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- W1622014662 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,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 2frac{sqrt{(Ab)^{m+1} (aB)^m}}{4^{2m} (m!)^2} left[ 1 + O left( frac{log^2 m}{m^2}right) right]. end{align*}" @default.
- W1622014662 created "2016-06-24" @default.
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- W1622014662 date "2013-12-08" @default.
- W1622014662 modified "2023-09-27" @default.
- W1622014662 title "Asymptotics of spectral gaps of 1D Dirac operator with two exponential terms potential" @default.
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