Dirac system

A Mixed Problem for a System of First Order Differential Equations with Continuous Potential

Authors: Burlutskaya M. S.

We study a mixed problem for a first order differential system with two independent variables and continuous potential when the initial condition is an arbitrary square summable vector-valued function. The corresponding spectral problem is the Dirac system. It sets the convergence almost everywhere of a formal decision, obtained by the Fourier method. It is shown that the sum of a formal decision is a generalized solution of a mixed problem, understood as the limit of classical solutions for the case of smooth approximation of the initial data of the problem.

Refined asymptotic formulas for eigenvalues and eigenfunctions of the Dirac system with nondifferentiable potential

Authors: Burlutskaya M. S., Kurdyumov V. P., Khromov A. P.

This paper investigates the Dirac system with the continuous potential. Asymptotic formulas for the eigenvalues (including refined) and eigenfunctions are established. As an application we obtain a theorem P. Dzhakova and B. S. Mityagin on the Riesz bases with brackets. 

Dirac System with Undifferentiable Potential and Antiperiodic Boundary Conditions

Authors: Kornev V. V., Khromov A. P.

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