Asymptotics of spectral gaps of 1D Dirac operator with two exponential terms potential

Abstract

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,π]), \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 $$P(x)=a{e}^{-2ix}+A{e}^{2ix},Q(x)=b{e}^{-2ix}+B{e}^{2ix},$$ where $a,A,b,B$ are nonzero complex numbers. We show, for large enough $m,$ that ${\gamma }_{\pm 2m}=0$ and \begin{align*} γ_{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*} γ_{-(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*}

References in corpus (1)

• Asymptotic formulas for spectral gaps and deviations of Hill and 1D Dirac operators