Abstract: The one-dimensional Dirac operator \[L=\mathrm{i}\pmatrix{1&0\cr0&-1}\frac{\mathrm{d}}{\mathrm{d}x}+\pmatrix{0&P(x)\crQ(x)&0},\quad P,Q\inL^{2}([0,\uppi ]),\] considered on $[0,\uppi ]$ with periodic or antiperiodic boundary conditions, has discrete spectra. For large enough |n|, n∈Z, there are two (counted with multiplicity) eigenvalues λn−, λn+ (periodic if n is even, or antiperiodic if n is odd) such that |λn±−n|<1/2. We study the asymptotics of spectral gaps γn=λn+−λn− in the case P(x)=ae−2ix+Ae2ix, Q(x)=be−2ix+Be2ix, where a, A, b, B are any complex numbers. We show, for large enough m, that γ±2m=0 and \begin{eqnarray*}\lefteqn{\gamma_{2m+1}=\pm2\frac{\sqrt{(Ab)^{m}(aB)^{m+1}}}{4^{2m}(m!)^{2}}\biggl[1+\mathrm{O}\biggl(\frac{\log^{2}m}{m^{2}}\biggr)\biggr],}\\\lefteqn{\gamma_{-(2m+1)}=\pm2\frac{\sqrt{(Ab)^{m+1}(aB)^{m}}}{4^{2m}(m!)^{2}}\biggl[1+\mathrm{O}\biggl(\frac{\log^{2}m}{m^{2}}\biggr)\biggr].}\end{eqnarray*}
Keywords: 1D Dirac operator, spectral gap asymptotics
