Spectral triangles of non-selfadjoint Hill and Dirac operators

Official URL: https://dx.doi.org/10.1070/RM9957

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, π]. If L is considered with Dirichlet, periodic, or antiperiodic boundary conditions, then the corresponding spectra are discrete and, for sufficiently large |n|, close to n2 in the Hill case or close to n in the Dirac case (n ϵ ℤ). There is one Dirichlet eigenvalue μ and two periodic (if n is even) or antiperiodic (if n is odd) eigenvalues λn- and λn+ (counted with multiplicity). Asymptotic estimates are given for the spectral gaps γn = λn+ - λn- and the deviations δn = μn - λn+ in terms of the Fourier coefficients of the potentials. Moreover, precise asymptotic expressions for γn and δn are found for special potentials that are trigonometric polynomials.