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 $y=\mathrm{col}(y_1,y_2)$: $$L_U(Q)y=-i{B}^{-1}{y}^{\prime }+Q(x)y=\lambda y,B=\left(\begin{array}{cc}{b}_1& 0\\ 0& b_2\end{array}\right),b_1<0<b_2,$$ 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 $$\{\Vert f_n-g_n{\Vert }_2{\}}_{n\in \mathbb{Z}}\in ({\ell }^p(\mathbb{Z}){)}^{\ast }\iff L_U(0)=L_U(0{)}^{\ast }.$$ These abstract results are applied to non-canonical initial-boundary value problem for a damped string equation.

28 pages, 49 references. arXiv admin note: text overlap with arXiv:2012.11170