Cite this article as:
Kornev V. V., Khromov A. P. Dirac System with Undifferentiable Potential and Antiperiodic Boundary Conditions. Izv. Saratov Univ. (N. S.), Ser. Math. Mech. Inform., 2013, vol. 13, iss. 3, pp. 28-35. DOI: https://doi.org/10.18500/1816-9791-2013-13-3-28-35
Dirac System with Undifferentiable Potential and Antiperiodic Boundary Conditions
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.
