Spectral triangles of non-selfadjoint Hill and Dirac operators

P. B. Djakov ; B. S. Mityagin

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 75 (2020) no. 4, pp. 587-626

Résumé

This is a survey of results from the last 10 to 12 years about the structure of the spectra of Hill–Schrödinger and Dirac operators. Let $L$ be a Hill operator or a one-dimensional Dirac operator on the interval $[0,\pi]$. If $L$ is considered with Dirichlet, periodic, or antiperiodic boundary conditions, then the corresponding spectra are discrete and, for sufficiently large $|n|$, close to $n^2$ in the Hill case or close to $n$ in the Dirac case ($n\in \mathbb{Z}$). There is one Dirichlet eigenvalue $\mu_n$ and two periodic (if $n$ is even) or antiperiodic (if $n$ is odd) eigenvalues $\lambda_n^-$ and $\lambda_n^+$ (counted with multiplicity). Asymptotic estimates are given for the spectral gaps $\gamma_n=\lambda_n^+-\lambda_n^-$ and the deviations $\delta_n=\mu_n-\lambda_n^+$ in terms of the Fourier coefficients of the potentials. Moreover, precise asymptotic expressions for $\gamma_n$ and $\delta_n$ are found for special potentials that are trigonometric polynomials. Bibliography: 45 titles.

Détail

Keywords: Hill operator, one-dimensional Dirac operator, periodic boundary conditions, antiperiodic boundary conditions, Dirichlet boundary conditions.