Spectral problems for the Dirac system with spectral parameter in local boundary conditions (Q5959530)

The author investigates a spectral boundary value problem in a 3-dimensional bounded domain \(\Omega\) for the Dirac system that describes the behavior of a relativistic particle in an electromagnetic field \[ [{\alpha}\cdot(-ich\nabla-e{\mathbf A}({\mathbf x})) +mc^2\beta -{\mathcal E} +eV({\mathbf x})]w = 0, \] where \(m\) and \(\mathcal E\) are the mass and energy, respectively, of the particle, \(c\) is the velocity of light, \( \alpha\) is ''vector'' of \(4\times 4\) Dirac matrices, \({\mathbf A}({\mathbf x})\) is the magnetic potential, \(V({\mathbf x})\) is the electric potential, \(e\) is the particle charge. The vector function \(w=(u_1,u_2,v_1,v_2)\) may be considered as a pair of two-dimensional parts \(u=(u_1,u_2)\) and \(v=(v_1,v_2)\). Let \[ a(\xi)=\begin{pmatrix} \xi_3 & \xi_1-i\xi_2 \cr \xi_1+i\xi_2 & -\xi_3Y\end{pmatrix} \] The following boundary condition is considered \[ ia(\xi)v^+ =\lambda u^+ \;\text{ on} \;\partial\Omega, \] where \(\nu\) is the unit outward normal vector at \({\mathbf x}\in\partial\Omega\), \(u^+, v^+\) are the boundary values of \(u\) and \(v\) on \(\partial\Omega\) and \(\lambda\) is the spectral parameter. It is proved that the eigenvalues of the problem have finite multiplicities and two points of accumulation, zero and infinity. The asymptotic behavior of the corresponding series of eigenvalues is obtained. Moreover, the existence of an orthonormal basis on the boundary consisting of two-dimensional parts of the four-dimensional eigenfunctions is proved.