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

Let $L$ be the Hill operator or the one dimensional Dirac operator on the interval $[0,\pi ]\mathrm{.}$ If $L$ is considered with Dirichlet, periodic or antiperiodic boundary conditions, then the corresponding spectra are discrete and for large enough $\mathrm{\mid }n\mathrm{\mid }$ close to $n^2$ in the Hill case, or close to $n, ; n\in $\mathbb{Z}$$intheDiraccase,thereareoneDirichleteigenvalue$\mu_n$andtwoperiodic(if$n$iseven)orantiperiodic(if$n$isodd)eigenvalues$\lambda_n^-, , \lambda_n^+ $ (counted with multiplicity). We give estimates for the asymptotics of the spectral gaps $\gamma_n = \lambda_n^+ - \lambda_n^-$anddeviations$ \delta_n =\mu_n - \lambda_n^+$ in terms of the Fourier coefficients of the potentials. Moreover, for special potentials that are trigonometric polynomials we provide precise asymptotics of ${\gamma }_n$ and ${\delta }_n\mathrm{.}$