2}-12\wp(x+z_0;τ)\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} λ^3+\frac{8}{5}η_1 λ^2-3g_2 λ+9g_3-6η_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.

33 pages

References in corpus (2)

• On the Spectra of Real and Complex Lamé Operators
• A necessary and sufficient condition for the Darboux-Treibich-Verdier potential with its spectrum contained in $\mathbb{R}$