Spectral boundary value problems for the Dirac equation with a singular potential
by
M. S. Agranovich
Moscow State Institute of Electronics and Mathematics (MIEM), Moscow 109028, Russia
Coauthors: G. V. Rozenblioum (Chalmers/Goteborg Univ., 41296 Goteborg, Sweden)

We consider a boundary value problem for the Dirac system with a singularity of Coulomb type in the potential in a 3-dimensional bounded domain \Omega with smooth boundary \Gamma. The problem is elliptic (if the singularity is absent) and formally self-adjoint. The system contains a parameter \lambda; the second parameter \mu enters into the boundary condition. Considering \lambda or \mu as a spectral parameter, one obtains two spectral problems, I and II, respectively. We describe their spectral properties.

The spectrum of Problem I is discrete, its eigenvalues have regular asymptotics, and there exists an orthonormal basis of eigenfunctions in L₂(\Omega) remaining an unconditional basis in Sobolev spaces H^s(\Omega) with small s > 0. The eigenvalues of Problem II have finite multiplicities and form two sequences tending to zero and infinity, with regular asymptotics. There exists an orthonormal basis of eigenfunctions in L₂(\Gamma) remaining an unconditional basis in all Sobolev spaces H^s(\Gamma).

Some physicists propose to use the eigenfunctions and the eigenvalues of each of these problems for the computation of the so-called R-matrix on the boundary. We discuss possibilities of justification for the corresponding procedures. The approach with the use of Problem II is preferable.

Date received: April 30, 2003