Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Asymptotics of spectral gaps of 1D Dirac operator with two exponential
  terms potential   The one-dimensional Dirac operator \begin{equation*} L = i \begin{pmatrix} 1
& 0 \\ 0 & -1 \end{pmatrix} \frac{d}{dx} +\begin{pmatrix} 0 & P(x) \\ Q(x) & 0
\end{pmatrix}, \quad P,Q \in L^2 ([0,\pi]), \end{equation*} considered on
$[0,\pi]$ with periodic and antiperiodic boundary conditions, has discrete
spectra. For large enough $|n|,\, n \in \mathbb{Z}, $ there are two (counted
with multiplicity) eigenvalues $\lambda_n^-,\lambda_n^+ $ (periodic if $n$ is
even, or antiperiodic if $n$ is odd) such that $|\lambda_n^\pm - n |<1/2.$
  We study the asymptotics of spectral gaps $\gamma_n =\lambda_n^+ -
\lambda_n^-$ in the case $$P(x)=a e^{-2ix} + A e^{2ix}, \quad Q(x)=b e^{-2ix} +
B e^{2ix},$$ where $a, A, b, B$ are nonzero complex numbers. We show, for large
enough $m,$ that $\gamma_{\pm 2m}=0 $ and \begin{align*} \gamma_{2m+1} = \pm 2
\frac{\sqrt{(Ab)^m (aB)^{m+1}}}{4^{2m} (m!)^2 } \left[ 1 + O \left(
\frac{\log^2 m}{m^2}\right) \right], \end{align*} \begin{align*}
\gamma_{-(2m+1)} = \pm 2\frac{\sqrt{(Ab)^{m+1} (aB)^m}}{4^{2m} (m!)^2} \left[ 1
+ O \left( \frac{\log^2 m}{m^2}\right) \right]. \end{align*}

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The categories of this paper are: nucl-th
37
The categories of this paper are: supr-con
36
The categories of this paper are: cmp-lg
0
The categories of this paper are: hep-ph
9
The categories of this paper are: nlin
19
The categories of this paper are: eess
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The categories of this paper are: chao-dyn
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The categories of this paper are: hep-ex
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The categories of this paper are: dg-ga
10
The categories of this paper are: q-bio
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The categories of this paper are: comp-gas
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The categories of this paper are: stat
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The categories of this paper are: plasm-ph
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The categories of this paper are: ao-sci
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The categories of this paper are: q-alg
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The categories of this paper are: physics