Asymptotics of spectral gaps of 1D Dirac operator whose potential is a linear combination of two exponential terms

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 λ_n⁻, λ_n⁺ (periodic if n is even, or antiperiodic if n is odd) such that |λ_n^±−n|<1/2.

We study the asymptotics of spectral gaps γ_n=λ_n⁺−λ_n⁻ in the case P(x)=ae^−2ix+Ae^2ix,  Q(x)=be^−2ix+Be^2ix, where a, A, b, B are any complex numbers. We show, for large enough m, that γ_±2m=0 and $\begin{eqnarray*}\lefteqn{\gamma_{2m+1}=\pm2\frac{\sqrt{(Ab)^{m}(aB)^{m+1}}}{4^{2m}(m!)^{2}}\biggl[1+\mathrm{O}\biggl(\frac{\log^{2}m}{m^{2}}\biggr)\biggr],}\\\lefteqn{\gamma_{-(2m+1)}=\pm2\frac{\sqrt{(Ab)^{m+1}(aB)^{m}}}{4^{2m}(m!)^{2}}\biggl[1+\mathrm{O}\biggl(\frac{\log^{2}m}{m^{2}}\biggr)\biggr].}\end{eqnarray*}$