The Riesz Basis Property with Brackets for Dirac Systems with Summable Potentials

In the space ℍ = (L2[0, π])2, we study the Dirac operator ℒ P , U generated by the differential expression ℓP(y) = By′ + Py, whereB=(−i00i),P(x)=(p1(x)p2(x)p3(x)p4(x)),y(x)=(y1(x)y2(x)), and the regular boundary conditionsU(y)=(u11u12u21u22)(y1(0)y2(0))+(u13u14u23u24)(y1(π)y2(π))=0. The elements of the matrix P are assumed to be complex-valued functions summable over [0, π]. We show that the spectrum of the operator ℒ P , U is discrete and consists of eigenvalues {λn}n ∈ ℤ such that λn=λn0+o(1) as |n| → ∞, where {λn0}n∈ℤ is the spectrum of the operator ℒ 0 , U with zero potential and the same boundary conditions. If the boundary conditions are strongly regular, then the spectrum of the operator ℒ P , U is asymptotically simple. We show that the system of eigenfunctions and associate functions of the operator ℒ P , U forms a Riesz base in the space ℍ provided that the eigenfunctions are normed. If the boundary conditions are regular, but not strongly regular, then all eigenvalues of the operator ℒ 0 , U are double, all eigenvalues of the operator ℒ P , U are asymptotically double, and the system formed by the corresponding two-dimensional root subspaces of the operator ℒ P , U is a Riesz base of subspaces (Riesz base with brackets) in the space ℍ. © 2018, Springer Science+Business Media, LLC, part of Springer Nature.