Criterion of Bari basis property for $2 \times 2$ Dirac-type operators with strictly regular boundary conditions

Abstract

The paper is concerned with the Bari basis property of a boundary value problem associated in $L^2([0,1]; \mathbb{C}^2)$ with the following $2 \times 2$ Dirac-type equation for $y = {\rm col}(y_1, y_2)$: $$L_U(Q) y =-i B^{-1} y' + Q(x) y = \lambda y , \quad B = \begin{pmatrix} b_1 & 0 \\ 0 & b_2 \end{pmatrix}, \quad b_1 < 0 < b_2, $$ with a potential matrix $Q \in L^2([0,1]; \mathbb{C}^{2 \times 2})$ and subject to the strictly regular boundary conditions $Uy :=\{U_1, U_2\}y=0$. If $b_2 = -b_1 =1$ this equation is equivalent to one dimensional Dirac equation. We show that the system of root vectors $\{f_n\}_{n \in \mathbb{Z}}$ of the operator $L_U(Q)$ forms a Bari basis in $L^2([0,1]; \mathbb{C}^2)$ if and only if the unperturbed operator $L_U(0)$ is self-adjoint. We also give explicit conditions for this in terms of coefficients in the boundary conditions. The Bari basis criterion is a consequence of our more general result: Let $Q \in L^p([0,1]; \mathbb{C}^{2 \times 2})$, $p \in [1,2]$, boundary conditions be strictly regular, and let $\{g_n\}_{n \in \mathbb{Z}}$ be the sequence biorthogonal to the system of root vectors $\{f_n\}_{n \in \mathbb{Z}}$ of the operator $L_U(Q)$. Then $$ \{\|f_n - g_n\|_2\}_{n \in \mathbb{Z}} \in (\ell^p(\mathbb{Z}))^* \quad\Leftrightarrow\quad L_U(0) = L_U(0)^*. $$ These abstract results are applied to non-canonical initial-boundary value problem for a damped string equation.