Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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.

13
The categories of this paper are: quant-ph
6
The categories of this paper are: astro-ph
18
The categories of this paper are: econ
10
The categories of this paper are: q-bio
22
The categories of this paper are: chao-dyn
35
The categories of this paper are: mtrl-th
34
The categories of this paper are: chem-ph
15
The categories of this paper are: nucl-ex
0
The categories of this paper are: hep-ph
17
The categories of this paper are: q-fin
25
The categories of this paper are: patt-sol
23
The categories of this paper are: solv-int
21
The categories of this paper are: adap-org
31
The categories of this paper are: ao-sci
27
The categories of this paper are: comp-gas
14
The categories of this paper are: math-ph