Izvestiya of Saratov University.

Mathematics. Mechanics. Informatics

ISSN 1816-9791 (Print)
ISSN 2541-9005 (Online)

For citation:

Kornev V. V., Khromov A. P. Dirac System with Undifferentiable Potential and Antiperiodic Boundary Conditions. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2013, vol. 13, iss. 3, pp. 28-35. DOI: 10.18500/1816-9791-2013-13-3-28-35

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Published online: 
Full text:
(downloads: 151)

Dirac System with Undifferentiable Potential and Antiperiodic Boundary Conditions

Kornev Vladimir Victorovich, Saratov State University
Khromov August Petrovich, Saratov State University

The object of the paper is Dirac system with antiperiodic boundary conditions and complex-valued conditions potential. A new method is suggested for investigating spectral properties of this boundary problem. The method is based on the formulas of the transform operators type. It is rather elementary and simple. Using this method asymptotic behaviour of eigenvalues is specificated and it is proved that eigen and associated functions form Riesz basis with brackets in the space of quadratic summerable two-dimensional vector-functions since eigenvalues may be multiple. The structure of Riesz projection operators is also studied. The results of the paper can be used in spectral problems for equations with partial derivatives of the 1-st order containing involution.

  1. Burlutskaya M. Sh., Kornev V. V., Khromov A. P. Dirac system with non-differentiable potential and periodic boundary conditions. Zh. Vychisl. Mat. Mat. Fiz., 2012, vol. 52, no. 9, pp. 1621–1632 (in Russian).
  2. Marchenko V. A. Operatory Shturma–Liuvillia i ikh prilozheniia [Sturm–Liouville operators and their applications]. Kiev, Naukova Dumka, 1977, 340 p. (in Russian).
  3. Djakov P., Mityagin B. Bari–Markus property for Riesz projections of ID periodic Dirac operators. Math. Nachr., 2010, vol. 283, no. 3, pp. 443–462.
  4. Djakov P., Mityagin B. S. Instability zones of periodic 1-dimensional Schr¨odinger and Dirac operators. Russian Math. Surveys, 2006, vol. 61, no. 4, pp. 663–766. DOI: 10.4213/rm2121.
  5. Burlutskaya M. Sh., Khromov A. P. Fourier method in an initial-boundary value problem for a first-order partial differential equation with involution. Comput. Math. Math. Phys., 2011, vol. 51, no. 12, pp. 2233–2246.
Short text (in English):
(downloads: 91)