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

Abstract

In the space ℍ = (L₂[0, π])², we study the Dirac operator \( {\mathrm{\mathcal{L}}}_{P,U} \) generated by the differential expression ℓ_P(y) = By′ + Py, where

\( B=\left(\begin{array}{cc}-i& 0\\ {}0& i\end{array}\right),\kern0.5em P(x)=\left(\begin{array}{cc}{p}_1(x)& {p}_2(x)\\ {}{p}_3(x)& {p}_4(x)\end{array}\right),\kern0.5em \mathbf{y}(x)=\left(\begin{array}{c}{y}_1(x)\\ {}{y}_2(x)\end{array}\right), \)

and the regular boundary conditions

\( U\left(\mathbf{y}\right)=\left(\begin{array}{cc}{u}_{11}& {u}_{12}\\ {}{u}_{21}& {u}_{22}\end{array}\right)\left(\begin{array}{c}{y}_1(0)\\ {}{y}_2(0)\end{array}\right)+\left(\begin{array}{cc}{u}_{13}& {u}_{14}\\ {}{u}_{23}& {u}_{24}\end{array}\right)\left(\begin{array}{c}{y}_1\left(\uppi \right)\\ {}{y}_2\left(\uppi \right)\end{array}\right)=0. \)

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 \( {\mathrm{\mathcal{L}}}_{P,U} \) is discrete and consists of eigenvalues {λ_n}_{n ∈ ℤ} such that \( {\uplambda}_n={\uplambda}_n^0+o(1) \) as |n| → ∞, where \( {\left\{{\uplambda}_n^0\right\}}_{n\in \mathrm{\mathbb{Z}}} \) is the spectrum of the operator \( {\mathrm{\mathcal{L}}}_{0,U} \) with zero potential and the same boundary conditions. If the boundary conditions are strongly regular, then the spectrum of the operator \( {\mathrm{\mathcal{L}}}_{P,U} \) is asymptotically simple. We show that the system of eigenfunctions and associate functions of the operator \( {\mathrm{\mathcal{L}}}_{P,U} \) 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 \( {\mathrm{\mathcal{L}}}_{0,U} \) are double, all eigenvalues of the operator \( {\mathrm{\mathcal{L}}}_{P,U} \) are asymptotically double, and the system formed by the corresponding two-dimensional root subspaces of the operator \( {\mathrm{\mathcal{L}}}_{P,U} \) is a Riesz base of subspaces (Riesz base with brackets) in the space ℍ.