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Let $L$ be the Hill operator or the one dimensional Dirac operator on theinterval $[0,\pi].$ If $L$ is considered with Dirichlet, periodic orantiperiodic boundary conditions, then the corresponding spectra are discreteand for large enough $|n|$ close to $n^2 $ in the Hill case, or close to $n, \;n\in \mathbb{Z}$ in the Dirac case, there are one Dirichlet eigenvalue $\mu_n$and two periodic (if $n$ is even) or antiperiodic (if $n$ is odd) eigenvalues$\lambda_n^-, \, \lambda_n^+ $ (counted with multiplicity).  We give estimates for the asymptotics of the spectral gaps $\gamma_n =\lambda_n^+ - \lambda_n^-$ and deviations $ \delta_n =\mu_n - \lambda_n^+$ interms of the Fourier coefficients of the potentials. Moreover, for specialpotentials that are trigonometric polynomials we provide precise asymptotics of$\gamma_n$ and $\delta_n.$

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