Spectral triangles of non-selfadjoint Hill and Dirac operators

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 c...

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Bibliographic Details
Published in:Russian mathematical surveys 2020-08, Vol.75 (4), p.587-626
Main Authors: Djakov, P. B., Mityagin, B. S.
Format: Article
Language:English
Subjects:
antiperiodic boundary conditions
Asymptotic properties
Boundary conditions
Dirichlet boundary conditions
Dirichlet problem
Eigenvalues
Hill operator
one-dimensional Dirac operator
Operators
periodic boundary conditions
Polynomials
Spectra
Triangles
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Summary: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.
ISSN:0036-0279
1468-4829
DOI:10.1070/RM9957