Spectral triangles of non-selfadjoint Hill and Dirac operators

Abstract

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 be a Hill operator or a one-dimensional Dirac operator on the interval . If is considered with Dirichlet, periodic, or antiperiodic boundary conditions, then the corresponding spectra are discrete and, for sufficiently large , close to in the Hill case or close to in the Dirac case (). There is one Dirichlet eigenvalue and two periodic (if is even) or antiperiodic (if is odd) eigenvalues and (counted with multiplicity). Asymptotic estimates are given for the spectral gaps and the deviations in terms of the Fourier coefficients of the potentials. Moreover, precise asymptotic expressions for and are found for special potentials that are trigonometric polynomials. Bibliography: 45 titles.