Abstract: | We study the spectral properties of the Dirac operator LP,U generated in the space (L20, π])2 by the differential expression By′ + P(x)y and by Birkhoff regular boundary conditions U, where y = (y1, y2)t, \(B = \left( {\begin{array}{*{20}{c}} { - i}&0 \\ 0&i \end{array}} \right)\), and the entries of the matrix P are complexvalued Lebesgue measurable functions on 0, π]. We also study the asymptotic properties of the eigenvalues {λn}n∈Z of the operator LP,U as n → ∞ depending on the “smoothness” degree of the potential P; i.e., we consider the scale of Besov spaces B1,∞θ, θ ∈ (0, 1). In the case of strongly regular boundary conditions, we study the asymptotic behavior of the system of normalized eigenfunctions of the operator LP,U, and in the case of regular but not strongly regular boundary conditions, we find the asymptotics of two-dimensional spectral projections. |