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

The one-dimensional Dirac operator L=i((1)(0) (-1)(0))d/dx + ((0)(Q(x)) (0) P-(x)), P,Q is an element of L-2 ([0, pi]), consider on [0, pi] with periodic or antiperiodic boundary conditions, has discrete spectra. For large enough |n|, n is an element of 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) - n| < 1/2. We study the asymptotics of spectral gaps gamma(n) = lambda(+)(n) - lambda(-)(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 gamma +/- 2m = 0 and gamma 2m+1 = +/- 2 root(Ab)(m)(aB)(m+1)/4(2m)(m!)(2) [1+O(log(2) m/m(2))], gamma-(2m+1) = +/- 2 root(Ab)(m+1)(aB)(m)/4(2m)(m!)(2) [1+O(log(2) m/m(2))].