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 $[0,π].$ If $L$ is considered with Dirichlet, periodic or antiperiodic boundary conditions, then the corresponding spectra are discrete and 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 $μ_n$ and two periodic (if $n$ is even) or antiperiodic (if $n$ is odd) eigenvalues $λ_n^-, \, λ_n^+$ (counted with multiplicity). We give estimates for the asymptotics of the spectral gaps $γ_n = λ_n^+ - λ_n^-$ and deviations $δ_n =μ_n - λ_n^+$ in terms of the Fourier coefficients of the potentials. Moreover, for special potentials that are trigonometric polynomials we provide precise asymptotics of $γ_n$ and $δ_n.$

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